id	sid	tid	token	lemma	pos
ejpam-6492	1	1	european	european	PROPN
ejpam-6492	1	2	journal	journal	PROPN
ejpam-6492	1	3	of	of	ADP
ejpam-6492	1	4	pure	pure	ADJ
ejpam-6492	1	5	and	and	CCONJ
ejpam-6492	1	6	applied	applied	ADJ
ejpam-6492	1	7	mathematics	mathematic	NOUN
ejpam-6492	1	8	2025	2025	NUM
ejpam-6492	1	9	,	,	PUNCT
ejpam-6492	1	10	vol	vol	NOUN
ejpam-6492	1	11	.	.	PROPN
ejpam-6492	1	12	18	18	NUM
ejpam-6492	1	13	,	,	PUNCT
ejpam-6492	1	14	issue	issue	NOUN
ejpam-6492	1	15	3	3	NUM
ejpam-6492	1	16	,	,	PUNCT
ejpam-6492	1	17	article	article	NOUN
ejpam-6492	1	18	number	number	NOUN
ejpam-6492	1	19	6492	6492	NUM
ejpam-6492	1	20	issn	issn	VERB
ejpam-6492	1	21	1307	1307	NUM
ejpam-6492	1	22	-	-	SYM
ejpam-6492	1	23	5543	5543	NUM
ejpam-6492	1	24	–	–	PUNCT
ejpam-6492	1	25	ejpam.com	ejpam.com	X
ejpam-6492	1	26	published	publish	VERB
ejpam-6492	1	27	by	by	ADP
ejpam-6492	1	28	new	new	PROPN
ejpam-6492	1	29	york	york	PROPN
ejpam-6492	1	30	business	business	PROPN
ejpam-6492	1	31	global	global	ADJ
ejpam-6492	1	32	product	product	NOUN
ejpam-6492	1	33	difference	difference	NOUN
ejpam-6492	1	34	fibonacci	fibonacci	NOUN
ejpam-6492	1	35	identities	identity	NOUN
ejpam-6492	1	36	revisited	revisit	VERB
ejpam-6492	1	37	:	:	PUNCT
ejpam-6492	1	38	quaternionic	quaternionic	ADJ
ejpam-6492	1	39	generalizations	generalization	NOUN
ejpam-6492	1	40	of	of	ADP
ejpam-6492	1	41	everman	everman	NOUN
ejpam-6492	1	42	and	and	CCONJ
ejpam-6492	1	43	koshy	koshy	ADJ
ejpam-6492	1	44	bahar	bahar	PROPN
ejpam-6492	1	45	demirtürk1,∗	demirtürk1,∗	PROPN
ejpam-6492	1	46	,	,	PUNCT
ejpam-6492	1	47	nazim	nazim	PROPN
ejpam-6492	1	48	topal2	topal2	PROPN
ejpam-6492	1	49	1	1	NUM
ejpam-6492	1	50	department	department	NOUN
ejpam-6492	1	51	of	of	ADP
ejpam-6492	1	52	fundamental	fundamental	ADJ
ejpam-6492	1	53	sciences	science	NOUN
ejpam-6492	1	54	,	,	PUNCT
ejpam-6492	1	55	faculty	faculty	NOUN
ejpam-6492	1	56	of	of	ADP
ejpam-6492	1	57	engineering	engineering	NOUN
ejpam-6492	1	58	and	and	CCONJ
ejpam-6492	1	59	architecture	architecture	NOUN
ejpam-6492	1	60	,	,	PUNCT
ejpam-6492	1	61	izmir	izmir	NOUN
ejpam-6492	1	62	bakırçay	bakırçay	ADJ
ejpam-6492	1	63	university	university	NOUN
ejpam-6492	1	64	,	,	PUNCT
ejpam-6492	1	65	izmir	izmir	PROPN
ejpam-6492	1	66	,	,	PUNCT
ejpam-6492	1	67	türkiye	türkiye	NOUN
ejpam-6492	1	68	abstract	abstract	NOUN
ejpam-6492	1	69	.	.	PUNCT
ejpam-6492	2	1	in	in	ADP
ejpam-6492	2	2	this	this	DET
ejpam-6492	2	3	paper	paper	NOUN
ejpam-6492	2	4	,	,	PUNCT
ejpam-6492	2	5	we	we	PRON
ejpam-6492	2	6	investigate	investigate	VERB
ejpam-6492	2	7	new	new	ADJ
ejpam-6492	2	8	identities	identity	NOUN
ejpam-6492	2	9	involving	involve	VERB
ejpam-6492	2	10	generalized	generalized	ADJ
ejpam-6492	2	11	fibonacci	fibonacci	NOUN
ejpam-6492	2	12	and	and	CCONJ
ejpam-6492	2	13	lucas	lucas	PROPN
ejpam-6492	2	14	sequences	sequence	NOUN
ejpam-6492	2	15	,	,	PUNCT
ejpam-6492	2	16	as	as	ADV
ejpam-6492	2	17	well	well	ADV
ejpam-6492	2	18	as	as	ADP
ejpam-6492	2	19	their	their	PRON
ejpam-6492	2	20	associated	associated	ADJ
ejpam-6492	2	21	quaternions	quaternion	NOUN
ejpam-6492	2	22	.	.	PUNCT
ejpam-6492	3	1	after	after	ADP
ejpam-6492	3	2	establishing	establish	VERB
ejpam-6492	3	3	the	the	DET
ejpam-6492	3	4	fundamental	fundamental	ADJ
ejpam-6492	3	5	properties	property	NOUN
ejpam-6492	3	6	of	of	ADP
ejpam-6492	3	7	these	these	DET
ejpam-6492	3	8	generalized	generalized	ADJ
ejpam-6492	3	9	number	number	NOUN
ejpam-6492	3	10	sequences	sequence	NOUN
ejpam-6492	3	11	,	,	PUNCT
ejpam-6492	3	12	we	we	PRON
ejpam-6492	3	13	derive	derive	VERB
ejpam-6492	3	14	quaternionic	quaternionic	ADJ
ejpam-6492	3	15	extensions	extension	NOUN
ejpam-6492	3	16	of	of	ADP
ejpam-6492	3	17	product	product	NOUN
ejpam-6492	3	18	-	-	PUNCT
ejpam-6492	3	19	difference	difference	NOUN
ejpam-6492	3	20	identities	identity	NOUN
ejpam-6492	3	21	originally	originally	ADV
ejpam-6492	3	22	introduced	introduce	VERB
ejpam-6492	3	23	by	by	ADP
ejpam-6492	3	24	everman	everman	NOUN
ejpam-6492	3	25	and	and	CCONJ
ejpam-6492	3	26	koshy	koshy	VERB
ejpam-6492	3	27	for	for	ADP
ejpam-6492	3	28	fibonacci	fibonacci	NOUN
ejpam-6492	3	29	numbers	number	NOUN
ejpam-6492	3	30	.	.	PUNCT
ejpam-6492	4	1	these	these	DET
ejpam-6492	4	2	results	result	NOUN
ejpam-6492	4	3	not	not	PART
ejpam-6492	4	4	only	only	ADV
ejpam-6492	4	5	generalize	generalize	VERB
ejpam-6492	4	6	classical	classical	ADJ
ejpam-6492	4	7	identities	identity	NOUN
ejpam-6492	4	8	,	,	PUNCT
ejpam-6492	4	9	but	but	CCONJ
ejpam-6492	4	10	also	also	ADV
ejpam-6492	4	11	reveal	reveal	VERB
ejpam-6492	4	12	new	new	ADJ
ejpam-6492	4	13	algebraic	algebraic	ADJ
ejpam-6492	4	14	structures	structure	NOUN
ejpam-6492	4	15	within	within	ADP
ejpam-6492	4	16	the	the	DET
ejpam-6492	4	17	framework	framework	NOUN
ejpam-6492	4	18	of	of	ADP
ejpam-6492	4	19	generalized	generalized	ADJ
ejpam-6492	4	20	quaternion	quaternion	NOUN
ejpam-6492	4	21	sequences	sequence	NOUN
ejpam-6492	4	22	.	.	PUNCT
ejpam-6492	5	1	2020	2020	NUM
ejpam-6492	5	2	mathematics	mathematic	NOUN
ejpam-6492	5	3	subject	subject	NOUN
ejpam-6492	5	4	classifications	classification	NOUN
ejpam-6492	5	5	:	:	PUNCT
ejpam-6492	5	6	15b33	15b33	NUM
ejpam-6492	5	7	,	,	PUNCT
ejpam-6492	5	8	11b39	11b39	NUM
ejpam-6492	5	9	,	,	PUNCT
ejpam-6492	5	10	11b75	11b75	NUM
ejpam-6492	5	11	,	,	PUNCT
ejpam-6492	5	12	11z05	11z05	NUM
ejpam-6492	5	13	key	key	ADJ
ejpam-6492	5	14	words	word	NOUN
ejpam-6492	5	15	and	and	CCONJ
ejpam-6492	5	16	phrases	phrase	NOUN
ejpam-6492	5	17	:	:	PUNCT
ejpam-6492	5	18	generalized	generalized	ADJ
ejpam-6492	5	19	fibonacci	fibonacci	NOUN
ejpam-6492	5	20	sequences	sequence	NOUN
ejpam-6492	5	21	,	,	PUNCT
ejpam-6492	5	22	generalized	generalized	ADJ
ejpam-6492	5	23	fibonacci	fibonacci	NOUN
ejpam-6492	5	24	quaternions	quaternion	NOUN
ejpam-6492	5	25	,	,	PUNCT
ejpam-6492	5	26	quaternions	quaternion	NOUN
ejpam-6492	5	27	,	,	PUNCT
ejpam-6492	5	28	product	product	NOUN
ejpam-6492	5	29	differences	difference	NOUN
ejpam-6492	5	30	1	1	NUM
ejpam-6492	5	31	.	.	PUNCT
ejpam-6492	6	1	introduction	introduction	NOUN
ejpam-6492	6	2	fibonacci	fibonacci	PROPN
ejpam-6492	6	3	and	and	CCONJ
ejpam-6492	6	4	lucas	lucas	PROPN
ejpam-6492	6	5	number	number	NOUN
ejpam-6492	6	6	sequences	sequence	NOUN
ejpam-6492	6	7	are	be	AUX
ejpam-6492	6	8	used	use	VERB
ejpam-6492	6	9	in	in	ADP
ejpam-6492	6	10	many	many	ADJ
ejpam-6492	6	11	areas	area	NOUN
ejpam-6492	6	12	of	of	ADP
ejpam-6492	6	13	mathematics	mathematic	NOUN
ejpam-6492	6	14	due	due	ADP
ejpam-6492	6	15	to	to	ADP
ejpam-6492	6	16	their	their	PRON
ejpam-6492	6	17	repetitive	repetitive	ADJ
ejpam-6492	6	18	structures	structure	NOUN
ejpam-6492	6	19	.	.	PUNCT
ejpam-6492	7	1	they	they	PRON
ejpam-6492	7	2	arise	arise	VERB
ejpam-6492	7	3	in	in	ADP
ejpam-6492	7	4	number	number	NOUN
ejpam-6492	7	5	theory	theory	NOUN
ejpam-6492	7	6	with	with	ADP
ejpam-6492	7	7	their	their	PRON
ejpam-6492	7	8	divisibility	divisibility	NOUN
ejpam-6492	7	9	properties	property	NOUN
ejpam-6492	7	10	,	,	PUNCT
ejpam-6492	7	11	in	in	ADP
ejpam-6492	7	12	analysis	analysis	NOUN
ejpam-6492	7	13	with	with	ADP
ejpam-6492	7	14	their	their	PRON
ejpam-6492	7	15	generator	generator	NOUN
ejpam-6492	7	16	functions	function	NOUN
ejpam-6492	7	17	,	,	PUNCT
ejpam-6492	7	18	in	in	ADP
ejpam-6492	7	19	linear	linear	ADJ
ejpam-6492	7	20	algebra	algebra	NOUN
ejpam-6492	7	21	with	with	ADP
ejpam-6492	7	22	their	their	PRON
ejpam-6492	7	23	matrix	matrix	NOUN
ejpam-6492	7	24	representations	representation	NOUN
ejpam-6492	7	25	,	,	PUNCT
ejpam-6492	7	26	in	in	ADP
ejpam-6492	7	27	combinatorics	combinatoric	NOUN
ejpam-6492	7	28	with	with	ADP
ejpam-6492	7	29	tiling	tile	VERB
ejpam-6492	7	30	problems	problem	NOUN
ejpam-6492	7	31	,	,	PUNCT
ejpam-6492	7	32	and	and	CCONJ
ejpam-6492	7	33	in	in	ADP
ejpam-6492	7	34	geometry	geometry	NOUN
ejpam-6492	7	35	with	with	ADP
ejpam-6492	7	36	their	their	PRON
ejpam-6492	7	37	fractal	fractal	ADJ
ejpam-6492	7	38	structures	structure	NOUN
ejpam-6492	7	39	,	,	PUNCT
ejpam-6492	7	40	etc	etc	X
ejpam-6492	7	41	.	.	X
ejpam-6492	8	1	because	because	SCONJ
ejpam-6492	8	2	of	of	ADP
ejpam-6492	8	3	such	such	DET
ejpam-6492	8	4	a	a	DET
ejpam-6492	8	5	wide	wide	ADJ
ejpam-6492	8	6	range	range	NOUN
ejpam-6492	8	7	of	of	ADP
ejpam-6492	8	8	applications	application	NOUN
ejpam-6492	8	9	,	,	PUNCT
ejpam-6492	8	10	many	many	ADJ
ejpam-6492	8	11	generalized	generalized	ADJ
ejpam-6492	8	12	versions	version	NOUN
ejpam-6492	8	13	of	of	ADP
ejpam-6492	8	14	these	these	DET
ejpam-6492	8	15	number	number	NOUN
ejpam-6492	8	16	sequences	sequence	NOUN
ejpam-6492	8	17	have	have	AUX
ejpam-6492	8	18	been	be	AUX
ejpam-6492	8	19	defined	define	VERB
ejpam-6492	8	20	;	;	PUNCT
ejpam-6492	8	21	some	some	PRON
ejpam-6492	8	22	by	by	ADP
ejpam-6492	8	23	changing	change	VERB
ejpam-6492	8	24	the	the	DET
ejpam-6492	8	25	recurrence	recurrence	NOUN
ejpam-6492	8	26	relation	relation	NOUN
ejpam-6492	8	27	,	,	PUNCT
ejpam-6492	8	28	and	and	CCONJ
ejpam-6492	8	29	others	other	NOUN
ejpam-6492	8	30	by	by	ADP
ejpam-6492	8	31	changing	change	VERB
ejpam-6492	8	32	the	the	DET
ejpam-6492	8	33	initial	initial	ADJ
ejpam-6492	8	34	conditions	condition	NOUN
ejpam-6492	8	35	.	.	PUNCT
ejpam-6492	9	1	these	these	DET
ejpam-6492	9	2	sequences	sequence	NOUN
ejpam-6492	9	3	offer	offer	VERB
ejpam-6492	9	4	more	more	ADV
ejpam-6492	9	5	complex	complex	ADJ
ejpam-6492	9	6	and	and	CCONJ
ejpam-6492	9	7	richer	rich	ADJ
ejpam-6492	9	8	mathematical	mathematical	ADJ
ejpam-6492	9	9	structures	structure	NOUN
ejpam-6492	9	10	,	,	PUNCT
ejpam-6492	9	11	especially	especially	ADV
ejpam-6492	9	12	when	when	SCONJ
ejpam-6492	9	13	combined	combine	VERB
ejpam-6492	9	14	with	with	ADP
ejpam-6492	9	15	quaternion	quaternion	NOUN
ejpam-6492	9	16	structures	structure	NOUN
ejpam-6492	9	17	.	.	PUNCT
ejpam-6492	10	1	furthermore	furthermore	ADV
ejpam-6492	10	2	,	,	PUNCT
ejpam-6492	10	3	the	the	DET
ejpam-6492	10	4	extension	extension	NOUN
ejpam-6492	10	5	of	of	ADP
ejpam-6492	10	6	these	these	DET
ejpam-6492	10	7	numbers	number	NOUN
ejpam-6492	10	8	to	to	ADP
ejpam-6492	10	9	quaternions	quaternion	NOUN
ejpam-6492	10	10	is	be	AUX
ejpam-6492	10	11	being	be	AUX
ejpam-6492	10	12	studied	study	VERB
ejpam-6492	10	13	by	by	ADP
ejpam-6492	10	14	many	many	ADJ
ejpam-6492	10	15	authors	author	NOUN
ejpam-6492	10	16	in	in	ADP
ejpam-6492	10	17	both	both	CCONJ
ejpam-6492	10	18	theoretical	theoretical	ADJ
ejpam-6492	10	19	and	and	CCONJ
ejpam-6492	10	20	applied	apply	VERB
ejpam-6492	10	21	fields	field	NOUN
ejpam-6492	10	22	.	.	PUNCT
ejpam-6492	11	1	fibonacci	fibonacci	NOUN
ejpam-6492	11	2	sequences	sequence	NOUN
ejpam-6492	11	3	are	be	AUX
ejpam-6492	11	4	seen	see	VERB
ejpam-6492	11	5	in	in	ADP
ejpam-6492	11	6	many	many	ADJ
ejpam-6492	11	7	structures	structure	NOUN
ejpam-6492	11	8	in	in	ADP
ejpam-6492	11	9	nature	nature	NOUN
ejpam-6492	11	10	,	,	PUNCT
ejpam-6492	11	11	such	such	ADJ
ejpam-6492	11	12	as	as	ADP
ejpam-6492	11	13	the	the	DET
ejpam-6492	11	14	arrangement	arrangement	NOUN
ejpam-6492	11	15	of	of	ADP
ejpam-6492	11	16	sunflower	sunflower	NOUN
ejpam-6492	11	17	seeds	seed	NOUN
ejpam-6492	11	18	,	,	PUNCT
ejpam-6492	11	19	the	the	DET
ejpam-6492	11	20	structure	structure	NOUN
ejpam-6492	11	21	of	of	ADP
ejpam-6492	11	22	a	a	DET
ejpam-6492	11	23	pine	pine	ADJ
ejpam-6492	11	24	cone	cone	NOUN
ejpam-6492	11	25	,	,	PUNCT
ejpam-6492	11	26	the	the	DET
ejpam-6492	11	27	crystallization	crystallization	NOUN
ejpam-6492	11	28	of	of	ADP
ejpam-6492	11	29	a	a	DET
ejpam-6492	11	30	snowflake	snowflake	NOUN
ejpam-6492	11	31	,	,	PUNCT
ejpam-6492	11	32	and	and	CCONJ
ejpam-6492	11	33	the	the	DET
ejpam-6492	11	34	spiral	spiral	ADJ
ejpam-6492	11	35	pattern	pattern	NOUN
ejpam-6492	11	36	in	in	ADP
ejpam-6492	11	37	seashells	seashell	NOUN
ejpam-6492	11	38	.	.	PUNCT
ejpam-6492	12	1	in	in	ADP
ejpam-6492	12	2	[	[	X
ejpam-6492	12	3	1	1	NUM
ejpam-6492	12	4	]	]	PUNCT
ejpam-6492	12	5	,	,	PUNCT
ejpam-6492	12	6	sharma	sharma	PROPN
ejpam-6492	12	7	explores	explore	VERB
ejpam-6492	12	8	the	the	DET
ejpam-6492	12	9	evolution	evolution	NOUN
ejpam-6492	12	10	of	of	ADP
ejpam-6492	12	11	the	the	DET
ejpam-6492	12	12	fibonacci	fibonacci	NOUN
ejpam-6492	12	13	sequence	sequence	NOUN
ejpam-6492	12	14	and	and	CCONJ
ejpam-6492	12	15	its	its	PRON
ejpam-6492	12	16	modern	modern	ADJ
ejpam-6492	12	17	applications	application	NOUN
ejpam-6492	12	18	,	,	PUNCT
ejpam-6492	12	19	especially	especially	ADV
ejpam-6492	12	20	in	in	ADP
ejpam-6492	12	21	fractal	fractal	ADJ
ejpam-6492	12	22	geometry	geometry	NOUN
ejpam-6492	12	23	.	.	PUNCT
ejpam-6492	13	1	generalized	generalize	VERB
ejpam-6492	13	2	sequences	sequence	NOUN
ejpam-6492	13	3	can	can	AUX
ejpam-6492	13	4	be	be	AUX
ejpam-6492	13	5	∗corresponding	∗corresponde	VERB
ejpam-6492	13	6	author	author	NOUN
ejpam-6492	13	7	.	.	PUNCT
ejpam-6492	14	1	doi	doi	NOUN
ejpam-6492	14	2	:	:	PUNCT
ejpam-6492	14	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6492	https://doi.org/10.29020/nybg.ejpam.v18i3.6492	PROPN
ejpam-6492	14	4	email	email	NOUN
ejpam-6492	14	5	addresses	address	NOUN
ejpam-6492	14	6	:	:	PUNCT
ejpam-6492	14	7	bahar.demirturk@bakircay.edu.tr	bahar.demirturk@bakircay.edu.tr	PROPN
ejpam-6492	14	8	(	(	PUNCT
ejpam-6492	14	9	b.	b.	PROPN
ejpam-6492	14	10	demirtürk	demirtürk	PROPN
ejpam-6492	14	11	)	)	PUNCT
ejpam-6492	14	12	,	,	PUNCT
ejpam-6492	14	13	nazimtopal@gmail.com	nazimtopal@gmail.com	X
ejpam-6492	14	14	(	(	PUNCT
ejpam-6492	14	15	n.	n.	NOUN
ejpam-6492	14	16	topal	topal	PROPN
ejpam-6492	14	17	)	)	PUNCT
ejpam-6492	14	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6492	15	1	1	1	NUM
ejpam-6492	15	2	copyright	copyright	NOUN
ejpam-6492	15	3	:	:	PUNCT
ejpam-6492	15	4	©	©	PROPN
ejpam-6492	15	5	2025	2025	NUM
ejpam-6492	15	6	the	the	DET
ejpam-6492	15	7	author(s	author(s	NOUN
ejpam-6492	15	8	)	)	PUNCT
ejpam-6492	15	9	.	.	PUNCT
ejpam-6492	16	1	(	(	PUNCT
ejpam-6492	16	2	cc	cc	NOUN
ejpam-6492	16	3	by	by	ADP
ejpam-6492	16	4	-	-	PUNCT
ejpam-6492	16	5	nc	nc	PROPN
ejpam-6492	16	6	4.0	4.0	NUM
ejpam-6492	16	7	)	)	PUNCT
ejpam-6492	16	8	b.	b.	NOUN
ejpam-6492	16	9	demirtürk	demirtürk	PROPN
ejpam-6492	16	10	,	,	PUNCT
ejpam-6492	16	11	n.	n.	NOUN
ejpam-6492	16	12	topal	topal	PROPN
ejpam-6492	16	13	/	/	SYM
ejpam-6492	16	14	eur	eur	PROPN
ejpam-6492	16	15	.	.	PUNCT
ejpam-6492	17	1	j.	j.	PROPN
ejpam-6492	17	2	pure	pure	PROPN
ejpam-6492	17	3	appl	appl	PROPN
ejpam-6492	17	4	.	.	PROPN
ejpam-6492	17	5	math	math	PROPN
ejpam-6492	17	6	,	,	PUNCT
ejpam-6492	17	7	18	18	NUM
ejpam-6492	17	8	(	(	PUNCT
ejpam-6492	17	9	3	3	NUM
ejpam-6492	17	10	)	)	PUNCT
ejpam-6492	17	11	(	(	PUNCT
ejpam-6492	17	12	2025	2025	NUM
ejpam-6492	17	13	)	)	PUNCT
ejpam-6492	17	14	,	,	PUNCT
ejpam-6492	17	15	6492	6492	NUM
ejpam-6492	17	16	2	2	NUM
ejpam-6492	17	17	of	of	ADP
ejpam-6492	17	18	22	22	NUM
ejpam-6492	17	19	used	use	VERB
ejpam-6492	17	20	to	to	PART
ejpam-6492	17	21	explain	explain	VERB
ejpam-6492	17	22	more	more	ADJ
ejpam-6492	17	23	complex	complex	ADJ
ejpam-6492	17	24	models	model	NOUN
ejpam-6492	17	25	of	of	ADP
ejpam-6492	17	26	these	these	DET
ejpam-6492	17	27	natural	natural	ADJ
ejpam-6492	17	28	patterns	pattern	NOUN
ejpam-6492	17	29	,	,	PUNCT
ejpam-6492	17	30	for	for	ADP
ejpam-6492	17	31	example	example	NOUN
ejpam-6492	17	32	the	the	DET
ejpam-6492	17	33	structure	structure	NOUN
ejpam-6492	17	34	of	of	ADP
ejpam-6492	17	35	dna	dna	PROPN
ejpam-6492	17	36	[	[	X
ejpam-6492	17	37	2	2	NUM
ejpam-6492	17	38	]	]	PUNCT
ejpam-6492	17	39	.	.	PUNCT
ejpam-6492	18	1	in	in	ADP
ejpam-6492	18	2	addition	addition	NOUN
ejpam-6492	18	3	to	to	ADP
ejpam-6492	18	4	being	be	AUX
ejpam-6492	18	5	used	use	VERB
ejpam-6492	18	6	to	to	PART
ejpam-6492	18	7	model	model	VERB
ejpam-6492	18	8	patterns	pattern	NOUN
ejpam-6492	18	9	in	in	ADP
ejpam-6492	18	10	quantum	quantum	ADJ
ejpam-6492	18	11	physics	physics	NOUN
ejpam-6492	18	12	and	and	CCONJ
ejpam-6492	18	13	other	other	ADJ
ejpam-6492	18	14	physical	physical	ADJ
ejpam-6492	18	15	systems	system	NOUN
ejpam-6492	18	16	,	,	PUNCT
ejpam-6492	18	17	quarks	quark	NOUN
ejpam-6492	18	18	,	,	PUNCT
ejpam-6492	18	19	which	which	PRON
ejpam-6492	18	20	we	we	PRON
ejpam-6492	18	21	know	know	VERB
ejpam-6492	18	22	from	from	ADP
ejpam-6492	18	23	physics	physics	NOUN
ejpam-6492	18	24	theory	theory	NOUN
ejpam-6492	18	25	,	,	PUNCT
ejpam-6492	18	26	can	can	AUX
ejpam-6492	18	27	also	also	ADV
ejpam-6492	18	28	be	be	AUX
ejpam-6492	18	29	given	give	VERB
ejpam-6492	18	30	as	as	ADP
ejpam-6492	18	31	an	an	DET
ejpam-6492	18	32	example[3	example[3	NOUN
ejpam-6492	18	33	]	]	PUNCT
ejpam-6492	18	34	.	.	PUNCT
ejpam-6492	19	1	fibonacci	fibonacci	PROPN
ejpam-6492	19	2	and	and	CCONJ
ejpam-6492	19	3	lucas	lucas	PROPN
ejpam-6492	19	4	number	number	NOUN
ejpam-6492	19	5	sequences	sequence	NOUN
ejpam-6492	19	6	are	be	AUX
ejpam-6492	19	7	notable	notable	ADJ
ejpam-6492	19	8	for	for	ADP
ejpam-6492	19	9	their	their	PRON
ejpam-6492	19	10	mathematical	mathematical	ADJ
ejpam-6492	19	11	structure	structure	NOUN
ejpam-6492	19	12	as	as	ADV
ejpam-6492	19	13	well	well	ADV
ejpam-6492	19	14	as	as	ADP
ejpam-6492	19	15	their	their	PRON
ejpam-6492	19	16	applications	application	NOUN
ejpam-6492	19	17	in	in	ADP
ejpam-6492	19	18	various	various	ADJ
ejpam-6492	19	19	disciplines	discipline	NOUN
ejpam-6492	19	20	.	.	PUNCT
ejpam-6492	20	1	the	the	DET
ejpam-6492	20	2	properties	property	NOUN
ejpam-6492	20	3	of	of	ADP
ejpam-6492	20	4	these	these	DET
ejpam-6492	20	5	number	number	NOUN
ejpam-6492	20	6	sequences	sequence	NOUN
ejpam-6492	20	7	are	be	AUX
ejpam-6492	20	8	utilized	utilize	VERB
ejpam-6492	20	9	in	in	ADP
ejpam-6492	20	10	computer	computer	NOUN
ejpam-6492	20	11	science	science	NOUN
ejpam-6492	20	12	,	,	PUNCT
ejpam-6492	20	13	algorithm	algorithm	NOUN
ejpam-6492	20	14	development	development	NOUN
ejpam-6492	20	15	,	,	PUNCT
ejpam-6492	20	16	cryptography	cryptography	NOUN
ejpam-6492	20	17	,	,	PUNCT
ejpam-6492	20	18	nature	nature	NOUN
ejpam-6492	20	19	science	science	NOUN
ejpam-6492	20	20	,	,	PUNCT
ejpam-6492	20	21	art	art	NOUN
ejpam-6492	20	22	,	,	PUNCT
ejpam-6492	20	23	engineering	engineering	NOUN
ejpam-6492	20	24	and	and	CCONJ
ejpam-6492	20	25	design	design	NOUN
ejpam-6492	20	26	,	,	PUNCT
ejpam-6492	20	27	finance	finance	NOUN
ejpam-6492	20	28	and	and	CCONJ
ejpam-6492	20	29	economics	economic	NOUN
ejpam-6492	20	30	[	[	X
ejpam-6492	20	31	4	4	NUM
ejpam-6492	20	32	]	]	PUNCT
ejpam-6492	20	33	.	.	PUNCT
ejpam-6492	21	1	among	among	ADP
ejpam-6492	21	2	the	the	DET
ejpam-6492	21	3	algorithms	algorithm	NOUN
ejpam-6492	21	4	used	use	VERB
ejpam-6492	21	5	in	in	ADP
ejpam-6492	21	6	computer	computer	NOUN
ejpam-6492	21	7	science	science	NOUN
ejpam-6492	21	8	,	,	PUNCT
ejpam-6492	21	9	the	the	DET
ejpam-6492	21	10	fibonacci	fibonacci	NOUN
ejpam-6492	21	11	search	search	NOUN
ejpam-6492	21	12	algorithm	algorithm	NOUN
ejpam-6492	21	13	and	and	CCONJ
ejpam-6492	21	14	the	the	DET
ejpam-6492	21	15	golden	golden	ADJ
ejpam-6492	21	16	ratio	ratio	NOUN
ejpam-6492	21	17	search	search	NOUN
ejpam-6492	21	18	algorithm	algorithm	NOUN
ejpam-6492	21	19	offer	offer	VERB
ejpam-6492	21	20	particularly	particularly	ADV
ejpam-6492	21	21	good	good	ADJ
ejpam-6492	21	22	solutions	solution	NOUN
ejpam-6492	21	23	for	for	ADP
ejpam-6492	21	24	searching	search	VERB
ejpam-6492	21	25	and	and	CCONJ
ejpam-6492	21	26	sorting	sorting	NOUN
ejpam-6492	21	27	.	.	PUNCT
ejpam-6492	22	1	the	the	DET
ejpam-6492	22	2	increasing	increase	VERB
ejpam-6492	22	3	number	number	NOUN
ejpam-6492	22	4	of	of	ADP
ejpam-6492	22	5	different	different	ADJ
ejpam-6492	22	6	generalizations	generalization	NOUN
ejpam-6492	22	7	of	of	ADP
ejpam-6492	22	8	these	these	DET
ejpam-6492	22	9	sequences	sequence	NOUN
ejpam-6492	22	10	of	of	ADP
ejpam-6492	22	11	numbers	number	NOUN
ejpam-6492	22	12	is	be	AUX
ejpam-6492	22	13	being	be	AUX
ejpam-6492	22	14	used	use	VERB
ejpam-6492	22	15	to	to	PART
ejpam-6492	22	16	develop	develop	VERB
ejpam-6492	22	17	new	new	ADJ
ejpam-6492	22	18	algorithms	algorithm	NOUN
ejpam-6492	22	19	[	[	X
ejpam-6492	22	20	5	5	NUM
ejpam-6492	22	21	]	]	PUNCT
ejpam-6492	22	22	.	.	PUNCT
ejpam-6492	23	1	thus	thus	ADV
ejpam-6492	23	2	,	,	PUNCT
ejpam-6492	23	3	more	more	ADV
ejpam-6492	23	4	flexible	flexible	ADJ
ejpam-6492	23	5	and	and	CCONJ
ejpam-6492	23	6	adaptable	adaptable	ADJ
ejpam-6492	23	7	algorithm	algorithm	NOUN
ejpam-6492	23	8	versions	version	NOUN
ejpam-6492	23	9	can	can	AUX
ejpam-6492	23	10	be	be	AUX
ejpam-6492	23	11	developed	develop	VERB
ejpam-6492	23	12	that	that	PRON
ejpam-6492	23	13	reduce	reduce	VERB
ejpam-6492	23	14	data	datum	NOUN
ejpam-6492	23	15	processing	processing	NOUN
ejpam-6492	23	16	costs	cost	NOUN
ejpam-6492	23	17	,	,	PUNCT
ejpam-6492	23	18	such	such	ADJ
ejpam-6492	23	19	as	as	ADP
ejpam-6492	23	20	the	the	DET
ejpam-6492	23	21	running	running	NOUN
ejpam-6492	23	22	time	time	NOUN
ejpam-6492	23	23	of	of	ADP
ejpam-6492	23	24	the	the	DET
ejpam-6492	23	25	algorithm	algorithm	NOUN
ejpam-6492	23	26	used	use	VERB
ejpam-6492	23	27	,	,	PUNCT
ejpam-6492	23	28	and	and	CCONJ
ejpam-6492	23	29	how	how	SCONJ
ejpam-6492	23	30	close	close	ADJ
ejpam-6492	23	31	the	the	DET
ejpam-6492	23	32	results	result	NOUN
ejpam-6492	23	33	obtained	obtain	VERB
ejpam-6492	23	34	are	be	AUX
ejpam-6492	23	35	to	to	ADP
ejpam-6492	23	36	the	the	DET
ejpam-6492	23	37	true	true	ADJ
ejpam-6492	23	38	value	value	NOUN
ejpam-6492	23	39	with	with	ADP
ejpam-6492	23	40	less	less	ADJ
ejpam-6492	23	41	error	error	NOUN
ejpam-6492	23	42	.	.	PUNCT
ejpam-6492	24	1	in	in	ADP
ejpam-6492	24	2	addition	addition	NOUN
ejpam-6492	24	3	,	,	PUNCT
ejpam-6492	24	4	these	these	DET
ejpam-6492	24	5	sequences	sequence	NOUN
ejpam-6492	24	6	of	of	ADP
ejpam-6492	24	7	numbers	number	NOUN
ejpam-6492	24	8	are	be	AUX
ejpam-6492	24	9	used	use	VERB
ejpam-6492	24	10	for	for	ADP
ejpam-6492	24	11	key	key	ADJ
ejpam-6492	24	12	exchange	exchange	NOUN
ejpam-6492	24	13	and	and	CCONJ
ejpam-6492	24	14	encryption	encryption	NOUN
ejpam-6492	24	15	in	in	ADP
ejpam-6492	24	16	secure	secure	ADJ
ejpam-6492	24	17	communication	communication	NOUN
ejpam-6492	24	18	systems	system	NOUN
ejpam-6492	24	19	.	.	PUNCT
ejpam-6492	25	1	moreover	moreover	ADV
ejpam-6492	25	2	,	,	PUNCT
ejpam-6492	25	3	fibonacci	fibonacci	NOUN
ejpam-6492	25	4	and	and	CCONJ
ejpam-6492	25	5	lucas	lucas	PROPN
ejpam-6492	25	6	sequences	sequence	NOUN
ejpam-6492	25	7	are	be	AUX
ejpam-6492	25	8	used	use	VERB
ejpam-6492	25	9	in	in	ADP
ejpam-6492	25	10	engineering	engineering	NOUN
ejpam-6492	25	11	for	for	ADP
ejpam-6492	25	12	signal	signal	NOUN
ejpam-6492	25	13	processing	processing	NOUN
ejpam-6492	25	14	and	and	CCONJ
ejpam-6492	25	15	structural	structural	ADJ
ejpam-6492	25	16	optimization	optimization	NOUN
ejpam-6492	25	17	,	,	PUNCT
ejpam-6492	25	18	as	as	ADV
ejpam-6492	25	19	well	well	ADV
ejpam-6492	25	20	as	as	ADP
ejpam-6492	25	21	architecture	architecture	NOUN
ejpam-6492	25	22	and	and	CCONJ
ejpam-6492	25	23	design	design	NOUN
ejpam-6492	25	24	[	[	X
ejpam-6492	25	25	6	6	NUM
ejpam-6492	25	26	]	]	PUNCT
ejpam-6492	25	27	.	.	PUNCT
ejpam-6492	26	1	the	the	DET
ejpam-6492	26	2	golden	golden	ADJ
ejpam-6492	26	3	ratio	ratio	NOUN
ejpam-6492	26	4	is	be	AUX
ejpam-6492	26	5	also	also	ADV
ejpam-6492	26	6	used	use	VERB
ejpam-6492	26	7	in	in	ADP
ejpam-6492	26	8	determining	determine	VERB
ejpam-6492	26	9	aesthetic	aesthetic	ADJ
ejpam-6492	26	10	proportions	proportion	NOUN
ejpam-6492	26	11	in	in	ADP
ejpam-6492	26	12	art	art	NOUN
ejpam-6492	26	13	and	and	CCONJ
ejpam-6492	26	14	architecture	architecture	NOUN
ejpam-6492	26	15	,	,	PUNCT
ejpam-6492	26	16	rhythmic	rhythmic	ADJ
ejpam-6492	26	17	structures	structure	NOUN
ejpam-6492	26	18	and	and	CCONJ
ejpam-6492	26	19	compositions	composition	NOUN
ejpam-6492	26	20	in	in	ADP
ejpam-6492	26	21	music	music	NOUN
ejpam-6492	26	22	.	.	PUNCT
ejpam-6492	27	1	in	in	ADP
ejpam-6492	27	2	addition	addition	NOUN
ejpam-6492	27	3	,	,	PUNCT
ejpam-6492	27	4	fibonacci	fibonacci	NOUN
ejpam-6492	27	5	sequences	sequence	NOUN
ejpam-6492	27	6	are	be	AUX
ejpam-6492	27	7	used	use	VERB
ejpam-6492	27	8	in	in	ADP
ejpam-6492	27	9	technical	technical	ADJ
ejpam-6492	27	10	analysis	analysis	NOUN
ejpam-6492	27	11	in	in	ADP
ejpam-6492	27	12	finance	finance	NOUN
ejpam-6492	27	13	and	and	CCONJ
ejpam-6492	27	14	economics	economic	NOUN
ejpam-6492	27	15	to	to	PART
ejpam-6492	27	16	determine	determine	VERB
ejpam-6492	27	17	support	support	NOUN
ejpam-6492	27	18	and	and	CCONJ
ejpam-6492	27	19	resistance	resistance	NOUN
ejpam-6492	27	20	levels	level	NOUN
ejpam-6492	27	21	[	[	X
ejpam-6492	27	22	7	7	NUM
ejpam-6492	27	23	]	]	PUNCT
ejpam-6492	27	24	.	.	PUNCT
ejpam-6492	28	1	generalized	generalize	VERB
ejpam-6492	28	2	sequences	sequence	NOUN
ejpam-6492	28	3	can	can	AUX
ejpam-6492	28	4	be	be	AUX
ejpam-6492	28	5	used	use	VERB
ejpam-6492	28	6	to	to	PART
ejpam-6492	28	7	build	build	VERB
ejpam-6492	28	8	more	more	ADJ
ejpam-6492	28	9	complex	complex	ADJ
ejpam-6492	28	10	models	model	NOUN
ejpam-6492	28	11	of	of	ADP
ejpam-6492	28	12	market	market	NOUN
ejpam-6492	28	13	movements	movement	NOUN
ejpam-6492	28	14	.	.	PUNCT
ejpam-6492	29	1	there	there	PRON
ejpam-6492	29	2	are	be	VERB
ejpam-6492	29	3	many	many	ADJ
ejpam-6492	29	4	studies	study	NOUN
ejpam-6492	29	5	on	on	ADP
ejpam-6492	29	6	the	the	DET
ejpam-6492	29	7	mathematical	mathematical	ADJ
ejpam-6492	29	8	properties	property	NOUN
ejpam-6492	29	9	and	and	CCONJ
ejpam-6492	29	10	applications	application	NOUN
ejpam-6492	29	11	of	of	ADP
ejpam-6492	29	12	quaternions	quaternion	NOUN
ejpam-6492	29	13	whose	whose	DET
ejpam-6492	29	14	elements	element	NOUN
ejpam-6492	29	15	are	be	AUX
ejpam-6492	29	16	generalized	generalized	ADJ
ejpam-6492	29	17	fibonacci	fibonacci	NOUN
ejpam-6492	29	18	and	and	CCONJ
ejpam-6492	29	19	lucas	lucas	PROPN
ejpam-6492	29	20	numbers	number	NOUN
ejpam-6492	29	21	.	.	PUNCT
ejpam-6492	30	1	research	research	NOUN
ejpam-6492	30	2	in	in	ADP
ejpam-6492	30	3	this	this	DET
ejpam-6492	30	4	area	area	NOUN
ejpam-6492	30	5	not	not	PART
ejpam-6492	30	6	only	only	ADV
ejpam-6492	30	7	provides	provide	VERB
ejpam-6492	30	8	new	new	ADJ
ejpam-6492	30	9	results	result	NOUN
ejpam-6492	30	10	in	in	ADP
ejpam-6492	30	11	theoretical	theoretical	ADJ
ejpam-6492	30	12	mathematics	mathematic	NOUN
ejpam-6492	30	13	,	,	PUNCT
ejpam-6492	30	14	but	but	CCONJ
ejpam-6492	30	15	also	also	ADV
ejpam-6492	30	16	new	new	ADJ
ejpam-6492	30	17	solutions	solution	NOUN
ejpam-6492	30	18	in	in	ADP
ejpam-6492	30	19	optimization	optimization	NOUN
ejpam-6492	30	20	,	,	PUNCT
ejpam-6492	30	21	cryptography	cryptography	NOUN
ejpam-6492	30	22	,	,	PUNCT
ejpam-6492	30	23	and	and	CCONJ
ejpam-6492	30	24	especially	especially	ADV
ejpam-6492	30	25	in	in	ADP
ejpam-6492	30	26	transformation	transformation	NOUN
ejpam-6492	30	27	and	and	CCONJ
ejpam-6492	30	28	rotation	rotation	NOUN
ejpam-6492	30	29	problems	problem	NOUN
ejpam-6492	30	30	in	in	ADP
ejpam-6492	30	31	physics	physics	NOUN
ejpam-6492	30	32	and	and	CCONJ
ejpam-6492	30	33	engineering	engineering	NOUN
ejpam-6492	30	34	.	.	PUNCT
ejpam-6492	31	1	halıcı	halıcı	PROPN
ejpam-6492	31	2	obtained	obtain	VERB
ejpam-6492	31	3	numerous	numerous	ADJ
ejpam-6492	31	4	new	new	ADJ
ejpam-6492	31	5	equations	equation	NOUN
ejpam-6492	31	6	by	by	ADP
ejpam-6492	31	7	generalizing	generalize	VERB
ejpam-6492	31	8	various	various	ADJ
ejpam-6492	31	9	fibonacci	fibonacci	NOUN
ejpam-6492	31	10	and	and	CCONJ
ejpam-6492	31	11	lucas	lucas	PROPN
ejpam-6492	31	12	quaternions	quaternion	NOUN
ejpam-6492	31	13	and	and	CCONJ
ejpam-6492	31	14	deriving	derive	VERB
ejpam-6492	31	15	their	their	PRON
ejpam-6492	31	16	binet	binet	NOUN
ejpam-6492	31	17	formulas	formula	NOUN
ejpam-6492	31	18	,	,	PUNCT
ejpam-6492	31	19	generator	generator	NOUN
ejpam-6492	31	20	functions	function	NOUN
ejpam-6492	31	21	,	,	PUNCT
ejpam-6492	31	22	and	and	CCONJ
ejpam-6492	31	23	matrix	matrix	NOUN
ejpam-6492	31	24	representations	representation	NOUN
ejpam-6492	31	25	[	[	X
ejpam-6492	31	26	8–10	8–10	NOUN
ejpam-6492	31	27	]	]	PUNCT
ejpam-6492	31	28	.	.	PUNCT
ejpam-6492	32	1	kesim	kesim	PROPN
ejpam-6492	32	2	obtained	obtain	VERB
ejpam-6492	32	3	exponential	exponential	ADJ
ejpam-6492	32	4	generator	generator	NOUN
ejpam-6492	32	5	functions	function	NOUN
ejpam-6492	32	6	for	for	ADP
ejpam-6492	32	7	the	the	DET
ejpam-6492	32	8	generalized	generalized	ADJ
ejpam-6492	32	9	fibonacci	fibonacci	NOUN
ejpam-6492	32	10	and	and	CCONJ
ejpam-6492	32	11	lucas	lucas	PROPN
ejpam-6492	32	12	quaternions	quaternion	NOUN
ejpam-6492	32	13	and	and	CCONJ
ejpam-6492	32	14	obtained	obtain	VERB
ejpam-6492	32	15	binomial	binomial	ADJ
ejpam-6492	32	16	sums	sum	NOUN
ejpam-6492	32	17	of	of	ADP
ejpam-6492	32	18	these	these	DET
ejpam-6492	32	19	quaternions	quaternion	NOUN
ejpam-6492	32	20	[	[	X
ejpam-6492	32	21	11	11	NUM
ejpam-6492	32	22	]	]	PUNCT
ejpam-6492	32	23	.	.	PUNCT
ejpam-6492	33	1	kome	kome	PROPN
ejpam-6492	33	2	et	et	PROPN
ejpam-6492	33	3	al	al	PROPN
ejpam-6492	33	4	.	.	PROPN
ejpam-6492	33	5	introduced	introduce	VERB
ejpam-6492	33	6	the	the	DET
ejpam-6492	33	7	modified	modify	VERB
ejpam-6492	33	8	generalized	generalize	VERB
ejpam-6492	33	9	fibonacci	fibonacci	NOUN
ejpam-6492	33	10	and	and	CCONJ
ejpam-6492	33	11	lucas	lucas	PROPN
ejpam-6492	33	12	quaternions	quaternion	NOUN
ejpam-6492	33	13	and	and	CCONJ
ejpam-6492	33	14	gave	give	VERB
ejpam-6492	33	15	the	the	DET
ejpam-6492	33	16	generator	generator	NOUN
ejpam-6492	33	17	functions	function	NOUN
ejpam-6492	33	18	,	,	PUNCT
ejpam-6492	33	19	binet	binet	NOUN
ejpam-6492	33	20	formulas	formula	NOUN
ejpam-6492	33	21	and	and	CCONJ
ejpam-6492	33	22	matrix	matrix	NOUN
ejpam-6492	33	23	representations	representation	NOUN
ejpam-6492	33	24	for	for	ADP
ejpam-6492	33	25	these	these	DET
ejpam-6492	33	26	quaternions	quaternion	NOUN
ejpam-6492	33	27	[	[	X
ejpam-6492	33	28	12	12	NUM
ejpam-6492	33	29	]	]	PUNCT
ejpam-6492	33	30	.	.	PUNCT
ejpam-6492	34	1	moreover	moreover	ADV
ejpam-6492	34	2	,	,	PUNCT
ejpam-6492	34	3	aydinyüz	aydinyüz	VERB
ejpam-6492	34	4	and	and	CCONJ
ejpam-6492	34	5	asci	asci	PROPN
ejpam-6492	34	6	introduced	introduce	VERB
ejpam-6492	34	7	the	the	DET
ejpam-6492	34	8	generalized	generalized	ADJ
ejpam-6492	34	9	k	k	ADJ
ejpam-6492	34	10	-	-	PUNCT
ejpam-6492	34	11	order	order	NOUN
ejpam-6492	34	12	fibonacci	fibonacci	NOUN
ejpam-6492	34	13	and	and	CCONJ
ejpam-6492	34	14	lucas	lucas	PROPN
ejpam-6492	34	15	quaternions	quaternion	NOUN
ejpam-6492	34	16	and	and	CCONJ
ejpam-6492	34	17	obtained	obtain	VERB
ejpam-6492	34	18	their	their	PRON
ejpam-6492	34	19	generating	generating	NOUN
ejpam-6492	34	20	functions	function	NOUN
ejpam-6492	34	21	and	and	CCONJ
ejpam-6492	34	22	matrix	matrix	NOUN
ejpam-6492	34	23	representations	representation	NOUN
ejpam-6492	34	24	[	[	X
ejpam-6492	34	25	13	13	NUM
ejpam-6492	34	26	]	]	PUNCT
ejpam-6492	34	27	.	.	PUNCT
ejpam-6492	35	1	kızılateş	kızılateş	PROPN
ejpam-6492	35	2	et	et	PROPN
ejpam-6492	35	3	al	al	PROPN
ejpam-6492	35	4	.	.	PROPN
ejpam-6492	36	1	define	define	VERB
ejpam-6492	36	2	higher	high	ADJ
ejpam-6492	36	3	-	-	PUNCT
ejpam-6492	36	4	order	order	NOUN
ejpam-6492	36	5	generalized	generalize	VERB
ejpam-6492	36	6	fibonacci	fibonacci	NOUN
ejpam-6492	36	7	quaternions	quaternion	NOUN
ejpam-6492	36	8	with	with	ADP
ejpam-6492	36	9	q	q	ADJ
ejpam-6492	36	10	-	-	PUNCT
ejpam-6492	36	11	integer	integer	NOUN
ejpam-6492	36	12	components	component	NOUN
ejpam-6492	36	13	using	use	VERB
ejpam-6492	36	14	q	q	NOUN
ejpam-6492	36	15	-	-	PUNCT
ejpam-6492	36	16	integers	integer	NOUN
ejpam-6492	36	17	and	and	CCONJ
ejpam-6492	36	18	higher	high	ADJ
ejpam-6492	36	19	order	order	NOUN
ejpam-6492	36	20	generalized	generalized	ADJ
ejpam-6492	36	21	fibonacci	fibonacci	NOUN
ejpam-6492	36	22	numbers	number	NOUN
ejpam-6492	36	23	and	and	CCONJ
ejpam-6492	36	24	obtain	obtain	VERB
ejpam-6492	36	25	binet	binet	NOUN
ejpam-6492	36	26	-	-	PUNCT
ejpam-6492	36	27	like	like	ADJ
ejpam-6492	36	28	formulas	formula	NOUN
ejpam-6492	36	29	,	,	PUNCT
ejpam-6492	36	30	generator	generator	NOUN
ejpam-6492	36	31	functions	function	NOUN
ejpam-6492	36	32	,	,	PUNCT
ejpam-6492	36	33	recurrence	recurrence	NOUN
ejpam-6492	36	34	relations	relation	NOUN
ejpam-6492	36	35	,	,	PUNCT
ejpam-6492	36	36	and	and	CCONJ
ejpam-6492	36	37	matrix	matrix	VERB
ejpam-6492	36	38	representations	representation	NOUN
ejpam-6492	36	39	for	for	ADP
ejpam-6492	36	40	these	these	DET
ejpam-6492	36	41	new	new	ADJ
ejpam-6492	36	42	quaternions	quaternion	NOUN
ejpam-6492	36	43	in	in	ADP
ejpam-6492	36	44	[	[	X
ejpam-6492	36	45	14	14	NUM
ejpam-6492	36	46	]	]	PUNCT
ejpam-6492	36	47	.	.	PUNCT
ejpam-6492	37	1	in	in	ADP
ejpam-6492	37	2	[	[	X
ejpam-6492	37	3	15	15	NUM
ejpam-6492	37	4	]	]	PUNCT
ejpam-6492	37	5	,	,	PUNCT
ejpam-6492	37	6	shpakivskyi	shpakivskyi	NOUN
ejpam-6492	37	7	investigates	investigate	VERB
ejpam-6492	37	8	some	some	DET
ejpam-6492	37	9	properties	property	NOUN
ejpam-6492	37	10	of	of	ADP
ejpam-6492	37	11	generalized	generalized	ADJ
ejpam-6492	37	12	fibonacci	fibonacci	NOUN
ejpam-6492	37	13	quaternions	quaternion	NOUN
ejpam-6492	37	14	and	and	CCONJ
ejpam-6492	37	15	fibonacci	fibonacci	PROPN
ejpam-6492	37	16	-	-	PUNCT
ejpam-6492	37	17	narayana	narayana	PROPN
ejpam-6492	37	18	quaternions	quaternion	NOUN
ejpam-6492	37	19	.	.	PUNCT
ejpam-6492	38	1	in	in	ADP
ejpam-6492	38	2	this	this	DET
ejpam-6492	38	3	paper	paper	NOUN
ejpam-6492	38	4	,	,	PUNCT
ejpam-6492	38	5	we	we	PRON
ejpam-6492	38	6	rederive	rederive	VERB
ejpam-6492	38	7	the	the	DET
ejpam-6492	38	8	classical	classical	ADJ
ejpam-6492	38	9	results	result	NOUN
ejpam-6492	38	10	of	of	ADP
ejpam-6492	38	11	koshy	koshy	ADJ
ejpam-6492	38	12	and	and	CCONJ
ejpam-6492	38	13	everman	everman	NOUN
ejpam-6492	38	14	using	use	VERB
ejpam-6492	38	15	generalized	generalized	ADJ
ejpam-6492	38	16	fibonacci	fibonacci	NOUN
ejpam-6492	38	17	and	and	CCONJ
ejpam-6492	38	18	lucas	lucas	PROPN
ejpam-6492	38	19	numbers	number	NOUN
ejpam-6492	38	20	,	,	PUNCT
ejpam-6492	38	21	and	and	CCONJ
ejpam-6492	38	22	subsequently	subsequently	ADV
ejpam-6492	38	23	apply	apply	VERB
ejpam-6492	38	24	these	these	DET
ejpam-6492	38	25	results	result	NOUN
ejpam-6492	38	26	to	to	ADP
ejpam-6492	38	27	quaternions	quaternion	NOUN
ejpam-6492	38	28	.	.	PUNCT
ejpam-6492	39	1	in	in	ADP
ejpam-6492	39	2	this	this	DET
ejpam-6492	39	3	context	context	NOUN
ejpam-6492	39	4	,	,	PUNCT
ejpam-6492	39	5	generalized	generalized	ADJ
ejpam-6492	39	6	fibonacci	fibonacci	NOUN
ejpam-6492	39	7	and	and	CCONJ
ejpam-6492	39	8	lucas	lucas	PROPN
ejpam-6492	39	9	quaternions	quaternion	NOUN
ejpam-6492	39	10	will	will	AUX
ejpam-6492	39	11	be	be	AUX
ejpam-6492	39	12	defined	define	VERB
ejpam-6492	39	13	,	,	PUNCT
ejpam-6492	39	14	and	and	CCONJ
ejpam-6492	39	15	their	their	PRON
ejpam-6492	39	16	properties	property	NOUN
ejpam-6492	39	17	will	will	AUX
ejpam-6492	39	18	be	be	AUX
ejpam-6492	39	19	studied	study	VERB
ejpam-6492	39	20	in	in	ADP
ejpam-6492	39	21	detail	detail	NOUN
ejpam-6492	39	22	.	.	PUNCT
ejpam-6492	40	1	in	in	ADP
ejpam-6492	40	2	addition	addition	NOUN
ejpam-6492	40	3	,	,	PUNCT
ejpam-6492	40	4	generalized	generalized	ADJ
ejpam-6492	40	5	versions	version	NOUN
ejpam-6492	40	6	of	of	ADP
ejpam-6492	40	7	some	some	DET
ejpam-6492	40	8	equations	equation	NOUN
ejpam-6492	40	9	in	in	ADP
ejpam-6492	40	10	b.	b.	PROPN
ejpam-6492	40	11	demirtürk	demirtürk	PROPN
ejpam-6492	40	12	,	,	PUNCT
ejpam-6492	40	13	n.	n.	NOUN
ejpam-6492	40	14	topal	topal	PROPN
ejpam-6492	40	15	/	/	SYM
ejpam-6492	40	16	eur	eur	PROPN
ejpam-6492	40	17	.	.	PUNCT
ejpam-6492	41	1	j.	j.	PROPN
ejpam-6492	41	2	pure	pure	PROPN
ejpam-6492	41	3	appl	appl	PROPN
ejpam-6492	41	4	.	.	PROPN
ejpam-6492	41	5	math	math	PROPN
ejpam-6492	41	6	,	,	PUNCT
ejpam-6492	41	7	18	18	NUM
ejpam-6492	41	8	(	(	PUNCT
ejpam-6492	41	9	3	3	NUM
ejpam-6492	41	10	)	)	PUNCT
ejpam-6492	41	11	(	(	PUNCT
ejpam-6492	41	12	2025	2025	NUM
ejpam-6492	41	13	)	)	PUNCT
ejpam-6492	41	14	,	,	PUNCT
ejpam-6492	41	15	6492	6492	NUM
ejpam-6492	41	16	3	3	NUM
ejpam-6492	41	17	of	of	ADP
ejpam-6492	41	18	22	22	NUM
ejpam-6492	41	19	the	the	DET
ejpam-6492	41	20	literature	literature	NOUN
ejpam-6492	41	21	will	will	AUX
ejpam-6492	41	22	also	also	ADV
ejpam-6492	41	23	be	be	AUX
ejpam-6492	41	24	obtained	obtain	VERB
ejpam-6492	41	25	in	in	ADP
ejpam-6492	41	26	this	this	DET
ejpam-6492	41	27	study	study	NOUN
ejpam-6492	41	28	.	.	PUNCT
ejpam-6492	42	1	the	the	DET
ejpam-6492	42	2	aim	aim	NOUN
ejpam-6492	42	3	of	of	ADP
ejpam-6492	42	4	this	this	DET
ejpam-6492	42	5	study	study	NOUN
ejpam-6492	42	6	is	be	AUX
ejpam-6492	42	7	to	to	PART
ejpam-6492	42	8	understand	understand	VERB
ejpam-6492	42	9	the	the	DET
ejpam-6492	42	10	mathematical	mathematical	ADJ
ejpam-6492	42	11	structures	structure	NOUN
ejpam-6492	42	12	of	of	ADP
ejpam-6492	42	13	quaternions	quaternion	NOUN
ejpam-6492	42	14	based	base	VERB
ejpam-6492	42	15	on	on	ADP
ejpam-6492	42	16	generalized	generalized	ADJ
ejpam-6492	42	17	fibonacci	fibonacci	NOUN
ejpam-6492	42	18	and	and	CCONJ
ejpam-6492	42	19	lucas	lucas	PROPN
ejpam-6492	42	20	numbers	number	NOUN
ejpam-6492	42	21	and	and	CCONJ
ejpam-6492	42	22	to	to	PART
ejpam-6492	42	23	show	show	VERB
ejpam-6492	42	24	their	their	PRON
ejpam-6492	42	25	connection	connection	NOUN
ejpam-6492	42	26	to	to	ADP
ejpam-6492	42	27	classical	classical	ADJ
ejpam-6492	42	28	results	result	NOUN
ejpam-6492	42	29	through	through	ADP
ejpam-6492	42	30	the	the	DET
ejpam-6492	42	31	derivation	derivation	NOUN
ejpam-6492	42	32	of	of	ADP
ejpam-6492	42	33	new	new	ADJ
ejpam-6492	42	34	identities	identity	NOUN
ejpam-6492	42	35	.	.	PUNCT
ejpam-6492	43	1	in	in	ADP
ejpam-6492	43	2	this	this	DET
ejpam-6492	43	3	section	section	NOUN
ejpam-6492	43	4	some	some	DET
ejpam-6492	43	5	literature	literature	NOUN
ejpam-6492	43	6	overwiev	overwiev	ADV
ejpam-6492	43	7	is	be	AUX
ejpam-6492	43	8	given	give	VERB
ejpam-6492	43	9	about	about	ADP
ejpam-6492	43	10	fibonacci	fibonacci	NOUN
ejpam-6492	43	11	and	and	CCONJ
ejpam-6492	43	12	lucas	lucas	PROPN
ejpam-6492	43	13	sequences	sequence	NOUN
ejpam-6492	43	14	,	,	PUNCT
ejpam-6492	43	15	their	their	PRON
ejpam-6492	43	16	generalizations	generalization	NOUN
ejpam-6492	43	17	,	,	PUNCT
ejpam-6492	43	18	and	and	CCONJ
ejpam-6492	43	19	quaternion	quaternion	NOUN
ejpam-6492	43	20	generalizations	generalization	NOUN
ejpam-6492	43	21	.	.	PUNCT
ejpam-6492	44	1	the	the	DET
ejpam-6492	44	2	rest	rest	NOUN
ejpam-6492	44	3	of	of	ADP
ejpam-6492	44	4	this	this	DET
ejpam-6492	44	5	paper	paper	NOUN
ejpam-6492	44	6	is	be	AUX
ejpam-6492	44	7	organized	organize	VERB
ejpam-6492	44	8	as	as	SCONJ
ejpam-6492	44	9	follows	follow	VERB
ejpam-6492	44	10	.	.	PUNCT
ejpam-6492	45	1	in	in	ADP
ejpam-6492	45	2	the	the	DET
ejpam-6492	45	3	next	next	ADJ
ejpam-6492	45	4	section	section	NOUN
ejpam-6492	45	5	,	,	PUNCT
ejpam-6492	45	6	the	the	DET
ejpam-6492	45	7	basic	basic	ADJ
ejpam-6492	45	8	properties	property	NOUN
ejpam-6492	45	9	of	of	ADP
ejpam-6492	45	10	generalized	generalized	ADJ
ejpam-6492	45	11	fibonacci	fibonacci	NOUN
ejpam-6492	45	12	and	and	CCONJ
ejpam-6492	45	13	lucas	lucas	PROPN
ejpam-6492	45	14	sequences	sequence	NOUN
ejpam-6492	45	15	will	will	AUX
ejpam-6492	45	16	be	be	AUX
ejpam-6492	45	17	discussed	discuss	VERB
ejpam-6492	45	18	and	and	CCONJ
ejpam-6492	45	19	new	new	ADJ
ejpam-6492	45	20	identities	identity	NOUN
ejpam-6492	45	21	related	relate	VERB
ejpam-6492	45	22	to	to	ADP
ejpam-6492	45	23	these	these	DET
ejpam-6492	45	24	sequences	sequence	NOUN
ejpam-6492	45	25	will	will	AUX
ejpam-6492	45	26	be	be	AUX
ejpam-6492	45	27	obtained	obtain	VERB
ejpam-6492	45	28	.	.	PUNCT
ejpam-6492	46	1	then	then	ADV
ejpam-6492	46	2	in	in	ADP
ejpam-6492	46	3	the	the	DET
ejpam-6492	46	4	third	third	ADJ
ejpam-6492	46	5	section	section	NOUN
ejpam-6492	46	6	,	,	PUNCT
ejpam-6492	46	7	the	the	DET
ejpam-6492	46	8	algebraic	algebraic	ADJ
ejpam-6492	46	9	properties	property	NOUN
ejpam-6492	46	10	of	of	ADP
ejpam-6492	46	11	quaternions	quaternion	NOUN
ejpam-6492	46	12	whose	whose	DET
ejpam-6492	46	13	terms	term	NOUN
ejpam-6492	46	14	are	be	AUX
ejpam-6492	46	15	composed	compose	VERB
ejpam-6492	46	16	of	of	ADP
ejpam-6492	46	17	generalized	generalized	ADJ
ejpam-6492	46	18	fibonacci	fibonacci	NOUN
ejpam-6492	46	19	and	and	CCONJ
ejpam-6492	46	20	lucas	lucas	PROPN
ejpam-6492	46	21	numbers	number	NOUN
ejpam-6492	46	22	will	will	AUX
ejpam-6492	46	23	be	be	AUX
ejpam-6492	46	24	given	give	VERB
ejpam-6492	46	25	,	,	PUNCT
ejpam-6492	46	26	and	and	CCONJ
ejpam-6492	46	27	new	new	ADJ
ejpam-6492	46	28	identities	identity	NOUN
ejpam-6492	46	29	related	relate	VERB
ejpam-6492	46	30	to	to	ADP
ejpam-6492	46	31	these	these	DET
ejpam-6492	46	32	quaternions	quaternion	NOUN
ejpam-6492	46	33	will	will	AUX
ejpam-6492	46	34	be	be	AUX
ejpam-6492	46	35	obtained	obtain	VERB
ejpam-6492	46	36	by	by	ADP
ejpam-6492	46	37	using	use	VERB
ejpam-6492	46	38	binet	binet	NOUN
ejpam-6492	46	39	formulas	formula	NOUN
ejpam-6492	46	40	.	.	PUNCT
ejpam-6492	47	1	finally	finally	ADV
ejpam-6492	47	2	,	,	PUNCT
ejpam-6492	47	3	we	we	PRON
ejpam-6492	47	4	will	will	AUX
ejpam-6492	47	5	give	give	VERB
ejpam-6492	47	6	quaternion	quaternion	NOUN
ejpam-6492	47	7	generalizations	generalization	NOUN
ejpam-6492	47	8	of	of	ADP
ejpam-6492	47	9	product	product	NOUN
ejpam-6492	47	10	differences	difference	NOUN
ejpam-6492	47	11	of	of	ADP
ejpam-6492	47	12	everman	everman	NOUN
ejpam-6492	47	13	and	and	CCONJ
ejpam-6492	47	14	koshy	koshy	ADJ
ejpam-6492	47	15	based	base	VERB
ejpam-6492	47	16	fibonacci	fibonacci	PROPN
ejpam-6492	47	17	identities	identity	NOUN
ejpam-6492	47	18	’	'	PUNCT
ejpam-6492	47	19	.	.	PUNCT
ejpam-6492	48	1	2	2	X
ejpam-6492	48	2	.	.	X
ejpam-6492	48	3	properties	property	NOUN
ejpam-6492	48	4	and	and	CCONJ
ejpam-6492	48	5	identities	identity	NOUN
ejpam-6492	48	6	of	of	ADP
ejpam-6492	48	7	generalized	generalized	ADJ
ejpam-6492	48	8	fibonacci	fibonacci	NOUN
ejpam-6492	48	9	and	and	CCONJ
ejpam-6492	48	10	lucas	lucas	PROPN
ejpam-6492	48	11	sequences	sequence	NOUN
ejpam-6492	48	12	definition	definition	NOUN
ejpam-6492	48	13	1	1	X
ejpam-6492	48	14	.	.	PUNCT
ejpam-6492	49	1	let	let	VERB
ejpam-6492	49	2	hn	hn	PRON
ejpam-6492	49	3	be	be	AUX
ejpam-6492	49	4	a	a	DET
ejpam-6492	49	5	sequence	sequence	NOUN
ejpam-6492	49	6	defined	define	VERB
ejpam-6492	49	7	by	by	ADP
ejpam-6492	49	8	the	the	DET
ejpam-6492	49	9	recurrence	recurrence	NOUN
ejpam-6492	49	10	relation	relation	NOUN
ejpam-6492	49	11	hn	hn	PROPN
ejpam-6492	49	12	=	=	PUNCT
ejpam-6492	49	13	hn−1	hn−1	PROPN
ejpam-6492	49	14	+	+	CCONJ
ejpam-6492	49	15	hn−2	hn−2	PROPN
ejpam-6492	49	16	,	,	PUNCT
ejpam-6492	49	17	for	for	ADP
ejpam-6492	49	18	n	n	PRON
ejpam-6492	49	19	≥	≥	NOUN
ejpam-6492	49	20	3	3	NUM
ejpam-6492	49	21	,	,	PUNCT
ejpam-6492	49	22	(	(	PUNCT
ejpam-6492	49	23	1	1	X
ejpam-6492	49	24	)	)	PUNCT
ejpam-6492	49	25	with	with	ADP
ejpam-6492	49	26	the	the	DET
ejpam-6492	49	27	initial	initial	ADJ
ejpam-6492	49	28	conditions	condition	NOUN
ejpam-6492	49	29	h1	h1	NOUN
ejpam-6492	49	30	=	=	SYM
ejpam-6492	49	31	p	p	PROPN
ejpam-6492	49	32	,	,	PUNCT
ejpam-6492	49	33	h2	h2	NOUN
ejpam-6492	50	1	=	=	PUNCT
ejpam-6492	50	2	p	p	X
ejpam-6492	50	3	+	+	CCONJ
ejpam-6492	50	4	q	q	ADJ
ejpam-6492	50	5	,	,	PUNCT
ejpam-6492	50	6	where	where	SCONJ
ejpam-6492	50	7	p	p	NOUN
ejpam-6492	50	8	and	and	CCONJ
ejpam-6492	50	9	q	q	NOUN
ejpam-6492	50	10	are	be	AUX
ejpam-6492	50	11	arbitrary	arbitrary	ADJ
ejpam-6492	50	12	integers	integer	NOUN
ejpam-6492	50	13	.	.	PUNCT
ejpam-6492	51	1	the	the	DET
ejpam-6492	51	2	terms	term	NOUN
ejpam-6492	51	3	of	of	ADP
ejpam-6492	51	4	this	this	DET
ejpam-6492	51	5	sequence	sequence	NOUN
ejpam-6492	51	6	are	be	AUX
ejpam-6492	51	7	p	p	ADJ
ejpam-6492	51	8	,	,	PUNCT
ejpam-6492	51	9	p	p	X
ejpam-6492	51	10	+	+	X
ejpam-6492	51	11	q	q	ADJ
ejpam-6492	51	12	,	,	PUNCT
ejpam-6492	51	13	2p	2p	NUM
ejpam-6492	51	14	+	+	CCONJ
ejpam-6492	51	15	q	q	ADJ
ejpam-6492	51	16	,	,	PUNCT
ejpam-6492	51	17	3p	3p	NUM
ejpam-6492	51	18	+	+	CCONJ
ejpam-6492	51	19	2q	2q	NOUN
ejpam-6492	51	20	,	,	PUNCT
ejpam-6492	51	21	5p	5p	NUM
ejpam-6492	51	22	+	+	CCONJ
ejpam-6492	51	23	3q	3q	NUM
ejpam-6492	51	24	,	,	PUNCT
ejpam-6492	51	25	8p	8p	NUM
ejpam-6492	51	26	+	+	CCONJ
ejpam-6492	51	27	5q	5q	NUM
ejpam-6492	51	28	,	,	PUNCT
ejpam-6492	51	29	13p	13p	NOUN
ejpam-6492	51	30	+	+	CCONJ
ejpam-6492	51	31	8q	8q	NOUN
ejpam-6492	51	32	,	,	PUNCT
ejpam-6492	51	33	.	.	PUNCT
ejpam-6492	51	34	.	.	PUNCT
ejpam-6492	51	35	.	.	PUNCT
ejpam-6492	52	1	.	.	PUNCT
ejpam-6492	53	1	this	this	DET
ejpam-6492	53	2	sequence	sequence	NOUN
ejpam-6492	53	3	is	be	AUX
ejpam-6492	53	4	called	call	VERB
ejpam-6492	53	5	horadam	horadam	PROPN
ejpam-6492	53	6	’s	’s	PART
ejpam-6492	53	7	generalized	generalize	VERB
ejpam-6492	53	8	fibonacci	fibonacci	NOUN
ejpam-6492	53	9	sequence	sequence	NOUN
ejpam-6492	53	10	[	[	X
ejpam-6492	53	11	16	16	NUM
ejpam-6492	53	12	,	,	PUNCT
ejpam-6492	53	13	17	17	NUM
ejpam-6492	53	14	]	]	PUNCT
ejpam-6492	53	15	.	.	PUNCT
ejpam-6492	54	1	(	(	PUNCT
ejpam-6492	54	2	fn	fn	NOUN
ejpam-6492	54	3	)	)	PUNCT
ejpam-6492	54	4	is	be	AUX
ejpam-6492	54	5	the	the	DET
ejpam-6492	54	6	fibonacci	fibonacci	NOUN
ejpam-6492	54	7	sequence	sequence	NOUN
ejpam-6492	54	8	with	with	ADP
ejpam-6492	54	9	the	the	DET
ejpam-6492	54	10	recurrence	recurrence	NOUN
ejpam-6492	54	11	relation	relation	NOUN
ejpam-6492	54	12	f0	f0	PROPN
ejpam-6492	54	13	=	=	SYM
ejpam-6492	54	14	0	0	NUM
ejpam-6492	54	15	,	,	PUNCT
ejpam-6492	54	16	f1	f1	NOUN
ejpam-6492	54	17	=	=	SYM
ejpam-6492	54	18	1	1	NUM
ejpam-6492	54	19	,	,	PUNCT
ejpam-6492	54	20	fn	fn	NOUN
ejpam-6492	54	21	=	=	PUNCT
ejpam-6492	54	22	fn−1	fn−1	PROPN
ejpam-6492	54	23	+	+	CCONJ
ejpam-6492	54	24	fn−2	fn−2	ADJ
ejpam-6492	54	25	for	for	ADP
ejpam-6492	54	26	n	n	PRON
ejpam-6492	54	27	≥	≥	NUM
ejpam-6492	54	28	2	2	NUM
ejpam-6492	54	29	,	,	PUNCT
ejpam-6492	54	30	defined	define	VERB
ejpam-6492	54	31	by	by	ADP
ejpam-6492	54	32	taking	take	VERB
ejpam-6492	54	33	p	p	NOUN
ejpam-6492	54	34	=	=	NOUN
ejpam-6492	54	35	1	1	NUM
ejpam-6492	54	36	and	and	CCONJ
ejpam-6492	54	37	q	q	NOUN
ejpam-6492	55	1	=	=	SYM
ejpam-6492	55	2	0	0	NUM
ejpam-6492	55	3	in	in	ADP
ejpam-6492	55	4	(	(	PUNCT
ejpam-6492	55	5	1	1	NUM
ejpam-6492	55	6	)	)	PUNCT
ejpam-6492	55	7	.	.	PUNCT
ejpam-6492	56	1	similarly	similarly	ADV
ejpam-6492	56	2	,	,	PUNCT
ejpam-6492	56	3	taking	take	VERB
ejpam-6492	56	4	p	p	NOUN
ejpam-6492	56	5	=	=	NOUN
ejpam-6492	56	6	1	1	NUM
ejpam-6492	56	7	and	and	CCONJ
ejpam-6492	56	8	q	q	NOUN
ejpam-6492	57	1	=	=	SYM
ejpam-6492	57	2	2	2	NUM
ejpam-6492	57	3	in	in	ADP
ejpam-6492	57	4	(	(	PUNCT
ejpam-6492	57	5	1	1	NUM
ejpam-6492	57	6	)	)	PUNCT
ejpam-6492	57	7	,	,	PUNCT
ejpam-6492	57	8	the	the	DET
ejpam-6492	57	9	lucas	lucas	PROPN
ejpam-6492	57	10	sequence	sequence	NOUN
ejpam-6492	57	11	(	(	PUNCT
ejpam-6492	57	12	ln	ln	ADJ
ejpam-6492	57	13	)	)	PUNCT
ejpam-6492	57	14	is	be	AUX
ejpam-6492	57	15	defined	define	VERB
ejpam-6492	57	16	with	with	ADP
ejpam-6492	57	17	the	the	DET
ejpam-6492	57	18	recurrence	recurrence	NOUN
ejpam-6492	57	19	relation	relation	NOUN
ejpam-6492	57	20	l0	l0	PROPN
ejpam-6492	57	21	=	=	SYM
ejpam-6492	57	22	2	2	NUM
ejpam-6492	57	23	,	,	PUNCT
ejpam-6492	57	24	l1	l1	PROPN
ejpam-6492	57	25	=	=	PROPN
ejpam-6492	57	26	1	1	NUM
ejpam-6492	57	27	,	,	PUNCT
ejpam-6492	57	28	ln	ln	NOUN
ejpam-6492	57	29	=	=	PROPN
ejpam-6492	57	30	ln−1	ln−1	PROPN
ejpam-6492	57	31	+	+	CCONJ
ejpam-6492	57	32	ln−2	ln−2	PROPN
ejpam-6492	57	33	for	for	ADP
ejpam-6492	57	34	n	n	X
ejpam-6492	57	35	≥	≥	NOUN
ejpam-6492	57	36	2	2	NUM
ejpam-6492	57	37	,	,	PUNCT
ejpam-6492	57	38	[	[	X
ejpam-6492	57	39	18	18	NUM
ejpam-6492	57	40	]	]	PUNCT
ejpam-6492	57	41	,	,	PUNCT
ejpam-6492	57	42	[	[	X
ejpam-6492	57	43	19	19	NUM
ejpam-6492	57	44	]	]	PUNCT
ejpam-6492	57	45	.	.	PUNCT
ejpam-6492	58	1	kalman	kalman	PROPN
ejpam-6492	58	2	and	and	CCONJ
ejpam-6492	58	3	mena	mena	PROPN
ejpam-6492	58	4	,	,	PUNCT
ejpam-6492	58	5	in	in	ADP
ejpam-6492	58	6	[	[	X
ejpam-6492	58	7	20	20	NUM
ejpam-6492	58	8	]	]	PUNCT
ejpam-6492	58	9	,	,	PUNCT
ejpam-6492	58	10	defined	define	VERB
ejpam-6492	58	11	generalized	generalized	ADJ
ejpam-6492	58	12	fibonacci	fibonacci	NOUN
ejpam-6492	58	13	and	and	CCONJ
ejpam-6492	58	14	lucas	lucas	PROPN
ejpam-6492	58	15	numbers	number	NOUN
ejpam-6492	58	16	with	with	ADP
ejpam-6492	58	17	the	the	DET
ejpam-6492	58	18	recurrence	recurrence	NOUN
ejpam-6492	58	19	relation	relation	NOUN
ejpam-6492	58	20	an+2	an+2	NUM
ejpam-6492	58	21	=	=	SYM
ejpam-6492	58	22	aan+1	aan+1	X
ejpam-6492	58	23	+	+	CCONJ
ejpam-6492	58	24	ban	ban	NOUN
ejpam-6492	58	25	,	,	PUNCT
ejpam-6492	58	26	for	for	ADP
ejpam-6492	58	27	all	all	DET
ejpam-6492	58	28	n	n	DET
ejpam-6492	58	29	≥	≥	NOUN
ejpam-6492	58	30	0	0	NUM
ejpam-6492	58	31	where	where	SCONJ
ejpam-6492	58	32	the	the	DET
ejpam-6492	58	33	sequences	sequence	NOUN
ejpam-6492	58	34	depend	depend	VERB
ejpam-6492	58	35	on	on	ADP
ejpam-6492	58	36	initial	initial	ADJ
ejpam-6492	58	37	conditions	condition	NOUN
ejpam-6492	58	38	a0	a0	NOUN
ejpam-6492	58	39	and	and	CCONJ
ejpam-6492	58	40	a1	a1	NOUN
ejpam-6492	58	41	.	.	PUNCT
ejpam-6492	59	1	they	they	PRON
ejpam-6492	59	2	identified	identify	VERB
ejpam-6492	59	3	several	several	ADJ
ejpam-6492	59	4	number	number	NOUN
ejpam-6492	59	5	sequences	sequence	NOUN
ejpam-6492	59	6	corresponding	correspond	VERB
ejpam-6492	59	7	to	to	ADP
ejpam-6492	59	8	different	different	ADJ
ejpam-6492	59	9	values	value	NOUN
ejpam-6492	59	10	of	of	ADP
ejpam-6492	59	11	r(a	r(a	PROPN
ejpam-6492	59	12	,	,	PUNCT
ejpam-6492	59	13	b	b	NOUN
ejpam-6492	59	14	)	)	PUNCT
ejpam-6492	59	15	,	,	PUNCT
ejpam-6492	59	16	as	as	SCONJ
ejpam-6492	59	17	illustrated	illustrate	VERB
ejpam-6492	59	18	below	below	ADV
ejpam-6492	59	19	.	.	PUNCT
ejpam-6492	60	1	b.	b.	PROPN
ejpam-6492	60	2	demirtürk	demirtürk	PROPN
ejpam-6492	60	3	,	,	PUNCT
ejpam-6492	60	4	n.	n.	NOUN
ejpam-6492	60	5	topal	topal	PROPN
ejpam-6492	60	6	/	/	SYM
ejpam-6492	60	7	eur	eur	PROPN
ejpam-6492	60	8	.	.	PUNCT
ejpam-6492	61	1	j.	j.	PROPN
ejpam-6492	61	2	pure	pure	PROPN
ejpam-6492	61	3	appl	appl	PROPN
ejpam-6492	61	4	.	.	PROPN
ejpam-6492	61	5	math	math	PROPN
ejpam-6492	61	6	,	,	PUNCT
ejpam-6492	61	7	18	18	NUM
ejpam-6492	61	8	(	(	PUNCT
ejpam-6492	61	9	3	3	NUM
ejpam-6492	61	10	)	)	PUNCT
ejpam-6492	61	11	(	(	PUNCT
ejpam-6492	61	12	2025	2025	NUM
ejpam-6492	61	13	)	)	PUNCT
ejpam-6492	61	14	,	,	PUNCT
ejpam-6492	61	15	6492	6492	NUM
ejpam-6492	61	16	4	4	NUM
ejpam-6492	61	17	of	of	ADP
ejpam-6492	61	18	22	22	NUM
ejpam-6492	61	19	table	table	NOUN
ejpam-6492	61	20	1	1	NUM
ejpam-6492	61	21	:	:	PUNCT
ejpam-6492	61	22	generalized	generalized	ADJ
ejpam-6492	61	23	fibonacci	fibonacci	NOUN
ejpam-6492	61	24	and	and	CCONJ
ejpam-6492	61	25	lucas	lucas	NOUN
ejpam-6492	61	26	-	-	PUNCT
ejpam-6492	61	27	type	type	NOUN
ejpam-6492	61	28	sequences	sequence	NOUN
ejpam-6492	61	29	[	[	X
ejpam-6492	61	30	20	20	NUM
ejpam-6492	61	31	]	]	SYM
ejpam-6492	61	32	sequence	sequence	NOUN
ejpam-6492	61	33	initial	initial	ADJ
ejpam-6492	61	34	conditions	condition	NOUN
ejpam-6492	61	35	first	first	ADJ
ejpam-6492	61	36	terms	term	NOUN
ejpam-6492	61	37	fibonacci	fibonacci	PROPN
ejpam-6492	61	38	numbers	number	NOUN
ejpam-6492	61	39	r(1	r(1	PROPN
ejpam-6492	61	40	,	,	PUNCT
ejpam-6492	61	41	1	1	NUM
ejpam-6492	61	42	)	)	PUNCT
ejpam-6492	61	43	,	,	PUNCT
ejpam-6492	61	44	a0	a0	PROPN
ejpam-6492	61	45	=	=	SYM
ejpam-6492	61	46	0	0	PROPN
ejpam-6492	61	47	,	,	PUNCT
ejpam-6492	61	48	a1	a1	NOUN
ejpam-6492	61	49	=	=	SYM
ejpam-6492	61	50	1	1	NUM
ejpam-6492	61	51	{	{	PUNCT
ejpam-6492	61	52	0	0	NUM
ejpam-6492	61	53	,	,	PUNCT
ejpam-6492	61	54	1	1	NUM
ejpam-6492	61	55	,	,	PUNCT
ejpam-6492	61	56	1	1	NUM
ejpam-6492	61	57	,	,	PUNCT
ejpam-6492	61	58	2	2	NUM
ejpam-6492	61	59	,	,	PUNCT
ejpam-6492	61	60	3	3	NUM
ejpam-6492	61	61	,	,	PUNCT
ejpam-6492	61	62	5	5	NUM
ejpam-6492	61	63	,	,	PUNCT
ejpam-6492	61	64	8	8	NUM
ejpam-6492	61	65	,	,	PUNCT
ejpam-6492	61	66	13	13	NUM
ejpam-6492	61	67	,	,	PUNCT
ejpam-6492	61	68	21	21	NUM
ejpam-6492	61	69	,	,	PUNCT
ejpam-6492	61	70	34	34	NUM
ejpam-6492	61	71	,	,	PUNCT
ejpam-6492	61	72	55	55	NUM
ejpam-6492	61	73	,	,	PUNCT
ejpam-6492	61	74	.	.	PUNCT
ejpam-6492	61	75	.	.	PUNCT
ejpam-6492	61	76	.	.	PUNCT
ejpam-6492	62	1	}	}	PUNCT
ejpam-6492	62	2	lucas	lucas	PROPN
ejpam-6492	62	3	numbers	number	NOUN
ejpam-6492	62	4	r(1	r(1	PROPN
ejpam-6492	62	5	,	,	PUNCT
ejpam-6492	62	6	1	1	NUM
ejpam-6492	62	7	)	)	PUNCT
ejpam-6492	62	8	,	,	PUNCT
ejpam-6492	62	9	a0	a0	PROPN
ejpam-6492	62	10	=	=	SYM
ejpam-6492	62	11	2	2	NUM
ejpam-6492	62	12	,	,	PUNCT
ejpam-6492	62	13	a1	a1	NOUN
ejpam-6492	62	14	=	=	PUNCT
ejpam-6492	62	15	a	a	PRON
ejpam-6492	62	16	{	{	PUNCT
ejpam-6492	62	17	2	2	NUM
ejpam-6492	62	18	,	,	PUNCT
ejpam-6492	62	19	1	1	NUM
ejpam-6492	62	20	,	,	PUNCT
ejpam-6492	62	21	3	3	NUM
ejpam-6492	62	22	,	,	PUNCT
ejpam-6492	62	23	4	4	NUM
ejpam-6492	62	24	,	,	PUNCT
ejpam-6492	62	25	7	7	NUM
ejpam-6492	62	26	,	,	PUNCT
ejpam-6492	62	27	11	11	NUM
ejpam-6492	62	28	,	,	PUNCT
ejpam-6492	62	29	18	18	NUM
ejpam-6492	62	30	,	,	PUNCT
ejpam-6492	62	31	29	29	NUM
ejpam-6492	62	32	,	,	PUNCT
ejpam-6492	62	33	47	47	NUM
ejpam-6492	62	34	,	,	PUNCT
ejpam-6492	62	35	76	76	NUM
ejpam-6492	62	36	,	,	PUNCT
ejpam-6492	62	37	.	.	PUNCT
ejpam-6492	62	38	.	.	PUNCT
ejpam-6492	62	39	.	.	PUNCT
ejpam-6492	63	1	}	}	PUNCT
ejpam-6492	63	2	pell	pell	VERB
ejpam-6492	63	3	numbers	number	NOUN
ejpam-6492	63	4	r(2	r(2	PROPN
ejpam-6492	63	5	,	,	PUNCT
ejpam-6492	63	6	1	1	NUM
ejpam-6492	63	7	)	)	PUNCT
ejpam-6492	63	8	,	,	PUNCT
ejpam-6492	63	9	a0	a0	PROPN
ejpam-6492	63	10	=	=	SYM
ejpam-6492	63	11	0	0	PROPN
ejpam-6492	63	12	,	,	PUNCT
ejpam-6492	63	13	a1	a1	NOUN
ejpam-6492	63	14	=	=	SYM
ejpam-6492	63	15	1	1	NUM
ejpam-6492	63	16	{	{	PUNCT
ejpam-6492	63	17	0	0	NUM
ejpam-6492	63	18	,	,	PUNCT
ejpam-6492	63	19	1	1	NUM
ejpam-6492	63	20	,	,	PUNCT
ejpam-6492	63	21	2	2	NUM
ejpam-6492	63	22	,	,	PUNCT
ejpam-6492	63	23	5	5	NUM
ejpam-6492	63	24	,	,	PUNCT
ejpam-6492	63	25	12	12	NUM
ejpam-6492	63	26	,	,	PUNCT
ejpam-6492	63	27	29	29	NUM
ejpam-6492	63	28	,	,	PUNCT
ejpam-6492	63	29	70	70	NUM
ejpam-6492	63	30	,	,	PUNCT
ejpam-6492	63	31	169	169	NUM
ejpam-6492	63	32	,	,	PUNCT
ejpam-6492	63	33	.	.	PUNCT
ejpam-6492	63	34	.	.	PUNCT
ejpam-6492	63	35	.	.	PUNCT
ejpam-6492	64	1	}	}	PUNCT
ejpam-6492	64	2	pell	pell	NOUN
ejpam-6492	64	3	-	-	PUNCT
ejpam-6492	64	4	lucas	lucas	NOUN
ejpam-6492	64	5	numbers	number	NOUN
ejpam-6492	64	6	r(2	r(2	PROPN
ejpam-6492	64	7	,	,	PUNCT
ejpam-6492	64	8	1	1	NUM
ejpam-6492	64	9	)	)	PUNCT
ejpam-6492	64	10	,	,	PUNCT
ejpam-6492	64	11	a0	a0	PROPN
ejpam-6492	64	12	=	=	SYM
ejpam-6492	64	13	2	2	NUM
ejpam-6492	64	14	,	,	PUNCT
ejpam-6492	64	15	a1	a1	NOUN
ejpam-6492	64	16	=	=	PUNCT
ejpam-6492	64	17	a	a	PRON
ejpam-6492	64	18	{	{	PUNCT
ejpam-6492	64	19	2	2	NUM
ejpam-6492	64	20	,	,	PUNCT
ejpam-6492	64	21	2	2	NUM
ejpam-6492	64	22	,	,	PUNCT
ejpam-6492	64	23	6	6	NUM
ejpam-6492	64	24	,	,	PUNCT
ejpam-6492	64	25	14	14	NUM
ejpam-6492	64	26	,	,	PUNCT
ejpam-6492	64	27	34	34	NUM
ejpam-6492	64	28	,	,	PUNCT
ejpam-6492	64	29	82	82	NUM
ejpam-6492	64	30	,	,	PUNCT
ejpam-6492	64	31	.	.	PUNCT
ejpam-6492	64	32	.	.	PUNCT
ejpam-6492	64	33	.	.	PUNCT
ejpam-6492	65	1	}	}	PUNCT
ejpam-6492	65	2	natural	natural	ADJ
ejpam-6492	65	3	numbers	number	NOUN
ejpam-6492	65	4	r(2	r(2	NOUN
ejpam-6492	65	5	,	,	PUNCT
ejpam-6492	65	6	−1	−1	NOUN
ejpam-6492	65	7	)	)	PUNCT
ejpam-6492	65	8	,	,	PUNCT
ejpam-6492	65	9	a0	a0	PROPN
ejpam-6492	65	10	=	=	SYM
ejpam-6492	65	11	0	0	PROPN
ejpam-6492	65	12	,	,	PUNCT
ejpam-6492	65	13	a1	a1	NOUN
ejpam-6492	65	14	=	=	SYM
ejpam-6492	65	15	1	1	NUM
ejpam-6492	65	16	{	{	PUNCT
ejpam-6492	65	17	0	0	NUM
ejpam-6492	65	18	,	,	PUNCT
ejpam-6492	65	19	1	1	NUM
ejpam-6492	65	20	,	,	PUNCT
ejpam-6492	65	21	2	2	NUM
ejpam-6492	65	22	,	,	PUNCT
ejpam-6492	65	23	3	3	NUM
ejpam-6492	65	24	,	,	PUNCT
ejpam-6492	65	25	4	4	NUM
ejpam-6492	65	26	,	,	PUNCT
ejpam-6492	65	27	5	5	NUM
ejpam-6492	65	28	,	,	PUNCT
ejpam-6492	65	29	6	6	NUM
ejpam-6492	65	30	,	,	PUNCT
ejpam-6492	65	31	.	.	PUNCT
ejpam-6492	65	32	.	.	PUNCT
ejpam-6492	65	33	.	.	PUNCT
ejpam-6492	66	1	}	}	PUNCT
ejpam-6492	66	2	constant	constant	ADJ
ejpam-6492	66	3	sequence	sequence	NOUN
ejpam-6492	66	4	r(2	r(2	NOUN
ejpam-6492	66	5	,	,	PUNCT
ejpam-6492	66	6	−1	−1	NOUN
ejpam-6492	66	7	)	)	PUNCT
ejpam-6492	66	8	,	,	PUNCT
ejpam-6492	66	9	a0	a0	PROPN
ejpam-6492	66	10	=	=	SYM
ejpam-6492	66	11	2	2	NUM
ejpam-6492	66	12	,	,	PUNCT
ejpam-6492	66	13	a1	a1	NOUN
ejpam-6492	66	14	=	=	PUNCT
ejpam-6492	66	15	a	a	PRON
ejpam-6492	66	16	{	{	PUNCT
ejpam-6492	66	17	2	2	NUM
ejpam-6492	66	18	,	,	PUNCT
ejpam-6492	66	19	2	2	NUM
ejpam-6492	66	20	,	,	PUNCT
ejpam-6492	66	21	2	2	NUM
ejpam-6492	66	22	,	,	PUNCT
ejpam-6492	66	23	2	2	NUM
ejpam-6492	66	24	,	,	PUNCT
ejpam-6492	66	25	2	2	NUM
ejpam-6492	66	26	,	,	PUNCT
ejpam-6492	66	27	2	2	NUM
ejpam-6492	66	28	,	,	PUNCT
ejpam-6492	66	29	2	2	NUM
ejpam-6492	66	30	,	,	PUNCT
ejpam-6492	66	31	.	.	PUNCT
ejpam-6492	66	32	.	.	PUNCT
ejpam-6492	66	33	.	.	PUNCT
ejpam-6492	67	1	}	}	PUNCT
ejpam-6492	67	2	mersenne	mersenne	NOUN
ejpam-6492	67	3	sequence	sequence	PROPN
ejpam-6492	67	4	r(3	r(3	PROPN
ejpam-6492	67	5	,	,	PUNCT
ejpam-6492	67	6	−2	−2	NOUN
ejpam-6492	67	7	)	)	PUNCT
ejpam-6492	67	8	,	,	PUNCT
ejpam-6492	67	9	a0	a0	PROPN
ejpam-6492	67	10	=	=	SYM
ejpam-6492	67	11	0	0	PROPN
ejpam-6492	67	12	,	,	PUNCT
ejpam-6492	67	13	a1	a1	NOUN
ejpam-6492	67	14	=	=	SYM
ejpam-6492	67	15	1	1	NUM
ejpam-6492	67	16	{	{	PUNCT
ejpam-6492	67	17	0	0	NUM
ejpam-6492	67	18	,	,	PUNCT
ejpam-6492	67	19	1	1	NUM
ejpam-6492	67	20	,	,	PUNCT
ejpam-6492	67	21	3	3	NUM
ejpam-6492	67	22	,	,	PUNCT
ejpam-6492	67	23	7	7	NUM
ejpam-6492	67	24	,	,	PUNCT
ejpam-6492	67	25	15	15	NUM
ejpam-6492	67	26	,	,	PUNCT
ejpam-6492	67	27	31	31	NUM
ejpam-6492	67	28	,	,	PUNCT
ejpam-6492	67	29	.	.	PUNCT
ejpam-6492	67	30	.	.	PUNCT
ejpam-6492	67	31	.	.	PUNCT
ejpam-6492	68	1	}	}	PUNCT
ejpam-6492	68	2	fermat	fermat	VERB
ejpam-6492	68	3	sequence	sequence	NOUN
ejpam-6492	68	4	r(3	r(3	PROPN
ejpam-6492	68	5	,	,	PUNCT
ejpam-6492	68	6	−2	−2	NOUN
ejpam-6492	68	7	)	)	PUNCT
ejpam-6492	68	8	,	,	PUNCT
ejpam-6492	68	9	a0	a0	PROPN
ejpam-6492	68	10	=	=	SYM
ejpam-6492	68	11	2	2	NUM
ejpam-6492	68	12	,	,	PUNCT
ejpam-6492	68	13	a1	a1	NOUN
ejpam-6492	68	14	=	=	PUNCT
ejpam-6492	68	15	a	a	PRON
ejpam-6492	68	16	{	{	PUNCT
ejpam-6492	68	17	2	2	NUM
ejpam-6492	68	18	,	,	PUNCT
ejpam-6492	68	19	3	3	NUM
ejpam-6492	68	20	,	,	PUNCT
ejpam-6492	68	21	5	5	NUM
ejpam-6492	68	22	,	,	PUNCT
ejpam-6492	68	23	9	9	NUM
ejpam-6492	68	24	,	,	PUNCT
ejpam-6492	68	25	16	16	NUM
ejpam-6492	68	26	,	,	PUNCT
ejpam-6492	68	27	33	33	NUM
ejpam-6492	68	28	,	,	PUNCT
ejpam-6492	68	29	.	.	PUNCT
ejpam-6492	68	30	.	.	PUNCT
ejpam-6492	68	31	.	.	PUNCT
ejpam-6492	69	1	}	}	PUNCT
ejpam-6492	69	2	periodic	periodic	NOUN
ejpam-6492	69	3	(	(	PUNCT
ejpam-6492	69	4	with	with	ADP
ejpam-6492	69	5	period=6	period=6	PROPN
ejpam-6492	69	6	)	)	PUNCT
ejpam-6492	69	7	r(1	r(1	PROPN
ejpam-6492	69	8	,	,	PUNCT
ejpam-6492	69	9	−1	−1	NOUN
ejpam-6492	69	10	)	)	PUNCT
ejpam-6492	69	11	,	,	PUNCT
ejpam-6492	69	12	a0	a0	PROPN
ejpam-6492	69	13	=	=	SYM
ejpam-6492	69	14	0	0	PROPN
ejpam-6492	69	15	,	,	PUNCT
ejpam-6492	69	16	a1	a1	NOUN
ejpam-6492	69	17	=	=	SYM
ejpam-6492	69	18	1	1	NUM
ejpam-6492	69	19	{	{	PUNCT
ejpam-6492	69	20	0	0	NUM
ejpam-6492	69	21	,	,	PUNCT
ejpam-6492	69	22	1	1	NUM
ejpam-6492	69	23	,	,	PUNCT
ejpam-6492	69	24	1	1	NUM
ejpam-6492	69	25	,	,	PUNCT
ejpam-6492	69	26	0	0	NUM
ejpam-6492	69	27	,	,	PUNCT
ejpam-6492	69	28	−1	−1	NOUN
ejpam-6492	69	29	,	,	PUNCT
ejpam-6492	69	30	−1	−1	NOUN
ejpam-6492	69	31	,	,	PUNCT
ejpam-6492	69	32	0	0	NUM
ejpam-6492	69	33	,	,	PUNCT
ejpam-6492	69	34	1	1	NUM
ejpam-6492	69	35	,	,	PUNCT
ejpam-6492	69	36	.	.	PUNCT
ejpam-6492	69	37	.	.	PUNCT
ejpam-6492	69	38	.	.	PUNCT
ejpam-6492	70	1	}	}	PUNCT
ejpam-6492	70	2	periodic	periodic	NOUN
ejpam-6492	70	3	(	(	PUNCT
ejpam-6492	70	4	with	with	ADP
ejpam-6492	70	5	period=6	period=6	PROPN
ejpam-6492	70	6	)	)	PUNCT
ejpam-6492	70	7	r(1	r(1	PROPN
ejpam-6492	70	8	,	,	PUNCT
ejpam-6492	70	9	−1	−1	NOUN
ejpam-6492	70	10	)	)	PUNCT
ejpam-6492	70	11	,	,	PUNCT
ejpam-6492	70	12	a0	a0	PROPN
ejpam-6492	70	13	=	=	SYM
ejpam-6492	70	14	2	2	NUM
ejpam-6492	70	15	,	,	PUNCT
ejpam-6492	70	16	a1	a1	NOUN
ejpam-6492	70	17	=	=	PUNCT
ejpam-6492	70	18	a	a	PRON
ejpam-6492	70	19	{	{	PUNCT
ejpam-6492	70	20	2	2	NUM
ejpam-6492	70	21	,	,	PUNCT
ejpam-6492	70	22	1	1	NUM
ejpam-6492	70	23	,	,	PUNCT
ejpam-6492	70	24	−1	−1	NOUN
ejpam-6492	70	25	,	,	PUNCT
ejpam-6492	70	26	−2	−2	NOUN
ejpam-6492	70	27	,	,	PUNCT
ejpam-6492	70	28	−1	−1	NOUN
ejpam-6492	70	29	,	,	PUNCT
ejpam-6492	70	30	1	1	NUM
ejpam-6492	70	31	,	,	PUNCT
ejpam-6492	70	32	2	2	NUM
ejpam-6492	70	33	,	,	PUNCT
ejpam-6492	70	34	1	1	NUM
ejpam-6492	70	35	,	,	PUNCT
ejpam-6492	70	36	.	.	PUNCT
ejpam-6492	70	37	.	.	PUNCT
ejpam-6492	70	38	.	.	PUNCT
ejpam-6492	71	1	}	}	PUNCT
ejpam-6492	71	2	even	even	ADV
ejpam-6492	71	3	-	-	PUNCT
ejpam-6492	71	4	indexed	index	VERB
ejpam-6492	71	5	fibonacci	fibonacci	NOUN
ejpam-6492	71	6	r(3	r(3	PROPN
ejpam-6492	71	7	,	,	PUNCT
ejpam-6492	71	8	−1	−1	NOUN
ejpam-6492	71	9	)	)	PUNCT
ejpam-6492	71	10	,	,	PUNCT
ejpam-6492	71	11	a0	a0	PROPN
ejpam-6492	71	12	=	=	SYM
ejpam-6492	71	13	0	0	PROPN
ejpam-6492	71	14	,	,	PUNCT
ejpam-6492	71	15	a1	a1	NOUN
ejpam-6492	71	16	=	=	SYM
ejpam-6492	71	17	1	1	NUM
ejpam-6492	71	18	{	{	PUNCT
ejpam-6492	71	19	0	0	NUM
ejpam-6492	71	20	,	,	PUNCT
ejpam-6492	71	21	1	1	NUM
ejpam-6492	71	22	,	,	PUNCT
ejpam-6492	71	23	3	3	NUM
ejpam-6492	71	24	,	,	PUNCT
ejpam-6492	71	25	8	8	NUM
ejpam-6492	71	26	,	,	PUNCT
ejpam-6492	71	27	21	21	NUM
ejpam-6492	71	28	,	,	PUNCT
ejpam-6492	71	29	55	55	NUM
ejpam-6492	71	30	,	,	PUNCT
ejpam-6492	71	31	.	.	PUNCT
ejpam-6492	71	32	.	.	PUNCT
ejpam-6492	71	33	.	.	PUNCT
ejpam-6492	72	1	}	}	PUNCT
ejpam-6492	72	2	even	even	ADV
ejpam-6492	72	3	-	-	PUNCT
ejpam-6492	72	4	indexed	index	VERB
ejpam-6492	72	5	lucas	lucas	PROPN
ejpam-6492	72	6	r(3	r(3	PROPN
ejpam-6492	72	7	,	,	PUNCT
ejpam-6492	72	8	−1	−1	NOUN
ejpam-6492	72	9	)	)	PUNCT
ejpam-6492	72	10	,	,	PUNCT
ejpam-6492	72	11	a0	a0	PROPN
ejpam-6492	72	12	=	=	SYM
ejpam-6492	72	13	2	2	NUM
ejpam-6492	72	14	,	,	PUNCT
ejpam-6492	72	15	a1	a1	NOUN
ejpam-6492	72	16	=	=	PUNCT
ejpam-6492	72	17	a	a	PRON
ejpam-6492	72	18	{	{	PUNCT
ejpam-6492	72	19	2	2	NUM
ejpam-6492	72	20	,	,	PUNCT
ejpam-6492	72	21	3	3	NUM
ejpam-6492	72	22	,	,	PUNCT
ejpam-6492	72	23	7	7	NUM
ejpam-6492	72	24	,	,	PUNCT
ejpam-6492	72	25	18	18	NUM
ejpam-6492	72	26	,	,	PUNCT
ejpam-6492	72	27	47	47	NUM
ejpam-6492	72	28	,	,	PUNCT
ejpam-6492	72	29	.	.	PUNCT
ejpam-6492	72	30	.	.	PUNCT
ejpam-6492	72	31	.	.	PUNCT
ejpam-6492	73	1	}	}	PUNCT
ejpam-6492	73	2	koshy	koshy	VERB
ejpam-6492	73	3	,	,	PUNCT
ejpam-6492	73	4	in	in	ADP
ejpam-6492	73	5	[	[	X
ejpam-6492	73	6	19	19	NUM
ejpam-6492	73	7	]	]	PUNCT
ejpam-6492	73	8	,	,	PUNCT
ejpam-6492	73	9	introduced	introduce	VERB
ejpam-6492	73	10	the	the	DET
ejpam-6492	73	11	notion	notion	NOUN
ejpam-6492	73	12	of	of	ADP
ejpam-6492	73	13	(	(	PUNCT
ejpam-6492	73	14	p	p	X
ejpam-6492	73	15	,	,	PUNCT
ejpam-6492	73	16	q	q	ADJ
ejpam-6492	73	17	)	)	PUNCT
ejpam-6492	73	18	generalized	generalized	ADJ
ejpam-6492	73	19	fibonacci	fibonacci	NOUN
ejpam-6492	73	20	numbers	number	NOUN
ejpam-6492	73	21	.	.	PUNCT
ejpam-6492	74	1	in	in	ADP
ejpam-6492	74	2	this	this	DET
ejpam-6492	74	3	study	study	NOUN
ejpam-6492	74	4	,	,	PUNCT
ejpam-6492	74	5	we	we	PRON
ejpam-6492	74	6	present	present	VERB
ejpam-6492	74	7	several	several	ADJ
ejpam-6492	74	8	properties	property	NOUN
ejpam-6492	74	9	of	of	ADP
ejpam-6492	74	10	the	the	DET
ejpam-6492	74	11	(	(	PUNCT
ejpam-6492	74	12	p	p	X
ejpam-6492	74	13	,	,	PUNCT
ejpam-6492	74	14	q)-generalized	q)-generalized	ADJ
ejpam-6492	74	15	fibonacci	fibonacci	NOUN
ejpam-6492	74	16	and	and	CCONJ
ejpam-6492	74	17	lucas	lucas	PROPN
ejpam-6492	74	18	numbers	number	NOUN
ejpam-6492	74	19	.	.	PUNCT
ejpam-6492	75	1	now	now	ADV
ejpam-6492	75	2	let	let	VERB
ejpam-6492	75	3	us	we	PRON
ejpam-6492	75	4	give	give	VERB
ejpam-6492	75	5	the	the	DET
ejpam-6492	75	6	definition	definition	NOUN
ejpam-6492	75	7	of	of	ADP
ejpam-6492	75	8	a	a	DET
ejpam-6492	75	9	(	(	PUNCT
ejpam-6492	75	10	p	p	X
ejpam-6492	75	11	,	,	PUNCT
ejpam-6492	75	12	q	q	ADJ
ejpam-6492	75	13	)	)	PUNCT
ejpam-6492	75	14	generalization	generalization	NOUN
ejpam-6492	75	15	of	of	ADP
ejpam-6492	75	16	fibonacci	fibonacci	PROPN
ejpam-6492	75	17	and	and	CCONJ
ejpam-6492	75	18	lucas	lucas	PROPN
ejpam-6492	75	19	sequences	sequence	NOUN
ejpam-6492	75	20	.	.	PUNCT
ejpam-6492	76	1	definition	definition	NOUN
ejpam-6492	76	2	2	2	NUM
ejpam-6492	76	3	.	.	PUNCT
ejpam-6492	77	1	(	(	PUNCT
ejpam-6492	77	2	[	[	X
ejpam-6492	77	3	16	16	NUM
ejpam-6492	77	4	,	,	PUNCT
ejpam-6492	77	5	17	17	NUM
ejpam-6492	77	6	]	]	PUNCT
ejpam-6492	77	7	)	)	PUNCT
ejpam-6492	77	8	let	let	VERB
ejpam-6492	77	9	p	p	PRON
ejpam-6492	77	10	,	,	PUNCT
ejpam-6492	77	11	q	q	PROPN
ejpam-6492	77	12	∈	∈	PROPN
ejpam-6492	77	13	z.	z.	X
ejpam-6492	78	1	the	the	DET
ejpam-6492	78	2	sequence	sequence	NOUN
ejpam-6492	78	3	(	(	PUNCT
ejpam-6492	78	4	un	un	PROPN
ejpam-6492	78	5	)	)	PUNCT
ejpam-6492	78	6	defined	define	VERB
ejpam-6492	78	7	by	by	ADP
ejpam-6492	78	8	the	the	DET
ejpam-6492	78	9	relation	relation	NOUN
ejpam-6492	78	10	un	un	PROPN
ejpam-6492	78	11	=	=	PROPN
ejpam-6492	78	12	pun−1	pun−1	PROPN
ejpam-6492	78	13	+	+	CCONJ
ejpam-6492	78	14	qun−2	qun−2	PROPN
ejpam-6492	78	15	,	,	PUNCT
ejpam-6492	78	16	for	for	ADP
ejpam-6492	78	17	all	all	DET
ejpam-6492	78	18	n	n	PRON
ejpam-6492	78	19	≥	≥	NOUN
ejpam-6492	78	20	2	2	NUM
ejpam-6492	78	21	with	with	ADP
ejpam-6492	78	22	initial	initial	ADJ
ejpam-6492	78	23	conditions	condition	NOUN
ejpam-6492	78	24	u0	u0	ADJ
ejpam-6492	78	25	=	=	SYM
ejpam-6492	78	26	0	0	NUM
ejpam-6492	78	27	,	,	PUNCT
ejpam-6492	78	28	u1	u1	NOUN
ejpam-6492	78	29	=	=	SYM
ejpam-6492	78	30	1	1	NUM
ejpam-6492	78	31	is	be	AUX
ejpam-6492	78	32	called	call	VERB
ejpam-6492	78	33	the	the	DET
ejpam-6492	78	34	generalized	generalized	ADJ
ejpam-6492	78	35	fibonacci	fibonacci	NOUN
ejpam-6492	78	36	sequence	sequence	NOUN
ejpam-6492	78	37	,	,	PUNCT
ejpam-6492	78	38	and	and	CCONJ
ejpam-6492	78	39	the	the	DET
ejpam-6492	78	40	number	number	NOUN
ejpam-6492	78	41	un	un	PROPN
ejpam-6492	78	42	is	be	AUX
ejpam-6492	78	43	called	call	VERB
ejpam-6492	79	1	the	the	DET
ejpam-6492	79	2	n	n	NOUN
ejpam-6492	79	3	−	−	ADV
ejpam-6492	79	4	th	th	X
ejpam-6492	79	5	generalized	generalize	VERB
ejpam-6492	79	6	fibonacci	fibonacci	NOUN
ejpam-6492	79	7	number	number	NOUN
ejpam-6492	79	8	.	.	PUNCT
ejpam-6492	80	1	definition	definition	NOUN
ejpam-6492	80	2	3	3	NUM
ejpam-6492	80	3	.	.	PUNCT
ejpam-6492	81	1	(	(	PUNCT
ejpam-6492	81	2	[	[	X
ejpam-6492	81	3	16	16	NUM
ejpam-6492	81	4	,	,	PUNCT
ejpam-6492	81	5	17	17	NUM
ejpam-6492	81	6	]	]	PUNCT
ejpam-6492	81	7	)	)	PUNCT
ejpam-6492	81	8	let	let	VERB
ejpam-6492	81	9	p	p	PRON
ejpam-6492	81	10	,	,	PUNCT
ejpam-6492	81	11	q	q	NOUN
ejpam-6492	81	12	∈	∈	PROPN
ejpam-6492	81	13	z.the	z.the	DET
ejpam-6492	81	14	sequence	sequence	NOUN
ejpam-6492	81	15	(	(	PUNCT
ejpam-6492	81	16	vn	vn	NOUN
ejpam-6492	81	17	)	)	PUNCT
ejpam-6492	81	18	defined	define	VERB
ejpam-6492	81	19	by	by	ADP
ejpam-6492	81	20	the	the	DET
ejpam-6492	81	21	relation	relation	NOUN
ejpam-6492	81	22	vn	vn	NOUN
ejpam-6492	81	23	=	=	PUNCT
ejpam-6492	82	1	pvn−1	pvn−1	PROPN
ejpam-6492	82	2	+	+	CCONJ
ejpam-6492	82	3	qvn−2	qvn−2	PROPN
ejpam-6492	82	4	,	,	PUNCT
ejpam-6492	82	5	for	for	ADP
ejpam-6492	82	6	all	all	DET
ejpam-6492	82	7	n	n	PRON
ejpam-6492	82	8	≥	≥	NOUN
ejpam-6492	82	9	2	2	NUM
ejpam-6492	82	10	with	with	ADP
ejpam-6492	82	11	initial	initial	ADJ
ejpam-6492	82	12	conditions	condition	NOUN
ejpam-6492	82	13	v0	v0	NOUN
ejpam-6492	82	14	=	=	SYM
ejpam-6492	82	15	2	2	NUM
ejpam-6492	82	16	,	,	PUNCT
ejpam-6492	82	17	v1	v1	NOUN
ejpam-6492	82	18	=	=	SYM
ejpam-6492	82	19	p	p	NOUN
ejpam-6492	82	20	,	,	PUNCT
ejpam-6492	82	21	is	be	AUX
ejpam-6492	82	22	called	call	VERB
ejpam-6492	82	23	the	the	DET
ejpam-6492	82	24	generalized	generalized	ADJ
ejpam-6492	82	25	lucas	lucas	NOUN
ejpam-6492	82	26	sequence	sequence	NOUN
ejpam-6492	82	27	,	,	PUNCT
ejpam-6492	82	28	and	and	CCONJ
ejpam-6492	82	29	the	the	DET
ejpam-6492	82	30	number	number	NOUN
ejpam-6492	82	31	vn	vn	PROPN
ejpam-6492	82	32	is	be	AUX
ejpam-6492	82	33	called	call	VERB
ejpam-6492	82	34	the	the	PRON
ejpam-6492	82	35	n	n	NOUN
ejpam-6492	82	36	−	−	ADV
ejpam-6492	82	37	th	th	X
ejpam-6492	82	38	generalized	generalize	VERB
ejpam-6492	82	39	lucas	lucas	NOUN
ejpam-6492	82	40	number	number	NOUN
ejpam-6492	82	41	.	.	PUNCT
ejpam-6492	83	1	the	the	DET
ejpam-6492	83	2	characteristic	characteristic	ADJ
ejpam-6492	83	3	equation	equation	NOUN
ejpam-6492	83	4	of	of	ADP
ejpam-6492	83	5	these	these	DET
ejpam-6492	83	6	generalized	generalize	VERB
ejpam-6492	83	7	sequences	sequence	NOUN
ejpam-6492	83	8	are	be	AUX
ejpam-6492	83	9	x2	x2	PRON
ejpam-6492	83	10	−	−	PROPN
ejpam-6492	84	1	px	px	INTJ
ejpam-6492	84	2	−	−	PROPN
ejpam-6492	84	3	q	q	NOUN
ejpam-6492	85	1	=	=	NOUN
ejpam-6492	85	2	0	0	X
ejpam-6492	85	3	.	.	PUNCT
ejpam-6492	86	1	let	let	VERB
ejpam-6492	86	2	∆	∆	PROPN
ejpam-6492	86	3	=	=	PUNCT
ejpam-6492	86	4	p2	p2	PROPN
ejpam-6492	86	5	+	+	X
ejpam-6492	86	6	4q	4q	NOUN
ejpam-6492	86	7	>	>	PUNCT
ejpam-6492	86	8	0	0	NUM
ejpam-6492	86	9	;	;	PUNCT
ejpam-6492	86	10	the	the	DET
ejpam-6492	86	11	roots	root	NOUN
ejpam-6492	86	12	of	of	ADP
ejpam-6492	86	13	this	this	DET
ejpam-6492	86	14	characteristic	characteristic	ADJ
ejpam-6492	86	15	equation	equation	NOUN
ejpam-6492	86	16	are	be	AUX
ejpam-6492	86	17	given	give	VERB
ejpam-6492	86	18	by	by	ADP
ejpam-6492	86	19	α	α	X
ejpam-6492	86	20	=	=	SYM
ejpam-6492	86	21	p	p	X
ejpam-6492	86	22	+	+	CCONJ
ejpam-6492	86	23	√	√	PROPN
ejpam-6492	86	24	∆	∆	PROPN
ejpam-6492	86	25	2	2	NUM
ejpam-6492	86	26	and	and	CCONJ
ejpam-6492	86	27	β	β	X
ejpam-6492	87	1	=	=	PUNCT
ejpam-6492	87	2	p	p	X
ejpam-6492	87	3	−	−	PROPN
ejpam-6492	87	4	√	√	PROPN
ejpam-6492	87	5	∆	∆	PROPN
ejpam-6492	87	6	2	2	NUM
ejpam-6492	87	7	,	,	PUNCT
ejpam-6492	87	8	which	which	PRON
ejpam-6492	87	9	satisfy	satisfy	VERB
ejpam-6492	87	10	α	α	PROPN
ejpam-6492	87	11	+	+	X
ejpam-6492	87	12	β	β	X
ejpam-6492	87	13	=	=	SYM
ejpam-6492	87	14	p	p	PROPN
ejpam-6492	87	15	and	and	CCONJ
ejpam-6492	87	16	αβ	αβ	NOUN
ejpam-6492	87	17	=	=	NOUN
ejpam-6492	87	18	−q	−q	NOUN
ejpam-6492	87	19	.	.	PUNCT
ejpam-6492	88	1	moreover	moreover	ADV
ejpam-6492	88	2	,	,	PUNCT
ejpam-6492	88	3	α2	α2	PROPN
ejpam-6492	88	4	=	=	PUNCT
ejpam-6492	88	5	pα	pα	NOUN
ejpam-6492	88	6	+	+	CCONJ
ejpam-6492	88	7	q	q	NOUN
ejpam-6492	88	8	and	and	CCONJ
ejpam-6492	88	9	β2	β2	NOUN
ejpam-6492	88	10	=	=	PRON
ejpam-6492	88	11	pβ	pβ	ADV
ejpam-6492	88	12	+	+	CCONJ
ejpam-6492	88	13	q.	q.	VERB
ejpam-6492	88	14	by	by	ADP
ejpam-6492	88	15	induction	induction	NOUN
ejpam-6492	88	16	,	,	PUNCT
ejpam-6492	88	17	it	it	PRON
ejpam-6492	88	18	can	can	AUX
ejpam-6492	88	19	be	be	AUX
ejpam-6492	88	20	shown	show	VERB
ejpam-6492	88	21	that	that	SCONJ
ejpam-6492	88	22	for	for	ADP
ejpam-6492	88	23	every	every	DET
ejpam-6492	88	24	n	n	PRON
ejpam-6492	88	25	∈	∈	PROPN
ejpam-6492	88	26	n	n	CCONJ
ejpam-6492	88	27	,	,	PUNCT
ejpam-6492	88	28	αn	αn	NOUN
ejpam-6492	88	29	=	=	NOUN
ejpam-6492	88	30	αun	αun	NOUN
ejpam-6492	88	31	+	+	CCONJ
ejpam-6492	88	32	qun−1	qun−1	PROPN
ejpam-6492	88	33	,	,	PUNCT
ejpam-6492	88	34	(	(	PUNCT
ejpam-6492	88	35	2	2	X
ejpam-6492	88	36	)	)	PUNCT
ejpam-6492	88	37	b.	b.	NOUN
ejpam-6492	88	38	demirtürk	demirtürk	PROPN
ejpam-6492	88	39	,	,	PUNCT
ejpam-6492	88	40	n.	n.	NOUN
ejpam-6492	88	41	topal	topal	PROPN
ejpam-6492	88	42	/	/	SYM
ejpam-6492	88	43	eur	eur	PROPN
ejpam-6492	88	44	.	.	PUNCT
ejpam-6492	89	1	j.	j.	PROPN
ejpam-6492	89	2	pure	pure	PROPN
ejpam-6492	89	3	appl	appl	PROPN
ejpam-6492	89	4	.	.	PROPN
ejpam-6492	89	5	math	math	PROPN
ejpam-6492	89	6	,	,	PUNCT
ejpam-6492	89	7	18	18	NUM
ejpam-6492	89	8	(	(	PUNCT
ejpam-6492	89	9	3	3	NUM
ejpam-6492	89	10	)	)	PUNCT
ejpam-6492	89	11	(	(	PUNCT
ejpam-6492	89	12	2025	2025	NUM
ejpam-6492	89	13	)	)	PUNCT
ejpam-6492	89	14	,	,	PUNCT
ejpam-6492	89	15	6492	6492	NUM
ejpam-6492	89	16	5	5	NUM
ejpam-6492	89	17	of	of	ADP
ejpam-6492	89	18	22	22	NUM
ejpam-6492	89	19	βn	βn	NOUN
ejpam-6492	89	20	=	=	SYM
ejpam-6492	89	21	βun	βun	PROPN
ejpam-6492	90	1	+	+	CCONJ
ejpam-6492	90	2	qun−1	qun−1	PROPN
ejpam-6492	90	3	.	.	PUNCT
ejpam-6492	91	1	(	(	PUNCT
ejpam-6492	91	2	3	3	X
ejpam-6492	91	3	)	)	PUNCT
ejpam-6492	91	4	similarly	similarly	ADV
ejpam-6492	91	5	,	,	PUNCT
ejpam-6492	91	6	it	it	PRON
ejpam-6492	91	7	can	can	AUX
ejpam-6492	91	8	be	be	AUX
ejpam-6492	91	9	shown	show	VERB
ejpam-6492	91	10	by	by	ADP
ejpam-6492	91	11	induction	induction	NOUN
ejpam-6492	91	12	that	that	SCONJ
ejpam-6492	91	13	for	for	ADP
ejpam-6492	91	14	every	every	DET
ejpam-6492	91	15	n	n	PRON
ejpam-6492	91	16	∈	∈	PROPN
ejpam-6492	91	17	n	n	CCONJ
ejpam-6492	91	18	,	,	PUNCT
ejpam-6492	91	19	the	the	DET
ejpam-6492	91	20	elements	element	NOUN
ejpam-6492	91	21	of	of	ADP
ejpam-6492	91	22	the	the	DET
ejpam-6492	91	23	generalized	generalized	ADJ
ejpam-6492	91	24	lucas	lucas	NOUN
ejpam-6492	91	25	sequence	sequence	NOUN
ejpam-6492	91	26	satisfy	satisfy	VERB
ejpam-6492	91	27	the	the	DET
ejpam-6492	91	28	identities	identity	NOUN
ejpam-6492	91	29	√	√	VERB
ejpam-6492	91	30	∆	∆	PROPN
ejpam-6492	91	31	αn	αn	NOUN
ejpam-6492	92	1	=	=	SYM
ejpam-6492	92	2	αvn	αvn	NOUN
ejpam-6492	92	3	+	+	CCONJ
ejpam-6492	92	4	qvn−1	qvn−1	PROPN
ejpam-6492	92	5	,	,	PUNCT
ejpam-6492	92	6	(	(	PUNCT
ejpam-6492	92	7	4	4	NUM
ejpam-6492	92	8	)	)	PUNCT
ejpam-6492	92	9	−	−	NOUN
ejpam-6492	92	10	√	√	PROPN
ejpam-6492	92	11	∆	∆	PROPN
ejpam-6492	92	12	βn	βn	NOUN
ejpam-6492	93	1	=	=	PUNCT
ejpam-6492	93	2	βvn	βvn	NOUN
ejpam-6492	94	1	+	+	CCONJ
ejpam-6492	94	2	qvn−1	qvn−1	PROPN
ejpam-6492	94	3	.	.	PROPN
ejpam-6492	95	1	(	(	PUNCT
ejpam-6492	95	2	5	5	NUM
ejpam-6492	95	3	)	)	PUNCT
ejpam-6492	95	4	moreover	moreover	ADV
ejpam-6492	95	5	,	,	PUNCT
ejpam-6492	95	6	the	the	DET
ejpam-6492	95	7	binet	binet	NOUN
ejpam-6492	95	8	formulas	formula	NOUN
ejpam-6492	95	9	for	for	ADP
ejpam-6492	95	10	generalized	generalized	ADJ
ejpam-6492	95	11	fibonacci	fibonacci	NOUN
ejpam-6492	95	12	and	and	CCONJ
ejpam-6492	95	13	lucas	lucas	PROPN
ejpam-6492	95	14	numbers	number	NOUN
ejpam-6492	95	15	are	be	AUX
ejpam-6492	95	16	given	give	VERB
ejpam-6492	95	17	by	by	ADP
ejpam-6492	95	18	un	un	PROPN
ejpam-6492	95	19	=	=	PROPN
ejpam-6492	95	20	αn	αn	NOUN
ejpam-6492	96	1	−	−	PROPN
ejpam-6492	96	2	βn	βn	NOUN
ejpam-6492	97	1	α	α	NOUN
ejpam-6492	97	2	−	−	NOUN
ejpam-6492	97	3	β	β	X
ejpam-6492	97	4	,	,	PUNCT
ejpam-6492	97	5	vn	vn	PROPN
ejpam-6492	97	6	=	=	SYM
ejpam-6492	97	7	αn	αn	NOUN
ejpam-6492	98	1	+	+	CCONJ
ejpam-6492	98	2	βn	βn	ADJ
ejpam-6492	98	3	(	(	PUNCT
ejpam-6492	98	4	6	6	NUM
ejpam-6492	98	5	)	)	PUNCT
ejpam-6492	98	6	for	for	ADP
ejpam-6492	98	7	all	all	DET
ejpam-6492	98	8	n	n	PRON
ejpam-6492	98	9	∈	∈	NOUN
ejpam-6492	98	10	z	z	NOUN
ejpam-6492	99	1	[	[	X
ejpam-6492	99	2	20	20	NUM
ejpam-6492	99	3	,	,	PUNCT
ejpam-6492	99	4	21	21	NUM
ejpam-6492	99	5	]	]	PUNCT
ejpam-6492	99	6	.	.	PUNCT
ejpam-6492	100	1	since	since	SCONJ
ejpam-6492	100	2	each	each	DET
ejpam-6492	100	3	term	term	NOUN
ejpam-6492	100	4	is	be	AUX
ejpam-6492	100	5	a	a	DET
ejpam-6492	100	6	linear	linear	ADJ
ejpam-6492	100	7	combination	combination	NOUN
ejpam-6492	100	8	of	of	ADP
ejpam-6492	100	9	the	the	DET
ejpam-6492	100	10	two	two	NUM
ejpam-6492	100	11	preceding	precede	VERB
ejpam-6492	100	12	terms	term	NOUN
ejpam-6492	100	13	,	,	PUNCT
ejpam-6492	100	14	we	we	PRON
ejpam-6492	100	15	have	have	VERB
ejpam-6492	100	16	u−1	u−1	PROPN
ejpam-6492	100	17	=	=	SYM
ejpam-6492	100	18	1	1	NUM
ejpam-6492	100	19	q	q	NOUN
ejpam-6492	100	20	,	,	PUNCT
ejpam-6492	100	21	u−2	u−2	NOUN
ejpam-6492	100	22	=	=	NOUN
ejpam-6492	101	1	−	−	PROPN
ejpam-6492	101	2	p	p	PROPN
ejpam-6492	101	3	q2	q2	NOUN
ejpam-6492	101	4	,	,	PUNCT
ejpam-6492	101	5	u−3	u−3	PROPN
ejpam-6492	101	6	=	=	PUNCT
ejpam-6492	101	7	p2	p2	PROPN
ejpam-6492	101	8	+	+	CCONJ
ejpam-6492	101	9	q	q	PROPN
ejpam-6492	101	10	q3	q3	NOUN
ejpam-6492	101	11	,	,	PUNCT
ejpam-6492	101	12	.	.	PUNCT
ejpam-6492	101	13	.	.	PUNCT
ejpam-6492	101	14	.	.	PUNCT
ejpam-6492	102	1	and	and	CCONJ
ejpam-6492	102	2	v−1	v−1	PROPN
ejpam-6492	102	3	=	=	PUNCT
ejpam-6492	102	4	−p	−p	ADJ
ejpam-6492	102	5	q	q	NOUN
ejpam-6492	102	6	,	,	PUNCT
ejpam-6492	102	7	v−2	v−2	NOUN
ejpam-6492	102	8	=	=	PUNCT
ejpam-6492	102	9	p2	p2	PROPN
ejpam-6492	102	10	+	+	CCONJ
ejpam-6492	102	11	2q	2q	NUM
ejpam-6492	102	12	q2	q2	NOUN
ejpam-6492	102	13	,	,	PUNCT
ejpam-6492	102	14	v−3	v−3	PROPN
ejpam-6492	102	15	=	=	SYM
ejpam-6492	102	16	−p3	−p3	PROPN
ejpam-6492	102	17	+	+	CCONJ
ejpam-6492	102	18	3pq	3pq	ADJ
ejpam-6492	102	19	q3	q3	NOUN
ejpam-6492	102	20	,	,	PUNCT
ejpam-6492	102	21	.	.	PUNCT
ejpam-6492	102	22	.	.	PUNCT
ejpam-6492	102	23	.	.	PUNCT
ejpam-6492	103	1	from	from	ADP
ejpam-6492	103	2	these	these	DET
ejpam-6492	103	3	results	result	NOUN
ejpam-6492	103	4	,	,	PUNCT
ejpam-6492	103	5	it	it	PRON
ejpam-6492	103	6	was	be	AUX
ejpam-6492	103	7	shown	show	VERB
ejpam-6492	103	8	in	in	ADP
ejpam-6492	103	9	[	[	X
ejpam-6492	103	10	22	22	NUM
ejpam-6492	103	11	]	]	PUNCT
ejpam-6492	103	12	,	,	PUNCT
ejpam-6492	103	13	using	use	VERB
ejpam-6492	103	14	the	the	DET
ejpam-6492	103	15	binet	binet	NOUN
ejpam-6492	103	16	formulas	formula	NOUN
ejpam-6492	103	17	,	,	PUNCT
ejpam-6492	103	18	that	that	SCONJ
ejpam-6492	103	19	the	the	DET
ejpam-6492	103	20	generalized	generalized	ADJ
ejpam-6492	103	21	fibonacci	fibonacci	NOUN
ejpam-6492	103	22	and	and	CCONJ
ejpam-6492	103	23	lucas	lucas	PROPN
ejpam-6492	103	24	numbers	number	NOUN
ejpam-6492	103	25	with	with	ADP
ejpam-6492	103	26	negative	negative	ADJ
ejpam-6492	103	27	indices	index	NOUN
ejpam-6492	103	28	satisfy	satisfy	VERB
ejpam-6492	103	29	u−n	u−n	ADV
ejpam-6492	103	30	=	=	SYM
ejpam-6492	103	31	−(−q)nun	−(−q)nun	PROPN
ejpam-6492	103	32	,	,	PUNCT
ejpam-6492	103	33	v−n	v−n	NOUN
ejpam-6492	103	34	=	=	SYM
ejpam-6492	103	35	(	(	PUNCT
ejpam-6492	103	36	−q)nvn	−q)nvn	X
ejpam-6492	103	37	.	.	PUNCT
ejpam-6492	104	1	many	many	ADJ
ejpam-6492	104	2	authors	author	NOUN
ejpam-6492	104	3	have	have	AUX
ejpam-6492	104	4	used	use	VERB
ejpam-6492	104	5	the	the	DET
ejpam-6492	104	6	properties	property	NOUN
ejpam-6492	104	7	of	of	ADP
ejpam-6492	104	8	these	these	DET
ejpam-6492	104	9	sequences	sequence	NOUN
ejpam-6492	104	10	and	and	CCONJ
ejpam-6492	104	11	have	have	AUX
ejpam-6492	104	12	proved	prove	VERB
ejpam-6492	104	13	numerious	numerious	ADJ
ejpam-6492	104	14	identities	identity	NOUN
ejpam-6492	104	15	,	,	PUNCT
ejpam-6492	104	16	sum	sum	NOUN
ejpam-6492	104	17	formulas	formula	NOUN
ejpam-6492	104	18	,	,	PUNCT
ejpam-6492	104	19	and	and	CCONJ
ejpam-6492	104	20	matrix	matrix	NOUN
ejpam-6492	104	21	structures	structure	NOUN
ejpam-6492	104	22	,	,	PUNCT
ejpam-6492	104	23	while	while	SCONJ
ejpam-6492	104	24	many	many	ADJ
ejpam-6492	104	25	others	other	NOUN
ejpam-6492	104	26	have	have	AUX
ejpam-6492	104	27	investigated	investigate	VERB
ejpam-6492	104	28	the	the	DET
ejpam-6492	104	29	product	product	NOUN
ejpam-6492	104	30	differences	difference	NOUN
ejpam-6492	104	31	of	of	ADP
ejpam-6492	104	32	fibonacci	fibonacci	NOUN
ejpam-6492	104	33	and	and	CCONJ
ejpam-6492	104	34	lucas	lucas	PROPN
ejpam-6492	104	35	numbers	number	NOUN
ejpam-6492	104	36	.	.	PUNCT
ejpam-6492	105	1	some	some	PRON
ejpam-6492	105	2	of	of	ADP
ejpam-6492	105	3	these	these	DET
ejpam-6492	105	4	classical	classical	ADJ
ejpam-6492	105	5	fibonacci	fibonacci	NOUN
ejpam-6492	105	6	identities	identity	NOUN
ejpam-6492	105	7	are	be	AUX
ejpam-6492	105	8	such	such	ADJ
ejpam-6492	105	9	as	as	ADP
ejpam-6492	105	10	cassini	cassini	NOUN
ejpam-6492	105	11	’s	’s	PART
ejpam-6492	105	12	identity	identity	NOUN
ejpam-6492	105	13	f	f	PROPN
ejpam-6492	105	14	2	2	NUM
ejpam-6492	105	15	n	n	NOUN
ejpam-6492	105	16	−	−	NOUN
ejpam-6492	105	17	fn−1	fn−1	ADJ
ejpam-6492	105	18	fn+1	fn+1	NOUN
ejpam-6492	105	19	=	=	SYM
ejpam-6492	105	20	(	(	PUNCT
ejpam-6492	105	21	−1)n−1	−1)n−1	PROPN
ejpam-6492	105	22	,	,	PUNCT
ejpam-6492	105	23	which	which	PRON
ejpam-6492	105	24	was	be	AUX
ejpam-6492	105	25	proved	prove	VERB
ejpam-6492	105	26	in	in	ADP
ejpam-6492	105	27	1680	1680	NUM
ejpam-6492	105	28	and	and	CCONJ
ejpam-6492	105	29	catalan	catalan	NOUN
ejpam-6492	105	30	’s	’s	PART
ejpam-6492	105	31	identity	identity	NOUN
ejpam-6492	105	32	f	f	PROPN
ejpam-6492	105	33	2	2	NUM
ejpam-6492	105	34	n	n	NUM
ejpam-6492	105	35	−	−	PROPN
ejpam-6492	105	36	fn−m	fn−m	NOUN
ejpam-6492	105	37	fn+m	fn+m	PROPN
ejpam-6492	105	38	=	=	SYM
ejpam-6492	105	39	(	(	PUNCT
ejpam-6492	105	40	−1	−1	NOUN
ejpam-6492	105	41	)	)	PUNCT
ejpam-6492	105	42	n−m	n−m	VERB
ejpam-6492	105	43	f	f	PROPN
ejpam-6492	105	44	2	2	NUM
ejpam-6492	105	45	m	m	PROPN
ejpam-6492	105	46	,	,	PUNCT
ejpam-6492	105	47	which	which	PRON
ejpam-6492	105	48	was	be	AUX
ejpam-6492	105	49	proved	prove	VERB
ejpam-6492	105	50	1879	1879	NUM
ejpam-6492	105	51	,	,	PUNCT
ejpam-6492	105	52	and	and	CCONJ
ejpam-6492	105	53	the	the	DET
ejpam-6492	105	54	d’ocagne	d’ocagne	PROPN
ejpam-6492	105	55	’s	’s	PART
ejpam-6492	105	56	identity	identity	NOUN
ejpam-6492	105	57	fm+n	fm+n	PROPN
ejpam-6492	105	58	=	=	SYM
ejpam-6492	105	59	fm−1	fm−1	PROPN
ejpam-6492	105	60	fn	fn	NOUN
ejpam-6492	106	1	+	+	CCONJ
ejpam-6492	106	2	fm	fm	PROPN
ejpam-6492	106	3	fn+1	fn+1	PROPN
ejpam-6492	106	4	,	,	PUNCT
ejpam-6492	106	5	was	be	AUX
ejpam-6492	106	6	proved	prove	VERB
ejpam-6492	106	7	in	in	ADP
ejpam-6492	106	8	1882	1882	NUM
ejpam-6492	106	9	.	.	PUNCT
ejpam-6492	107	1	since	since	SCONJ
ejpam-6492	107	2	we	we	PRON
ejpam-6492	107	3	will	will	AUX
ejpam-6492	107	4	focus	focus	VERB
ejpam-6492	107	5	on	on	ADP
ejpam-6492	107	6	the	the	DET
ejpam-6492	107	7	product	product	NOUN
ejpam-6492	107	8	differences	difference	NOUN
ejpam-6492	107	9	of	of	ADP
ejpam-6492	107	10	generalized	generalized	ADJ
ejpam-6492	107	11	everman	everman	NOUN
ejpam-6492	107	12	and	and	CCONJ
ejpam-6492	107	13	koshy	koshy	ADJ
ejpam-6492	107	14	based	base	VERB
ejpam-6492	107	15	identities	identity	NOUN
ejpam-6492	107	16	,	,	PUNCT
ejpam-6492	107	17	let	let	VERB
ejpam-6492	107	18	us	we	PRON
ejpam-6492	107	19	first	first	ADV
ejpam-6492	107	20	recall	recall	VERB
ejpam-6492	107	21	the	the	DET
ejpam-6492	107	22	term	term	NOUN
ejpam-6492	107	23	“	"	PUNCT
ejpam-6492	107	24	product	product	NOUN
ejpam-6492	107	25	difference	difference	NOUN
ejpam-6492	107	26	fibonacci	fibonacci	NOUN
ejpam-6492	107	27	identity	identity	NOUN
ejpam-6492	107	28	”	"	PUNCT
ejpam-6492	107	29	with	with	ADP
ejpam-6492	107	30	the	the	DET
ejpam-6492	107	31	following	follow	VERB
ejpam-6492	107	32	definition	definition	NOUN
ejpam-6492	107	33	.	.	PUNCT
ejpam-6492	108	1	b.	b.	PROPN
ejpam-6492	108	2	demirtürk	demirtürk	PROPN
ejpam-6492	108	3	,	,	PUNCT
ejpam-6492	108	4	n.	n.	NOUN
ejpam-6492	108	5	topal	topal	PROPN
ejpam-6492	108	6	/	/	SYM
ejpam-6492	108	7	eur	eur	PROPN
ejpam-6492	108	8	.	.	PUNCT
ejpam-6492	109	1	j.	j.	PROPN
ejpam-6492	109	2	pure	pure	PROPN
ejpam-6492	109	3	appl	appl	PROPN
ejpam-6492	109	4	.	.	PROPN
ejpam-6492	109	5	math	math	PROPN
ejpam-6492	109	6	,	,	PUNCT
ejpam-6492	109	7	18	18	NUM
ejpam-6492	109	8	(	(	PUNCT
ejpam-6492	109	9	3	3	NUM
ejpam-6492	109	10	)	)	PUNCT
ejpam-6492	109	11	(	(	PUNCT
ejpam-6492	109	12	2025	2025	NUM
ejpam-6492	109	13	)	)	PUNCT
ejpam-6492	109	14	,	,	PUNCT
ejpam-6492	109	15	6492	6492	NUM
ejpam-6492	109	16	6	6	NUM
ejpam-6492	109	17	of	of	ADP
ejpam-6492	109	18	22	22	NUM
ejpam-6492	109	19	definition	definition	NOUN
ejpam-6492	109	20	4	4	NUM
ejpam-6492	109	21	.	.	PUNCT
ejpam-6492	110	1	(	(	PUNCT
ejpam-6492	110	2	product	product	NOUN
ejpam-6492	110	3	difference	difference	NOUN
ejpam-6492	110	4	fibonacci	fibonacci	NOUN
ejpam-6492	110	5	identity	identity	NOUN
ejpam-6492	110	6	)	)	PUNCT
ejpam-6492	110	7	let	let	VERB
ejpam-6492	110	8	s	s	PRON
ejpam-6492	110	9	≥	≥	NOUN
ejpam-6492	110	10	1	1	NUM
ejpam-6492	110	11	,	,	PUNCT
ejpam-6492	110	12	and	and	CCONJ
ejpam-6492	110	13	the	the	DET
ejpam-6492	110	14	ai	ai	NOUN
ejpam-6492	110	15	and	and	CCONJ
ejpam-6492	110	16	bi	bi	NOUN
ejpam-6492	110	17	be	be	AUX
ejpam-6492	110	18	specified	specify	VERB
ejpam-6492	110	19	integers	integer	NOUN
ejpam-6492	110	20	and	and	CCONJ
ejpam-6492	110	21	dn	dn	PROPN
ejpam-6492	110	22	be	be	AUX
ejpam-6492	110	23	of	of	ADP
ejpam-6492	110	24	some	some	DET
ejpam-6492	110	25	interesting	interesting	ADJ
ejpam-6492	110	26	form	form	NOUN
ejpam-6492	110	27	for	for	ADP
ejpam-6492	110	28	all	all	DET
ejpam-6492	110	29	integers	integer	NOUN
ejpam-6492	110	30	n.	n.	VERB
ejpam-6492	110	31	then	then	ADV
ejpam-6492	110	32	the	the	DET
ejpam-6492	110	33	product	product	NOUN
ejpam-6492	110	34	of	of	ADP
ejpam-6492	110	35	the	the	DET
ejpam-6492	110	36	form	form	NOUN
ejpam-6492	110	37	s∏	s∏	PROPN
ejpam-6492	110	38	i=1	i=1	PRON
ejpam-6492	111	1	fn+ai	fn+ai	NOUN
ejpam-6492	112	1	−	−	PROPN
ejpam-6492	112	2	s∏	s∏	PROPN
ejpam-6492	112	3	i=1	i=1	X
ejpam-6492	113	1	fn+bi	fn+bi	PROPN
ejpam-6492	113	2	=	=	SYM
ejpam-6492	113	3	dn(ai	dn(ai	PROPN
ejpam-6492	113	4	,	,	PUNCT
ejpam-6492	113	5	bi	bi	NOUN
ejpam-6492	113	6	;	;	PUNCT
ejpam-6492	113	7	s	s	X
ejpam-6492	113	8	)	)	PUNCT
ejpam-6492	113	9	=	=	SYM
ejpam-6492	114	1	dn	dn	PROPN
ejpam-6492	114	2	.	.	PROPN
ejpam-6492	114	3	(	(	PUNCT
ejpam-6492	114	4	7	7	X
ejpam-6492	114	5	)	)	PUNCT
ejpam-6492	114	6	is	be	AUX
ejpam-6492	114	7	introduced	introduce	VERB
ejpam-6492	114	8	by	by	ADP
ejpam-6492	114	9	fairgrieve	fairgrieve	NOUN
ejpam-6492	114	10	and	and	CCONJ
ejpam-6492	114	11	gould	gould	X
ejpam-6492	114	12	in	in	ADP
ejpam-6492	114	13	[	[	X
ejpam-6492	114	14	23	23	NUM
ejpam-6492	114	15	]	]	PUNCT
ejpam-6492	114	16	.	.	PUNCT
ejpam-6492	115	1	it	it	PRON
ejpam-6492	115	2	can	can	AUX
ejpam-6492	115	3	be	be	AUX
ejpam-6492	115	4	seen	see	VERB
ejpam-6492	115	5	that	that	SCONJ
ejpam-6492	115	6	cassini	cassini	PROPN
ejpam-6492	115	7	’s	’s	PART
ejpam-6492	115	8	and	and	CCONJ
ejpam-6492	115	9	catalan	catalan	NOUN
ejpam-6492	115	10	’s	’s	PART
ejpam-6492	115	11	identities	identity	NOUN
ejpam-6492	115	12	are	be	AUX
ejpam-6492	115	13	some	some	DET
ejpam-6492	115	14	versions	version	NOUN
ejpam-6492	115	15	of	of	ADP
ejpam-6492	115	16	the	the	DET
ejpam-6492	115	17	equation	equation	NOUN
ejpam-6492	115	18	(	(	PUNCT
ejpam-6492	115	19	7	7	NUM
ejpam-6492	115	20	)	)	PUNCT
ejpam-6492	115	21	.	.	PUNCT
ejpam-6492	116	1	there	there	PRON
ejpam-6492	116	2	are	be	VERB
ejpam-6492	116	3	many	many	ADJ
ejpam-6492	116	4	identities	identity	NOUN
ejpam-6492	116	5	related	relate	VERB
ejpam-6492	116	6	to	to	ADP
ejpam-6492	116	7	(	(	PUNCT
ejpam-6492	116	8	7	7	NUM
ejpam-6492	116	9	)	)	PUNCT
ejpam-6492	116	10	.	.	PUNCT
ejpam-6492	117	1	for	for	ADP
ejpam-6492	117	2	example	example	NOUN
ejpam-6492	117	3	,	,	PUNCT
ejpam-6492	117	4	morgado	morgado	PROPN
ejpam-6492	117	5	used	use	VERB
ejpam-6492	117	6	the	the	DET
ejpam-6492	117	7	catalan	catalan	NOUN
ejpam-6492	117	8	identity	identity	NOUN
ejpam-6492	117	9	to	to	PART
ejpam-6492	117	10	prove	prove	VERB
ejpam-6492	117	11	the	the	DET
ejpam-6492	117	12	following	follow	VERB
ejpam-6492	117	13	equation	equation	NOUN
ejpam-6492	117	14	fn−2	fn−2	PROPN
ejpam-6492	117	15	fn−1	fn−1	PROPN
ejpam-6492	117	16	fn+1	fn+1	NOUN
ejpam-6492	117	17	fn+2	fn+2	NUM
ejpam-6492	117	18	−	−	PROPN
ejpam-6492	117	19	f	f	NOUN
ejpam-6492	117	20	4	4	NUM
ejpam-6492	117	21	n	n	NOUN
ejpam-6492	117	22	=	=	PUNCT
ejpam-6492	117	23	−1	−1	NOUN
ejpam-6492	117	24	.	.	PUNCT
ejpam-6492	118	1	in	in	ADP
ejpam-6492	118	2	[	[	X
ejpam-6492	118	3	24	24	NUM
ejpam-6492	118	4	]	]	PUNCT
ejpam-6492	118	5	.	.	PUNCT
ejpam-6492	119	1	recently	recently	ADV
ejpam-6492	119	2	,	,	PUNCT
ejpam-6492	119	3	in	in	ADP
ejpam-6492	119	4	[	[	X
ejpam-6492	119	5	25	25	NUM
ejpam-6492	119	6	]	]	PUNCT
ejpam-6492	119	7	,	,	PUNCT
ejpam-6492	119	8	melham	melham	PROPN
ejpam-6492	119	9	discovered	discover	VERB
ejpam-6492	119	10	the	the	DET
ejpam-6492	119	11	following	follow	VERB
ejpam-6492	119	12	formula	formula	NOUN
ejpam-6492	119	13	fn+1fn+2fn+6	fn+1fn+2fn+6	NOUN
ejpam-6492	119	14	−	−	NOUN
ejpam-6492	119	15	f	f	NOUN
ejpam-6492	119	16	3	3	NUM
ejpam-6492	119	17	n+3	n+3	PROPN
ejpam-6492	119	18	=	=	SYM
ejpam-6492	119	19	(	(	PUNCT
ejpam-6492	119	20	−1)nfn	−1)nfn	PROPN
ejpam-6492	119	21	.	.	PUNCT
ejpam-6492	120	1	more	more	ADV
ejpam-6492	120	2	generally	generally	ADV
ejpam-6492	120	3	the	the	DET
ejpam-6492	120	4	following	follow	VERB
ejpam-6492	120	5	equation	equation	NOUN
ejpam-6492	120	6	fn+a	fn+a	PROPN
ejpam-6492	120	7	fn+b	fn+b	PROPN
ejpam-6492	121	1	−	−	PROPN
ejpam-6492	121	2	fn	fn	NOUN
ejpam-6492	121	3	fn+a+b	fn+a+b	NOUN
ejpam-6492	121	4	=	=	SYM
ejpam-6492	121	5	(	(	PUNCT
ejpam-6492	121	6	−1)n	−1)n	X
ejpam-6492	121	7	fa	fa	INTJ
ejpam-6492	121	8	fb	fb	INTJ
ejpam-6492	121	9	.	.	PUNCT
ejpam-6492	122	1	(	(	PUNCT
ejpam-6492	122	2	8)	8)	NUM
ejpam-6492	122	3	was	be	AUX
ejpam-6492	122	4	stated	state	VERB
ejpam-6492	122	5	by	by	ADP
ejpam-6492	122	6	everman	everman	NOUN
ejpam-6492	122	7	at	at	ADP
ejpam-6492	122	8	al	al	PROPN
ejpam-6492	122	9	.	.	PROPN
ejpam-6492	123	1	as	as	ADP
ejpam-6492	123	2	a	a	DET
ejpam-6492	123	3	problem	problem	NOUN
ejpam-6492	123	4	in	in	ADP
ejpam-6492	123	5	the	the	DET
ejpam-6492	123	6	american	american	PROPN
ejpam-6492	123	7	mathematical	mathematical	PROPN
ejpam-6492	123	8	monthly	monthly	ADJ
ejpam-6492	123	9	[	[	X
ejpam-6492	123	10	26	26	NUM
ejpam-6492	123	11	]	]	PUNCT
ejpam-6492	124	1	and	and	CCONJ
ejpam-6492	124	2	appears	appear	VERB
ejpam-6492	124	3	in	in	ADP
ejpam-6492	124	4	vajda	vajda	PROPN
ejpam-6492	125	1	[	[	X
ejpam-6492	125	2	21	21	NUM
ejpam-6492	125	3	,	,	PUNCT
ejpam-6492	125	4	p.	p.	NOUN
ejpam-6492	125	5	177	177	NUM
ejpam-6492	125	6	,	,	PUNCT
ejpam-6492	125	7	eq	eq	NOUN
ejpam-6492	125	8	.	.	PROPN
ejpam-6492	126	1	(	(	PUNCT
ejpam-6492	126	2	20a	20a	NOUN
ejpam-6492	126	3	)	)	PUNCT
ejpam-6492	126	4	]	]	PUNCT
ejpam-6492	126	5	.	.	PUNCT
ejpam-6492	127	1	(	(	PUNCT
ejpam-6492	127	2	8)	8)	NUM
ejpam-6492	127	3	can	can	AUX
ejpam-6492	127	4	be	be	AUX
ejpam-6492	127	5	called	call	VERB
ejpam-6492	127	6	the	the	DET
ejpam-6492	127	7	extended	extended	ADJ
ejpam-6492	127	8	version	version	NOUN
ejpam-6492	127	9	of	of	ADP
ejpam-6492	127	10	these	these	DET
ejpam-6492	127	11	classical	classical	ADJ
ejpam-6492	127	12	fibonacci	fibonacci	NOUN
ejpam-6492	127	13	identities	identity	NOUN
ejpam-6492	127	14	.	.	PUNCT
ejpam-6492	128	1	actually	actually	ADV
ejpam-6492	128	2	,	,	PUNCT
ejpam-6492	128	3	horadam	horadam	PROPN
ejpam-6492	128	4	had	have	AUX
ejpam-6492	128	5	already	already	ADV
ejpam-6492	128	6	expressed	express	VERB
ejpam-6492	128	7	some	some	DET
ejpam-6492	128	8	identities	identity	NOUN
ejpam-6492	128	9	similar	similar	ADJ
ejpam-6492	128	10	to	to	ADP
ejpam-6492	128	11	equality	equality	NOUN
ejpam-6492	128	12	(	(	PUNCT
ejpam-6492	128	13	8)	8)	NUM
ejpam-6492	128	14	with	with	ADP
ejpam-6492	128	15	some	some	DET
ejpam-6492	128	16	generalizations	generalization	NOUN
ejpam-6492	128	17	and	and	CCONJ
ejpam-6492	128	18	he	he	PRON
ejpam-6492	128	19	had	have	AUX
ejpam-6492	128	20	stated	state	VERB
ejpam-6492	128	21	the	the	DET
ejpam-6492	128	22	following	follow	VERB
ejpam-6492	128	23	equation	equation	NOUN
ejpam-6492	128	24	hn	hn	PROPN
ejpam-6492	128	25	hn+r+1	hn+r+1	PROPN
ejpam-6492	128	26	−	−	PROPN
ejpam-6492	129	1	h	h	NOUN
ejpam-6492	129	2	n−s	n−s	NOUN
ejpam-6492	129	3	h	h	NOUN
ejpam-6492	129	4	n+r+s+1	n+r+s+1	NOUN
ejpam-6492	129	5	=	=	PUNCT
ejpam-6492	129	6	(	(	PUNCT
ejpam-6492	129	7	−1	−1	NOUN
ejpam-6492	129	8	)	)	PUNCT
ejpam-6492	129	9	n+s	n+s	PUNCT
ejpam-6492	130	1	[	[	X
ejpam-6492	130	2	p2	p2	X
ejpam-6492	130	3	−	−	NOUN
ejpam-6492	130	4	p	p	X
ejpam-6492	130	5	q	q	X
ejpam-6492	130	6	−	−	PROPN
ejpam-6492	130	7	q2	q2	NOUN
ejpam-6492	130	8	]	]	X
ejpam-6492	130	9	f	f	PROPN
ejpam-6492	130	10	r+s+1	r+s+1	PROPN
ejpam-6492	130	11	(	(	PUNCT
ejpam-6492	130	12	9	9	NUM
ejpam-6492	130	13	)	)	PUNCT
ejpam-6492	130	14	in	in	ADP
ejpam-6492	130	15	[	[	X
ejpam-6492	130	16	16	16	NUM
ejpam-6492	130	17	]	]	PUNCT
ejpam-6492	130	18	.	.	PUNCT
ejpam-6492	131	1	then	then	ADV
ejpam-6492	131	2	in	in	ADP
ejpam-6492	131	3	[	[	X
ejpam-6492	131	4	25	25	NUM
ejpam-6492	131	5	]	]	PUNCT
ejpam-6492	131	6	melham	melham	NOUN
ejpam-6492	131	7	proved	prove	VERB
ejpam-6492	131	8	the	the	DET
ejpam-6492	131	9	identity	identity	NOUN
ejpam-6492	131	10	fn+a+b−c	fn+a+b−c	PROPN
ejpam-6492	131	11	fn−a+c	fn−a+c	ADP
ejpam-6492	131	12	fn−b+c	fn−b+c	ADJ
ejpam-6492	131	13	−	−	PROPN
ejpam-6492	131	14	fn−a−b+c	fn−a−b+c	ADJ
ejpam-6492	131	15	fn+a	fn+a	PROPN
ejpam-6492	131	16	fn+b	fn+b	PROPN
ejpam-6492	131	17	=	=	SYM
ejpam-6492	131	18	(	(	PUNCT
ejpam-6492	131	19	−1	−1	NOUN
ejpam-6492	131	20	)	)	PUNCT
ejpam-6492	132	1	n+a+b+c	n+a+b+c	PROPN
ejpam-6492	132	2	fa+b−c	fa+b−c	X
ejpam-6492	132	3	(	(	PUNCT
ejpam-6492	132	4	fc	fc	INTJ
ejpam-6492	132	5	fn+a+b−c+(−1)c	fn+a+b−c+(−1)c	NOUN
ejpam-6492	132	6	fa−c	fa−c	ADJ
ejpam-6492	132	7	fb−c	fb−c	ADJ
ejpam-6492	132	8	ln	ln	NOUN
ejpam-6492	132	9	)	)	PUNCT
ejpam-6492	132	10	.	.	PUNCT
ejpam-6492	133	1	since	since	SCONJ
ejpam-6492	133	2	cassini	cassini	PROPN
ejpam-6492	133	3	’s	’s	PART
ejpam-6492	133	4	,	,	PUNCT
ejpam-6492	133	5	catalan	catalan	NOUN
ejpam-6492	133	6	’s	’s	PART
ejpam-6492	133	7	,	,	PUNCT
ejpam-6492	133	8	d’ocagne	d’ocagne	PROPN
ejpam-6492	133	9	’s	’s	PART
ejpam-6492	133	10	identities	identity	NOUN
ejpam-6492	133	11	,	,	PUNCT
ejpam-6492	133	12	and	and	CCONJ
ejpam-6492	133	13	all	all	PRON
ejpam-6492	133	14	of	of	ADP
ejpam-6492	133	15	the	the	DET
ejpam-6492	133	16	identities	identity	NOUN
ejpam-6492	133	17	given	give	VERB
ejpam-6492	133	18	in	in	ADP
ejpam-6492	133	19	equation	equation	NOUN
ejpam-6492	133	20	(	(	PUNCT
ejpam-6492	133	21	7	7	NUM
ejpam-6492	133	22	)	)	PUNCT
ejpam-6492	133	23	,	,	PUNCT
ejpam-6492	133	24	(	(	PUNCT
ejpam-6492	133	25	8)	8)	NUM
ejpam-6492	133	26	,	,	PUNCT
ejpam-6492	133	27	(	(	PUNCT
ejpam-6492	133	28	9	9	NUM
ejpam-6492	133	29	)	)	PUNCT
ejpam-6492	133	30	and	and	CCONJ
ejpam-6492	133	31	their	their	PRON
ejpam-6492	133	32	variations	variation	NOUN
ejpam-6492	133	33	are	be	AUX
ejpam-6492	133	34	also	also	ADV
ejpam-6492	133	35	given	give	VERB
ejpam-6492	133	36	in	in	ADP
ejpam-6492	133	37	koshy	koshy	PROPN
ejpam-6492	133	38	’s	’s	PART
ejpam-6492	133	39	book	book	NOUN
ejpam-6492	134	1	[	[	X
ejpam-6492	134	2	19	19	NUM
ejpam-6492	134	3	]	]	PUNCT
ejpam-6492	134	4	,	,	PUNCT
ejpam-6492	134	5	we	we	PRON
ejpam-6492	134	6	refer	refer	VERB
ejpam-6492	134	7	to	to	ADP
ejpam-6492	134	8	the	the	DET
ejpam-6492	134	9	new	new	ADJ
ejpam-6492	134	10	identities	identity	NOUN
ejpam-6492	134	11	that	that	PRON
ejpam-6492	134	12	we	we	PRON
ejpam-6492	134	13	prove	prove	VERB
ejpam-6492	134	14	in	in	ADP
ejpam-6492	134	15	this	this	DET
ejpam-6492	134	16	article	article	NOUN
ejpam-6492	134	17	as	as	ADP
ejpam-6492	134	18	“	"	PUNCT
ejpam-6492	134	19	quaternionic	quaternionic	ADJ
ejpam-6492	134	20	generalizations	generalization	NOUN
ejpam-6492	134	21	of	of	ADP
ejpam-6492	134	22	everman	everman	NOUN
ejpam-6492	134	23	and	and	CCONJ
ejpam-6492	134	24	koshy	koshy	NOUN
ejpam-6492	134	25	”	"	PUNCT
ejpam-6492	134	26	.	.	PUNCT
ejpam-6492	135	1	using	use	VERB
ejpam-6492	135	2	the	the	DET
ejpam-6492	135	3	binet	binet	NOUN
ejpam-6492	135	4	formulas	formula	NOUN
ejpam-6492	135	5	given	give	VERB
ejpam-6492	135	6	in	in	ADP
ejpam-6492	135	7	(	(	PUNCT
ejpam-6492	135	8	6	6	NUM
ejpam-6492	135	9	)	)	PUNCT
ejpam-6492	135	10	,	,	PUNCT
ejpam-6492	135	11	cassini	cassini	PROPN
ejpam-6492	135	12	’s	’s	PART
ejpam-6492	135	13	and	and	CCONJ
ejpam-6492	135	14	catalan	catalan	PROPN
ejpam-6492	135	15	’s	’s	PART
ejpam-6492	135	16	identities	identity	NOUN
ejpam-6492	135	17	can	can	AUX
ejpam-6492	135	18	also	also	ADV
ejpam-6492	135	19	be	be	AUX
ejpam-6492	135	20	generalized	generalize	VERB
ejpam-6492	135	21	.	.	PUNCT
ejpam-6492	136	1	theorem	theorem	NOUN
ejpam-6492	136	2	1	1	NUM
ejpam-6492	136	3	.	.	PUNCT
ejpam-6492	137	1	(	(	PUNCT
ejpam-6492	137	2	cassini	cassini	PROPN
ejpam-6492	137	3	’s	’s	PART
ejpam-6492	137	4	identity	identity	NOUN
ejpam-6492	137	5	)	)	PUNCT
ejpam-6492	137	6	for	for	ADP
ejpam-6492	137	7	all	all	DET
ejpam-6492	137	8	n	n	PRON
ejpam-6492	137	9	∈	∈	PROPN
ejpam-6492	137	10	z	z	PROPN
ejpam-6492	137	11	,	,	PUNCT
ejpam-6492	137	12	the	the	DET
ejpam-6492	137	13	following	follow	VERB
ejpam-6492	137	14	identity	identity	NOUN
ejpam-6492	137	15	holds	hold	VERB
ejpam-6492	137	16	un−1un+1	un−1un+1	ADJ
ejpam-6492	137	17	−	−	PROPN
ejpam-6492	137	18	u2	u2	PROPN
ejpam-6492	137	19	n	n	NOUN
ejpam-6492	137	20	=	=	NOUN
ejpam-6492	137	21	−(−q)n−1	−(−q)n−1	PROPN
ejpam-6492	137	22	.	.	PUNCT
ejpam-6492	138	1	b.	b.	PROPN
ejpam-6492	138	2	demirtürk	demirtürk	PROPN
ejpam-6492	138	3	,	,	PUNCT
ejpam-6492	138	4	n.	n.	NOUN
ejpam-6492	138	5	topal	topal	PROPN
ejpam-6492	138	6	/	/	SYM
ejpam-6492	138	7	eur	eur	PROPN
ejpam-6492	138	8	.	.	PUNCT
ejpam-6492	139	1	j.	j.	PROPN
ejpam-6492	139	2	pure	pure	PROPN
ejpam-6492	139	3	appl	appl	PROPN
ejpam-6492	139	4	.	.	PROPN
ejpam-6492	139	5	math	math	PROPN
ejpam-6492	139	6	,	,	PUNCT
ejpam-6492	139	7	18	18	NUM
ejpam-6492	139	8	(	(	PUNCT
ejpam-6492	139	9	3	3	NUM
ejpam-6492	139	10	)	)	PUNCT
ejpam-6492	139	11	(	(	PUNCT
ejpam-6492	139	12	2025	2025	NUM
ejpam-6492	139	13	)	)	PUNCT
ejpam-6492	139	14	,	,	PUNCT
ejpam-6492	139	15	6492	6492	NUM
ejpam-6492	139	16	7	7	NUM
ejpam-6492	139	17	of	of	ADP
ejpam-6492	139	18	22	22	NUM
ejpam-6492	139	19	proof	proof	NOUN
ejpam-6492	139	20	.	.	PUNCT
ejpam-6492	140	1	using	use	VERB
ejpam-6492	140	2	the	the	DET
ejpam-6492	140	3	binet	binet	NOUN
ejpam-6492	140	4	formula	formula	NOUN
ejpam-6492	140	5	and	and	CCONJ
ejpam-6492	140	6	the	the	DET
ejpam-6492	140	7	relation	relation	NOUN
ejpam-6492	141	1	αβ	αβ	INTJ
ejpam-6492	141	2	=	=	SYM
ejpam-6492	141	3	−q	−q	NOUN
ejpam-6492	141	4	,	,	PUNCT
ejpam-6492	141	5	we	we	PRON
ejpam-6492	141	6	have	have	VERB
ejpam-6492	141	7	un−1un+1	un−1un+1	ADJ
ejpam-6492	141	8	−	−	PROPN
ejpam-6492	141	9	u2	u2	PROPN
ejpam-6492	141	10	n	n	NOUN
ejpam-6492	141	11	=	=	PUNCT
ejpam-6492	141	12	(	(	PUNCT
ejpam-6492	141	13	αn−1	αn−1	ADV
ejpam-6492	141	14	−	−	PROPN
ejpam-6492	141	15	βn−1)(αn+1	βn−1)(αn+1	PUNCT
ejpam-6492	141	16	−	−	NOUN
ejpam-6492	141	17	βn+1	βn+1	NUM
ejpam-6492	141	18	)	)	PUNCT
ejpam-6492	141	19	(	(	PUNCT
ejpam-6492	141	20	α	α	NOUN
ejpam-6492	141	21	−	−	NOUN
ejpam-6492	141	22	β)2	β)2	NOUN
ejpam-6492	141	23	−	−	PROPN
ejpam-6492	141	24	(	(	PUNCT
ejpam-6492	141	25	αn	αn	NOUN
ejpam-6492	141	26	−	−	PROPN
ejpam-6492	142	1	βn)2	βn)2	PROPN
ejpam-6492	142	2	(	(	PUNCT
ejpam-6492	142	3	α	α	NOUN
ejpam-6492	142	4	−	−	NOUN
ejpam-6492	142	5	β)2	β)2	NOUN
ejpam-6492	142	6	=	=	SYM
ejpam-6492	142	7	α2n	α2n	PROPN
ejpam-6492	143	1	+	+	CCONJ
ejpam-6492	143	2	β2n	β2n	PUNCT
ejpam-6492	143	3	−	−	PROPN
ejpam-6492	143	4	αn−1βn+1	αn−1βn+1	ADV
ejpam-6492	143	5	−	−	PROPN
ejpam-6492	144	1	αn+1βn−1	αn+1βn−1	PROPN
ejpam-6492	144	2	(	(	PUNCT
ejpam-6492	144	3	α	α	NOUN
ejpam-6492	144	4	−	−	NOUN
ejpam-6492	144	5	β)2	β)2	ADV
ejpam-6492	144	6	−	−	PROPN
ejpam-6492	144	7	α2n	α2n	PROPN
ejpam-6492	145	1	+	+	CCONJ
ejpam-6492	145	2	β2n	β2n	PUNCT
ejpam-6492	145	3	−	−	PROPN
ejpam-6492	145	4	2αnβn	2αnβn	NUM
ejpam-6492	145	5	(	(	PUNCT
ejpam-6492	145	6	α	α	NOUN
ejpam-6492	145	7	−	−	NOUN
ejpam-6492	145	8	β)2	β)2	ADV
ejpam-6492	145	9	=	=	PROPN
ejpam-6492	145	10	−αn+1βn−1	−αn+1βn−1	PROPN
ejpam-6492	145	11	−	−	PROPN
ejpam-6492	145	12	αn−1βn+1	αn−1βn+1	NOUN
ejpam-6492	145	13	+	+	CCONJ
ejpam-6492	145	14	2αnβn	2αnβn	NUM
ejpam-6492	145	15	(	(	PUNCT
ejpam-6492	145	16	α	α	NOUN
ejpam-6492	145	17	−	−	NOUN
ejpam-6492	145	18	β)2	β)2	NOUN
ejpam-6492	145	19	=	=	NOUN
ejpam-6492	145	20	−(αβ)n−1	−(αβ)n−1	NOUN
ejpam-6492	145	21	α2	α2	PROPN
ejpam-6492	145	22	+	+	CCONJ
ejpam-6492	145	23	β2	β2	VERB
ejpam-6492	145	24	−	−	PROPN
ejpam-6492	145	25	2αβ	2αβ	NOUN
ejpam-6492	145	26	(	(	PUNCT
ejpam-6492	145	27	α	α	NOUN
ejpam-6492	145	28	−	−	NOUN
ejpam-6492	145	29	β)2	β)2	NOUN
ejpam-6492	145	30	=	=	NOUN
ejpam-6492	145	31	−(αβ)n−1	−(αβ)n−1	ADJ
ejpam-6492	145	32	=	=	PUNCT
ejpam-6492	145	33	−(−q)n−1	−(−q)n−1	NOUN
ejpam-6492	145	34	.	.	PUNCT
ejpam-6492	146	1	theorem	theorem	NOUN
ejpam-6492	146	2	2	2	NUM
ejpam-6492	146	3	.	.	X
ejpam-6492	146	4	for	for	ADP
ejpam-6492	146	5	all	all	DET
ejpam-6492	146	6	n	n	PRON
ejpam-6492	146	7	∈	∈	PROPN
ejpam-6492	146	8	z	z	PROPN
ejpam-6492	146	9	,	,	PUNCT
ejpam-6492	146	10	the	the	DET
ejpam-6492	146	11	following	follow	VERB
ejpam-6492	146	12	identity	identity	NOUN
ejpam-6492	146	13	holds	hold	VERB
ejpam-6492	146	14	vn−1vn+1	vn−1vn+1	ADV
ejpam-6492	146	15	−	−	NOUN
ejpam-6492	146	16	v	v	ADP
ejpam-6492	146	17	2	2	NUM
ejpam-6492	146	18	n	n	NOUN
ejpam-6492	146	19	=	=	SYM
ejpam-6492	146	20	(	(	PUNCT
ejpam-6492	146	21	−q)n−1∆.	−q)n−1∆.	PROPN
ejpam-6492	146	22	proof	proof	NOUN
ejpam-6492	146	23	.	.	PUNCT
ejpam-6492	147	1	using	use	VERB
ejpam-6492	147	2	the	the	DET
ejpam-6492	147	3	binet	binet	NOUN
ejpam-6492	147	4	formula	formula	NOUN
ejpam-6492	147	5	and	and	CCONJ
ejpam-6492	147	6	identities	identity	NOUN
ejpam-6492	147	7	αβ	αβ	INTJ
ejpam-6492	147	8	=	=	SYM
ejpam-6492	147	9	−q	−q	NOUN
ejpam-6492	147	10	,	,	PUNCT
ejpam-6492	147	11	α	α	NOUN
ejpam-6492	147	12	−	−	NOUN
ejpam-6492	147	13	β	β	NOUN
ejpam-6492	147	14	=	=	SYM
ejpam-6492	147	15	√	√	PROPN
ejpam-6492	147	16	∆	∆	PROPN
ejpam-6492	147	17	,	,	PUNCT
ejpam-6492	147	18	we	we	PRON
ejpam-6492	147	19	obtain	obtain	VERB
ejpam-6492	147	20	vn−1vn+1	vn−1vn+1	ADV
ejpam-6492	147	21	−	−	NOUN
ejpam-6492	147	22	v	v	ADP
ejpam-6492	147	23	2	2	NUM
ejpam-6492	147	24	n	n	NOUN
ejpam-6492	147	25	=	=	PUNCT
ejpam-6492	147	26	(	(	PUNCT
ejpam-6492	147	27	αn−1	αn−1	ADJ
ejpam-6492	147	28	+	+	CCONJ
ejpam-6492	147	29	βn−1)(αn+1	βn−1)(αn+1	SYM
ejpam-6492	147	30	+	+	NUM
ejpam-6492	147	31	βn+1	βn+1	NUM
ejpam-6492	147	32	)	)	PUNCT
ejpam-6492	147	33	−	−	PROPN
ejpam-6492	148	1	(	(	PUNCT
ejpam-6492	148	2	αn	αn	NOUN
ejpam-6492	149	1	+	+	CCONJ
ejpam-6492	149	2	βn)2	βn)2	PROPN
ejpam-6492	149	3	=	=	SYM
ejpam-6492	149	4	(	(	PUNCT
ejpam-6492	149	5	α2n	α2n	PROPN
ejpam-6492	149	6	+	+	CCONJ
ejpam-6492	149	7	β2n	β2n	PUNCT
ejpam-6492	149	8	+	+	PUNCT
ejpam-6492	149	9	αn−1βn+1	αn−1βn+1	X
ejpam-6492	149	10	+	+	CCONJ
ejpam-6492	149	11	αn+1βn−1	αn+1βn−1	NUM
ejpam-6492	149	12	)	)	PUNCT
ejpam-6492	149	13	−	−	PROPN
ejpam-6492	150	1	(	(	PUNCT
ejpam-6492	150	2	α2n	α2n	PROPN
ejpam-6492	150	3	+	+	CCONJ
ejpam-6492	150	4	β2n	β2n	PUNCT
ejpam-6492	150	5	+	+	NUM
ejpam-6492	150	6	2αnβn	2αnβn	NUM
ejpam-6492	150	7	)	)	PUNCT
ejpam-6492	150	8	=	=	PUNCT
ejpam-6492	151	1	αn+1βn−1	αn+1βn−1	PROPN
ejpam-6492	151	2	+	+	CCONJ
ejpam-6492	151	3	αn−1βn+1	αn−1βn+1	NOUN
ejpam-6492	151	4	−	−	PROPN
ejpam-6492	151	5	2αnβn	2αnβn	NUM
ejpam-6492	151	6	=	=	SYM
ejpam-6492	151	7	(	(	PUNCT
ejpam-6492	151	8	αβ)n−1(α2	αβ)n−1(α2	NOUN
ejpam-6492	151	9	+	+	CCONJ
ejpam-6492	151	10	β2	β2	NOUN
ejpam-6492	151	11	−	−	PROPN
ejpam-6492	151	12	2αβ	2αβ	NOUN
ejpam-6492	151	13	)	)	PUNCT
ejpam-6492	152	1	=	=	SYM
ejpam-6492	152	2	(	(	PUNCT
ejpam-6492	152	3	αβ)n−1(α	αβ)n−1(α	PROPN
ejpam-6492	152	4	−	−	PROPN
ejpam-6492	152	5	β)2	β)2	NOUN
ejpam-6492	152	6	=	=	X
ejpam-6492	152	7	(	(	PUNCT
ejpam-6492	152	8	−q)n−1∆.	−q)n−1∆.	PROPN
ejpam-6492	152	9	theorem	theorem	VERB
ejpam-6492	152	10	3	3	NUM
ejpam-6492	152	11	.	.	PUNCT
ejpam-6492	153	1	(	(	PUNCT
ejpam-6492	153	2	catalan	catalan	NOUN
ejpam-6492	153	3	’s	’s	PART
ejpam-6492	153	4	identity	identity	NOUN
ejpam-6492	153	5	)	)	PUNCT
ejpam-6492	153	6	for	for	ADP
ejpam-6492	153	7	all	all	DET
ejpam-6492	153	8	n	n	NOUN
ejpam-6492	153	9	,	,	PUNCT
ejpam-6492	153	10	r	r	NOUN
ejpam-6492	153	11	∈	∈	PROPN
ejpam-6492	153	12	z	z	X
ejpam-6492	153	13	,	,	PUNCT
ejpam-6492	153	14	it	it	PRON
ejpam-6492	153	15	follows	follow	VERB
ejpam-6492	153	16	that	that	SCONJ
ejpam-6492	153	17	un−run+r	un−run+r	PROPN
ejpam-6492	153	18	−	−	NOUN
ejpam-6492	153	19	u2	u2	NOUN
ejpam-6492	153	20	n	n	NOUN
ejpam-6492	153	21	=	=	PUNCT
ejpam-6492	153	22	−(−q)n−ru2	−(−q)n−ru2	NOUN
ejpam-6492	153	23	r	r	NOUN
ejpam-6492	153	24	.	.	PUNCT
ejpam-6492	154	1	proof	proof	NOUN
ejpam-6492	154	2	.	.	PUNCT
ejpam-6492	155	1	using	use	VERB
ejpam-6492	155	2	the	the	DET
ejpam-6492	155	3	binet	binet	NOUN
ejpam-6492	155	4	formula	formula	NOUN
ejpam-6492	155	5	and	and	CCONJ
ejpam-6492	155	6	αβ	αβ	NOUN
ejpam-6492	155	7	=	=	NOUN
ejpam-6492	155	8	−q	−q	NOUN
ejpam-6492	155	9	,	,	PUNCT
ejpam-6492	155	10	we	we	PRON
ejpam-6492	155	11	have	have	VERB
ejpam-6492	155	12	un−run+r	un−run+r	NOUN
ejpam-6492	155	13	−	−	NOUN
ejpam-6492	155	14	u2	u2	NOUN
ejpam-6492	155	15	n	n	NOUN
ejpam-6492	155	16	=	=	PUNCT
ejpam-6492	155	17	(	(	PUNCT
ejpam-6492	155	18	αn−r	αn−r	NOUN
ejpam-6492	155	19	−	−	PROPN
ejpam-6492	156	1	βn−r)(αn+r	βn−r)(αn+r	PROPN
ejpam-6492	156	2	−	−	PROPN
ejpam-6492	156	3	βn+r	βn+r	NOUN
ejpam-6492	156	4	)	)	PUNCT
ejpam-6492	156	5	(	(	PUNCT
ejpam-6492	156	6	α	α	NOUN
ejpam-6492	156	7	−	−	NOUN
ejpam-6492	156	8	β)2	β)2	NOUN
ejpam-6492	156	9	−	−	PROPN
ejpam-6492	156	10	(	(	PUNCT
ejpam-6492	156	11	αn	αn	NOUN
ejpam-6492	156	12	−	−	PROPN
ejpam-6492	156	13	βn)2	βn)2	PROPN
ejpam-6492	156	14	(	(	PUNCT
ejpam-6492	156	15	α	α	NOUN
ejpam-6492	156	16	−	−	NOUN
ejpam-6492	156	17	β)2	β)2	NOUN
ejpam-6492	156	18	=	=	SYM
ejpam-6492	156	19	α2n	α2n	PROPN
ejpam-6492	157	1	+	+	CCONJ
ejpam-6492	157	2	β2n	β2n	PUNCT
ejpam-6492	157	3	−	−	X
ejpam-6492	157	4	αn−rβn+r	αn−rβn+r	PROPN
ejpam-6492	157	5	−	−	PROPN
ejpam-6492	157	6	αn+rβn−r	αn+rβn−r	NOUN
ejpam-6492	157	7	(	(	PUNCT
ejpam-6492	157	8	α	α	NOUN
ejpam-6492	157	9	−	−	NOUN
ejpam-6492	157	10	β)2	β)2	ADV
ejpam-6492	157	11	−	−	PROPN
ejpam-6492	157	12	α2n	α2n	PROPN
ejpam-6492	158	1	+	+	CCONJ
ejpam-6492	158	2	β2n	β2n	PUNCT
ejpam-6492	158	3	−	−	PROPN
ejpam-6492	158	4	2αnβn	2αnβn	NUM
ejpam-6492	158	5	(	(	PUNCT
ejpam-6492	158	6	α	α	NOUN
ejpam-6492	158	7	−	−	NOUN
ejpam-6492	158	8	β)2	β)2	NOUN
ejpam-6492	158	9	=	=	NOUN
ejpam-6492	158	10	−αn+rβn−r	−αn+rβn−r	PROPN
ejpam-6492	158	11	−	−	X
ejpam-6492	158	12	αn−rβn+r	αn−rβn+r	X
ejpam-6492	158	13	+	+	CCONJ
ejpam-6492	158	14	2αnβn	2αnβn	NUM
ejpam-6492	158	15	(	(	PUNCT
ejpam-6492	158	16	α	α	NOUN
ejpam-6492	158	17	−	−	NOUN
ejpam-6492	158	18	β)2	β)2	NOUN
ejpam-6492	158	19	=	=	NOUN
ejpam-6492	158	20	−(αβ)n−r	−(αβ)n−r	NOUN
ejpam-6492	158	21	α2r	α2r	NUM
ejpam-6492	158	22	+	+	CCONJ
ejpam-6492	158	23	β2r	β2r	SYM
ejpam-6492	158	24	−	−	PROPN
ejpam-6492	158	25	2αrβr	2αrβr	PROPN
ejpam-6492	158	26	(	(	PUNCT
ejpam-6492	158	27	α	α	NOUN
ejpam-6492	158	28	−	−	NOUN
ejpam-6492	158	29	β)2	β)2	NOUN
ejpam-6492	158	30	=	=	X
ejpam-6492	158	31	−(αβ)n−r	−(αβ)n−r	PROPN
ejpam-6492	158	32	(	(	PUNCT
ejpam-6492	158	33	αr	αr	INTJ
ejpam-6492	158	34	−	−	PROPN
ejpam-6492	158	35	βr	βr	ADP
ejpam-6492	159	1	α	α	PRON
ejpam-6492	159	2	−	−	NOUN
ejpam-6492	159	3	β	β	X
ejpam-6492	159	4	)	)	PUNCT
ejpam-6492	159	5	2	2	NUM
ejpam-6492	159	6	=	=	SYM
ejpam-6492	159	7	−(αβ)n−r(ur)2	−(αβ)n−r(ur)2	PUNCT
ejpam-6492	159	8	=	=	PUNCT
ejpam-6492	159	9	−(−q)n−ru2	−(−q)n−ru2	NOUN
ejpam-6492	159	10	r	r	NOUN
ejpam-6492	159	11	.	.	PUNCT
ejpam-6492	160	1	b.	b.	PROPN
ejpam-6492	160	2	demirtürk	demirtürk	PROPN
ejpam-6492	160	3	,	,	PUNCT
ejpam-6492	160	4	n.	n.	NOUN
ejpam-6492	160	5	topal	topal	PROPN
ejpam-6492	160	6	/	/	SYM
ejpam-6492	160	7	eur	eur	PROPN
ejpam-6492	160	8	.	.	PUNCT
ejpam-6492	161	1	j.	j.	PROPN
ejpam-6492	161	2	pure	pure	PROPN
ejpam-6492	161	3	appl	appl	PROPN
ejpam-6492	161	4	.	.	PROPN
ejpam-6492	161	5	math	math	PROPN
ejpam-6492	161	6	,	,	PUNCT
ejpam-6492	161	7	18	18	NUM
ejpam-6492	161	8	(	(	PUNCT
ejpam-6492	161	9	3	3	NUM
ejpam-6492	161	10	)	)	PUNCT
ejpam-6492	161	11	(	(	PUNCT
ejpam-6492	161	12	2025	2025	NUM
ejpam-6492	161	13	)	)	PUNCT
ejpam-6492	161	14	,	,	PUNCT
ejpam-6492	161	15	6492	6492	NUM
ejpam-6492	161	16	8	8	NUM
ejpam-6492	161	17	of	of	ADP
ejpam-6492	161	18	22	22	NUM
ejpam-6492	161	19	theorem	theorem	NOUN
ejpam-6492	161	20	4	4	NUM
ejpam-6492	161	21	.	.	PUNCT
ejpam-6492	162	1	for	for	ADP
ejpam-6492	162	2	all	all	DET
ejpam-6492	162	3	n	n	NOUN
ejpam-6492	162	4	,	,	PUNCT
ejpam-6492	162	5	r	r	NOUN
ejpam-6492	162	6	∈	∈	PROPN
ejpam-6492	162	7	z	z	PROPN
ejpam-6492	162	8	,	,	PUNCT
ejpam-6492	162	9	vn−rvn+r	vn−rvn+r	PROPN
ejpam-6492	162	10	−	−	NOUN
ejpam-6492	162	11	v	v	ADP
ejpam-6492	162	12	2	2	NUM
ejpam-6492	162	13	n	n	NOUN
ejpam-6492	162	14	=	=	PUNCT
ejpam-6492	162	15	(	(	PUNCT
ejpam-6492	162	16	−q)n−r∆u2	−q)n−r∆u2	NOUN
ejpam-6492	162	17	r	r	NOUN
ejpam-6492	162	18	.	.	PUNCT
ejpam-6492	163	1	proof	proof	NOUN
ejpam-6492	163	2	.	.	PUNCT
ejpam-6492	164	1	considering	consider	VERB
ejpam-6492	164	2	the	the	DET
ejpam-6492	164	3	binet	binet	NOUN
ejpam-6492	164	4	formula	formula	NOUN
ejpam-6492	164	5	,	,	PUNCT
ejpam-6492	164	6	identities	identity	NOUN
ejpam-6492	164	7	αβ	αβ	NOUN
ejpam-6492	164	8	=	=	NOUN
ejpam-6492	164	9	−q	−q	NOUN
ejpam-6492	164	10	and	and	CCONJ
ejpam-6492	164	11	α	α	NOUN
ejpam-6492	164	12	−	−	NOUN
ejpam-6492	164	13	β	β	NOUN
ejpam-6492	164	14	=	=	SYM
ejpam-6492	164	15	√	√	PROPN
ejpam-6492	164	16	∆	∆	PROPN
ejpam-6492	164	17	,	,	PUNCT
ejpam-6492	164	18	we	we	PRON
ejpam-6492	164	19	have	have	VERB
ejpam-6492	164	20	vn−rvn+r	vn−rvn+r	PROPN
ejpam-6492	164	21	−	−	PROPN
ejpam-6492	164	22	v	v	ADP
ejpam-6492	164	23	2	2	NUM
ejpam-6492	164	24	n	n	NOUN
ejpam-6492	164	25	=	=	PUNCT
ejpam-6492	164	26	(	(	PUNCT
ejpam-6492	164	27	αn−r	αn−r	NOUN
ejpam-6492	164	28	+	+	CCONJ
ejpam-6492	164	29	βn−r)(αn+r	βn−r)(αn+r	PROPN
ejpam-6492	164	30	+	+	CCONJ
ejpam-6492	164	31	βn+r	βn+r	NOUN
ejpam-6492	164	32	)	)	PUNCT
ejpam-6492	164	33	−	−	PROPN
ejpam-6492	165	1	(	(	PUNCT
ejpam-6492	165	2	αn	αn	NOUN
ejpam-6492	166	1	+	+	CCONJ
ejpam-6492	166	2	βn)2	βn)2	PROPN
ejpam-6492	166	3	=	=	SYM
ejpam-6492	166	4	(	(	PUNCT
ejpam-6492	166	5	α2n	α2n	PROPN
ejpam-6492	166	6	+	+	CCONJ
ejpam-6492	166	7	β2n	β2n	PUNCT
ejpam-6492	166	8	+	+	CCONJ
ejpam-6492	166	9	αn−rβn+r	αn−rβn+r	PROPN
ejpam-6492	166	10	+	+	CCONJ
ejpam-6492	166	11	αn+rβn−r	αn+rβn−r	NOUN
ejpam-6492	166	12	)	)	PUNCT
ejpam-6492	166	13	−	−	PROPN
ejpam-6492	167	1	(	(	PUNCT
ejpam-6492	167	2	α2n	α2n	PROPN
ejpam-6492	167	3	+	+	CCONJ
ejpam-6492	167	4	β2n	β2n	PUNCT
ejpam-6492	167	5	+	+	NUM
ejpam-6492	167	6	2αnβn	2αnβn	NUM
ejpam-6492	167	7	)	)	PUNCT
ejpam-6492	167	8	=	=	SYM
ejpam-6492	167	9	αn+rβn−r	αn+rβn−r	NOUN
ejpam-6492	167	10	+	+	CCONJ
ejpam-6492	168	1	αn−rβn+r	αn−rβn+r	PROPN
ejpam-6492	168	2	−	−	PROPN
ejpam-6492	168	3	2αnβn	2αnβn	NUM
ejpam-6492	168	4	=	=	SYM
ejpam-6492	168	5	(	(	PUNCT
ejpam-6492	168	6	αβ)n−r(α2r	αβ)n−r(α2r	ADV
ejpam-6492	168	7	+	+	CCONJ
ejpam-6492	168	8	β2r	β2r	SYM
ejpam-6492	168	9	−	−	PROPN
ejpam-6492	168	10	2αrβr	2αrβr	NUM
ejpam-6492	168	11	)	)	PUNCT
ejpam-6492	168	12	=	=	PUNCT
ejpam-6492	168	13	(	(	PUNCT
ejpam-6492	168	14	αβ)n−r(αr	αβ)n−r(αr	ADV
ejpam-6492	168	15	−	−	ADP
ejpam-6492	168	16	βr)2	βr)2	PROPN
ejpam-6492	169	1	=	=	SYM
ejpam-6492	170	1	(	(	PUNCT
ejpam-6492	170	2	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	170	3	(	(	PUNCT
ejpam-6492	170	4	αr	αr	INTJ
ejpam-6492	170	5	−	−	PROPN
ejpam-6492	170	6	βr	βr	ADP
ejpam-6492	170	7	α	α	PRON
ejpam-6492	170	8	−	−	NOUN
ejpam-6492	170	9	β	β	X
ejpam-6492	170	10	)	)	PUNCT
ejpam-6492	170	11	2	2	NUM
ejpam-6492	170	12	(	(	PUNCT
ejpam-6492	170	13	α	α	NOUN
ejpam-6492	170	14	−	−	NOUN
ejpam-6492	170	15	β)2	β)2	NOUN
ejpam-6492	170	16	=	=	X
ejpam-6492	170	17	(	(	PUNCT
ejpam-6492	170	18	αβ)n−ru2	αβ)n−ru2	X
ejpam-6492	170	19	r	r	NOUN
ejpam-6492	170	20	(	(	PUNCT
ejpam-6492	170	21	α	α	NOUN
ejpam-6492	170	22	−	−	NOUN
ejpam-6492	170	23	β)2	β)2	NOUN
ejpam-6492	170	24	=	=	X
ejpam-6492	170	25	(	(	PUNCT
ejpam-6492	170	26	−q)n−r∆u2	−q)n−r∆u2	NOUN
ejpam-6492	170	27	r	r	NOUN
ejpam-6492	170	28	.	.	PUNCT
ejpam-6492	171	1	corollary	corollary	ADJ
ejpam-6492	171	2	1	1	NUM
ejpam-6492	171	3	.	.	PUNCT
ejpam-6492	172	1	for	for	ADP
ejpam-6492	172	2	all	all	DET
ejpam-6492	172	3	n	n	NOUN
ejpam-6492	172	4	,	,	PUNCT
ejpam-6492	172	5	r	r	NOUN
ejpam-6492	172	6	∈	∈	PROPN
ejpam-6492	172	7	z	z	NOUN
ejpam-6492	172	8	,	,	PUNCT
ejpam-6492	172	9	we	we	PRON
ejpam-6492	172	10	have	have	VERB
ejpam-6492	172	11	vn−rvn+r	vn−rvn+r	PROPN
ejpam-6492	172	12	−	−	PROPN
ejpam-6492	172	13	v	v	ADP
ejpam-6492	172	14	2	2	NUM
ejpam-6492	172	15	n	n	NOUN
ejpam-6492	172	16	=	=	PUNCT
ejpam-6492	172	17	−∆(un−run+r	−∆(un−run+r	NOUN
ejpam-6492	172	18	−	−	PROPN
ejpam-6492	172	19	u2	u2	PROPN
ejpam-6492	172	20	n	n	CCONJ
ejpam-6492	172	21	)	)	PUNCT
ejpam-6492	172	22	.	.	PUNCT
ejpam-6492	173	1	proof	proof	NOUN
ejpam-6492	173	2	.	.	PUNCT
ejpam-6492	174	1	follows	follow	VERB
ejpam-6492	174	2	directly	directly	ADV
ejpam-6492	174	3	from	from	ADP
ejpam-6492	174	4	theorem	theorem	ADJ
ejpam-6492	174	5	3	3	NUM
ejpam-6492	174	6	and	and	CCONJ
ejpam-6492	174	7	theorem	theorem	VERB
ejpam-6492	174	8	4	4	NUM
ejpam-6492	174	9	.	.	PUNCT
ejpam-6492	174	10	şiar	şiar	NOUN
ejpam-6492	174	11	and	and	CCONJ
ejpam-6492	174	12	keskin	keskin	X
ejpam-6492	175	1	[	[	X
ejpam-6492	175	2	27	27	NUM
ejpam-6492	175	3	]	]	PUNCT
ejpam-6492	175	4	established	establish	VERB
ejpam-6492	175	5	several	several	ADJ
ejpam-6492	175	6	identities	identity	NOUN
ejpam-6492	175	7	using	use	VERB
ejpam-6492	175	8	the	the	DET
ejpam-6492	175	9	matrices	matrix	NOUN
ejpam-6492	175	10	[	[	PUNCT
ejpam-6492	175	11	p	p	X
ejpam-6492	175	12	q	q	PROPN
ejpam-6492	175	13	1	1	NUM
ejpam-6492	175	14	0	0	NUM
ejpam-6492	175	15	]	]	PUNCT
ejpam-6492	175	16	and	and	CCONJ
ejpam-6492	175	17	[	[	PUNCT
ejpam-6492	175	18	un+1	un+1	PROPN
ejpam-6492	175	19	qun	qun	PROPN
ejpam-6492	175	20	un	un	PROPN
ejpam-6492	175	21	qun−1	qun−1	PROPN
ejpam-6492	175	22	]	]	PUNCT
ejpam-6492	175	23	,	,	PUNCT
ejpam-6492	175	24	and	and	CCONJ
ejpam-6492	175	25	proved	prove	VERB
ejpam-6492	175	26	the	the	DET
ejpam-6492	175	27	following	follow	VERB
ejpam-6492	175	28	results	result	NOUN
ejpam-6492	175	29	v	v	ADP
ejpam-6492	175	30	2	2	NUM
ejpam-6492	175	31	n	n	NOUN
ejpam-6492	175	32	−	−	PROPN
ejpam-6492	175	33	(	(	PUNCT
ejpam-6492	175	34	p2	p2	X
ejpam-6492	175	35	+	+	CCONJ
ejpam-6492	175	36	4q)u2	4q)u2	NUM
ejpam-6492	175	37	n	n	NOUN
ejpam-6492	175	38	=	=	SYM
ejpam-6492	175	39	4(−q)n	4(−q)n	NUM
ejpam-6492	175	40	,	,	PUNCT
ejpam-6492	175	41	∆umun	∆umun	PROPN
ejpam-6492	175	42	=	=	SYM
ejpam-6492	175	43	vm+n	vm+n	NUM
ejpam-6492	175	44	−	−	NOUN
ejpam-6492	175	45	(	(	PUNCT
ejpam-6492	175	46	−q)nvm−n	−q)nvm−n	PROPN
ejpam-6492	175	47	,	,	PUNCT
ejpam-6492	175	48	(	(	PUNCT
ejpam-6492	175	49	−q)nvm−n	−q)nvm−n	NOUN
ejpam-6492	175	50	=	=	PUNCT
ejpam-6492	175	51	um+1vn	um+1vn	PROPN
ejpam-6492	175	52	−	−	PROPN
ejpam-6492	175	53	vn+1um	vn+1um	NOUN
ejpam-6492	175	54	,	,	PUNCT
ejpam-6492	175	55	urum+n+r	urum+n+r	PROPN
ejpam-6492	175	56	=	=	SYM
ejpam-6492	175	57	um+run+r	um+run+r	NUM
ejpam-6492	175	58	−	−	PROPN
ejpam-6492	175	59	(	(	PUNCT
ejpam-6492	175	60	−q)rumun	−q)rumun	X
ejpam-6492	175	61	,	,	PUNCT
ejpam-6492	175	62	urum+n−r	urum+n−r	NOUN
ejpam-6492	175	63	=	=	SYM
ejpam-6492	175	64	umun	umun	PROPN
ejpam-6492	176	1	−	−	PROPN
ejpam-6492	176	2	(	(	PUNCT
ejpam-6492	176	3	−q)rum−run−r	−q)rum−run−r	PROPN
ejpam-6492	176	4	,	,	PUNCT
ejpam-6492	176	5	urum+n	urum+n	X
ejpam-6492	176	6	=	=	PUNCT
ejpam-6492	176	7	umun+r	umun+r	NOUN
ejpam-6492	177	1	−	−	PROPN
ejpam-6492	177	2	(	(	PUNCT
ejpam-6492	177	3	−q)rum−run	−q)rum−run	NOUN
ejpam-6492	177	4	,	,	PUNCT
ejpam-6492	177	5	vrvm+n+r	vrvm+n+r	PROPN
ejpam-6492	177	6	=	=	SYM
ejpam-6492	178	1	vm+rvn+r	vm+rvn+r	PROPN
ejpam-6492	178	2	+	+	CCONJ
ejpam-6492	178	3	(	(	PUNCT
ejpam-6492	178	4	−q)r∆umun	−q)r∆umun	PROPN
ejpam-6492	178	5	,	,	PUNCT
ejpam-6492	178	6	vrvm+n−r	vrvm+n−r	NOUN
ejpam-6492	178	7	=	=	PUNCT
ejpam-6492	178	8	(	(	PUNCT
ejpam-6492	178	9	−q)rvm−rvn−r	−q)rvm−rvn−r	NOUN
ejpam-6492	178	10	+	+	CCONJ
ejpam-6492	178	11	∆umun	∆umun	ADJ
ejpam-6492	178	12	,	,	PUNCT
ejpam-6492	178	13	vrum+n	vrum+n	NOUN
ejpam-6492	178	14	=	=	SYM
ejpam-6492	178	15	unvm+r	unvm+r	PROPN
ejpam-6492	178	16	+	+	CCONJ
ejpam-6492	178	17	(	(	PUNCT
ejpam-6492	178	18	−q)rvn−rum	−q)rvn−rum	ADP
ejpam-6492	178	19	,	,	PUNCT
ejpam-6492	178	20	and	and	CCONJ
ejpam-6492	178	21	more	more	ADJ
ejpam-6492	178	22	.	.	PUNCT
ejpam-6492	179	1	now	now	ADV
ejpam-6492	179	2	we	we	PRON
ejpam-6492	179	3	will	will	AUX
ejpam-6492	179	4	prove	prove	VERB
ejpam-6492	179	5	other	other	ADJ
ejpam-6492	179	6	identities	identity	NOUN
ejpam-6492	179	7	as	as	SCONJ
ejpam-6492	179	8	follows	follow	VERB
ejpam-6492	179	9	.	.	PUNCT
ejpam-6492	180	1	theorem	theorem	ADJ
ejpam-6492	180	2	5	5	NUM
ejpam-6492	180	3	.	.	PUNCT
ejpam-6492	180	4	for	for	ADP
ejpam-6492	180	5	all	all	DET
ejpam-6492	180	6	n	n	NOUN
ejpam-6492	180	7	,	,	PUNCT
ejpam-6492	180	8	r	r	NOUN
ejpam-6492	180	9	∈	∈	PROPN
ejpam-6492	180	10	z	z	NOUN
ejpam-6492	180	11	,	,	PUNCT
ejpam-6492	180	12	we	we	PRON
ejpam-6492	180	13	have	have	VERB
ejpam-6492	180	14	u2	u2	PROPN
ejpam-6492	180	15	n+r	n+r	PROPN
ejpam-6492	180	16	−	−	ADP
ejpam-6492	180	17	q2ru2	q2ru2	NOUN
ejpam-6492	180	18	n−r	n−r	NOUN
ejpam-6492	180	19	=	=	PUNCT
ejpam-6492	180	20	u2nu2r	u2nu2r	X
ejpam-6492	180	21	.	.	PUNCT
ejpam-6492	180	22	b.	b.	PROPN
ejpam-6492	180	23	demirtürk	demirtürk	PROPN
ejpam-6492	180	24	,	,	PUNCT
ejpam-6492	180	25	n.	n.	NOUN
ejpam-6492	180	26	topal	topal	PROPN
ejpam-6492	180	27	/	/	SYM
ejpam-6492	180	28	eur	eur	PROPN
ejpam-6492	180	29	.	.	PUNCT
ejpam-6492	181	1	j.	j.	PROPN
ejpam-6492	181	2	pure	pure	PROPN
ejpam-6492	181	3	appl	appl	PROPN
ejpam-6492	181	4	.	.	PROPN
ejpam-6492	181	5	math	math	PROPN
ejpam-6492	181	6	,	,	PUNCT
ejpam-6492	181	7	18	18	NUM
ejpam-6492	181	8	(	(	PUNCT
ejpam-6492	181	9	3	3	NUM
ejpam-6492	181	10	)	)	PUNCT
ejpam-6492	181	11	(	(	PUNCT
ejpam-6492	181	12	2025	2025	NUM
ejpam-6492	181	13	)	)	PUNCT
ejpam-6492	181	14	,	,	PUNCT
ejpam-6492	181	15	6492	6492	NUM
ejpam-6492	181	16	9	9	NUM
ejpam-6492	181	17	of	of	ADP
ejpam-6492	181	18	22	22	NUM
ejpam-6492	181	19	proof	proof	NOUN
ejpam-6492	181	20	.	.	PUNCT
ejpam-6492	182	1	u2	u2	PROPN
ejpam-6492	182	2	n+r	n+r	PROPN
ejpam-6492	182	3	−	−	ADP
ejpam-6492	182	4	q2ru2	q2ru2	NOUN
ejpam-6492	182	5	n−r	n−r	NOUN
ejpam-6492	182	6	=	=	SYM
ejpam-6492	182	7	(	(	PUNCT
ejpam-6492	182	8	αn+r	αn+r	PROPN
ejpam-6492	182	9	−	−	PROPN
ejpam-6492	182	10	βn+r	βn+r	NOUN
ejpam-6492	182	11	α	α	NOUN
ejpam-6492	182	12	−	−	NOUN
ejpam-6492	182	13	β	β	NOUN
ejpam-6492	182	14	)	)	PUNCT
ejpam-6492	182	15	2	2	NUM
ejpam-6492	182	16	−	−	PROPN
ejpam-6492	182	17	(	(	PUNCT
ejpam-6492	182	18	−q)2r	−q)2r	INTJ
ejpam-6492	182	19	(	(	PUNCT
ejpam-6492	182	20	αn−r	αn−r	NOUN
ejpam-6492	182	21	−	−	PROPN
ejpam-6492	182	22	βn−r	βn−r	NOUN
ejpam-6492	182	23	α	α	NOUN
ejpam-6492	182	24	−	−	NOUN
ejpam-6492	182	25	β	β	NOUN
ejpam-6492	182	26	)	)	PUNCT
ejpam-6492	182	27	2	2	NUM
ejpam-6492	182	28	=	=	SYM
ejpam-6492	182	29	α2n+2r	α2n+2r	NOUN
ejpam-6492	182	30	+	+	CCONJ
ejpam-6492	182	31	β2n+2r	β2n+2r	NOUN
ejpam-6492	182	32	−	−	NOUN
ejpam-6492	182	33	2	2	NUM
ejpam-6492	182	34	(	(	PUNCT
ejpam-6492	182	35	αβ)n+r	αβ)n+r	PROPN
ejpam-6492	182	36	(	(	PUNCT
ejpam-6492	182	37	α	α	NOUN
ejpam-6492	182	38	−	−	NOUN
ejpam-6492	182	39	β)2	β)2	NOUN
ejpam-6492	182	40	−	−	PROPN
ejpam-6492	182	41	(	(	PUNCT
ejpam-6492	182	42	αβ)2r	αβ)2r	INTJ
ejpam-6492	182	43	α2n−2r	α2n−2r	PROPN
ejpam-6492	182	44	+	+	CCONJ
ejpam-6492	182	45	β2n−2r	β2n−2r	NOUN
ejpam-6492	182	46	−	−	NOUN
ejpam-6492	182	47	2	2	NUM
ejpam-6492	182	48	(	(	PUNCT
ejpam-6492	182	49	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	182	50	(	(	PUNCT
ejpam-6492	182	51	α	α	NOUN
ejpam-6492	182	52	−	−	NOUN
ejpam-6492	182	53	β)2	β)2	NOUN
ejpam-6492	182	54	=	=	PRON
ejpam-6492	182	55	[	[	PUNCT
ejpam-6492	182	56	α2n+2r	α2n+2r	NOUN
ejpam-6492	182	57	+	+	CCONJ
ejpam-6492	182	58	β2n+2r	β2n+2r	NOUN
ejpam-6492	182	59	−	−	NOUN
ejpam-6492	182	60	2	2	NUM
ejpam-6492	182	61	(	(	PUNCT
ejpam-6492	182	62	αβ)n+r	αβ)n+r	NOUN
ejpam-6492	182	63	]	]	PUNCT
ejpam-6492	182	64	−	−	PROPN
ejpam-6492	183	1	(	(	PUNCT
ejpam-6492	183	2	αβ)2r	αβ)2r	X
ejpam-6492	183	3	[	[	PUNCT
ejpam-6492	183	4	α2n−2r	α2n−2r	NOUN
ejpam-6492	183	5	+	+	CCONJ
ejpam-6492	183	6	β2n−2r	β2n−2r	NOUN
ejpam-6492	183	7	−	−	NOUN
ejpam-6492	183	8	2	2	NUM
ejpam-6492	183	9	(	(	PUNCT
ejpam-6492	183	10	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	183	11	]	]	PUNCT
ejpam-6492	183	12	(	(	PUNCT
ejpam-6492	183	13	α	α	NOUN
ejpam-6492	183	14	−	−	NOUN
ejpam-6492	183	15	β)2	β)2	NOUN
ejpam-6492	183	16	=	=	PUNCT
ejpam-6492	183	17	α2n+2r	α2n+2r	PROPN
ejpam-6492	183	18	+	+	CCONJ
ejpam-6492	183	19	β2n+2r	β2n+2r	NOUN
ejpam-6492	183	20	−	−	PROPN
ejpam-6492	183	21	α2nβ2r	α2nβ2r	NUM
ejpam-6492	183	22	−	−	NOUN
ejpam-6492	183	23	α2rβ2n	α2rβ2n	NUM
ejpam-6492	183	24	(	(	PUNCT
ejpam-6492	183	25	α	α	NOUN
ejpam-6492	183	26	−	−	NOUN
ejpam-6492	183	27	β)2	β)2	NOUN
ejpam-6492	183	28	=	=	PROPN
ejpam-6492	183	29	α2n+2r	α2n+2r	PROPN
ejpam-6492	183	30	−	−	PROPN
ejpam-6492	183	31	α2nβ2r	α2nβ2r	NUM
ejpam-6492	183	32	+	+	NUM
ejpam-6492	183	33	β2n+2r	β2n+2r	NOUN
ejpam-6492	183	34	−	−	NOUN
ejpam-6492	183	35	α2rβ2n	α2rβ2n	NUM
ejpam-6492	183	36	(	(	PUNCT
ejpam-6492	183	37	α	α	NOUN
ejpam-6492	183	38	−	−	PROPN
ejpam-6492	183	39	β)2	β)2	NOUN
ejpam-6492	183	40	=	=	PROPN
ejpam-6492	183	41	α2n	α2n	PROPN
ejpam-6492	183	42	(	(	PUNCT
ejpam-6492	183	43	α2r	α2r	PROPN
ejpam-6492	183	44	−	−	NOUN
ejpam-6492	183	45	β2r	β2r	PUNCT
ejpam-6492	183	46	)	)	PUNCT
ejpam-6492	184	1	+	+	CCONJ
ejpam-6492	184	2	β2n	β2n	PUNCT
ejpam-6492	184	3	(	(	PUNCT
ejpam-6492	184	4	β2r	β2r	PUNCT
ejpam-6492	184	5	−	−	PROPN
ejpam-6492	184	6	α2r	α2r	PROPN
ejpam-6492	184	7	)	)	PUNCT
ejpam-6492	184	8	(	(	PUNCT
ejpam-6492	184	9	α	α	NOUN
ejpam-6492	184	10	−	−	PROPN
ejpam-6492	184	11	β)2	β)2	NOUN
ejpam-6492	184	12	=	=	PROPN
ejpam-6492	184	13	α2n	α2n	PROPN
ejpam-6492	184	14	(	(	PUNCT
ejpam-6492	184	15	α2r	α2r	PROPN
ejpam-6492	184	16	−	−	NOUN
ejpam-6492	184	17	β2r	β2r	PUNCT
ejpam-6492	184	18	)	)	PUNCT
ejpam-6492	184	19	−	−	PROPN
ejpam-6492	185	1	β2n	β2n	INTJ
ejpam-6492	185	2	(	(	PUNCT
ejpam-6492	185	3	α2r	α2r	PROPN
ejpam-6492	185	4	−	−	NOUN
ejpam-6492	185	5	β2r	β2r	NUM
ejpam-6492	185	6	)	)	PUNCT
ejpam-6492	185	7	(	(	PUNCT
ejpam-6492	185	8	α	α	NOUN
ejpam-6492	185	9	−	−	PROPN
ejpam-6492	185	10	β)2	β)2	PROPN
ejpam-6492	185	11	(	(	PUNCT
ejpam-6492	185	12	α2n	α2n	PROPN
ejpam-6492	185	13	−	−	PROPN
ejpam-6492	185	14	β2n	β2n	PUNCT
ejpam-6492	185	15	)	)	PUNCT
ejpam-6492	185	16	(	(	PUNCT
ejpam-6492	185	17	α2r	α2r	PROPN
ejpam-6492	185	18	−	−	NOUN
ejpam-6492	185	19	β2r	β2r	NUM
ejpam-6492	185	20	)	)	PUNCT
ejpam-6492	185	21	(	(	PUNCT
ejpam-6492	185	22	α	α	NOUN
ejpam-6492	185	23	−	−	NOUN
ejpam-6492	185	24	β)2	β)2	NOUN
ejpam-6492	185	25	=	=	X
ejpam-6492	185	26	(	(	PUNCT
ejpam-6492	185	27	α2n	α2n	PROPN
ejpam-6492	185	28	−	−	PROPN
ejpam-6492	185	29	β2n	β2n	PUNCT
ejpam-6492	185	30	)	)	PUNCT
ejpam-6492	186	1	α	α	INTJ
ejpam-6492	186	2	−	−	NOUN
ejpam-6492	187	1	β	β	X
ejpam-6492	187	2	(	(	PUNCT
ejpam-6492	187	3	α2r	α2r	PROPN
ejpam-6492	187	4	−	−	NOUN
ejpam-6492	187	5	β2r	β2r	NUM
ejpam-6492	187	6	)	)	PUNCT
ejpam-6492	188	1	α	α	PRON
ejpam-6492	188	2	−	−	NOUN
ejpam-6492	188	3	β	β	X
ejpam-6492	188	4	=	=	SYM
ejpam-6492	188	5	u2nu2r	u2nu2r	X
ejpam-6492	188	6	.	.	PUNCT
ejpam-6492	189	1	theorem	theorem	VERB
ejpam-6492	189	2	6	6	NUM
ejpam-6492	189	3	.	.	PUNCT
ejpam-6492	190	1	for	for	ADP
ejpam-6492	190	2	all	all	DET
ejpam-6492	190	3	n	n	NOUN
ejpam-6492	190	4	,	,	PUNCT
ejpam-6492	190	5	r	r	NOUN
ejpam-6492	190	6	∈	∈	PROPN
ejpam-6492	190	7	z	z	X
ejpam-6492	190	8	,	,	PUNCT
ejpam-6492	190	9	it	it	PRON
ejpam-6492	190	10	follows	follow	VERB
ejpam-6492	190	11	that	that	SCONJ
ejpam-6492	190	12	v	v	ADP
ejpam-6492	190	13	2	2	NUM
ejpam-6492	190	14	n+r	n+r	NUM
ejpam-6492	190	15	−	−	NOUN
ejpam-6492	190	16	q2rv	q2rv	NOUN
ejpam-6492	190	17	2	2	NUM
ejpam-6492	190	18	n−r	n−r	NOUN
ejpam-6492	190	19	=	=	PUNCT
ejpam-6492	190	20	∆u2nu2r	∆u2nu2r	NOUN
ejpam-6492	190	21	.	.	PUNCT
ejpam-6492	191	1	proof	proof	NOUN
ejpam-6492	191	2	.	.	PUNCT
ejpam-6492	192	1	v	v	ADP
ejpam-6492	192	2	2	2	NUM
ejpam-6492	192	3	n+r	n+r	NUM
ejpam-6492	192	4	−	−	NOUN
ejpam-6492	192	5	q2rv	q2rv	NOUN
ejpam-6492	192	6	2	2	NUM
ejpam-6492	192	7	n−r	n−r	NOUN
ejpam-6492	192	8	=	=	SYM
ejpam-6492	192	9	(	(	PUNCT
ejpam-6492	192	10	αn+r	αn+r	PROPN
ejpam-6492	192	11	+	+	CCONJ
ejpam-6492	192	12	βn+r	βn+r	PROPN
ejpam-6492	192	13	)	)	PUNCT
ejpam-6492	192	14	2	2	NUM
ejpam-6492	192	15	−	−	NOUN
ejpam-6492	192	16	(	(	PUNCT
ejpam-6492	192	17	−q)2r	−q)2r	INTJ
ejpam-6492	192	18	(	(	PUNCT
ejpam-6492	192	19	αn−r	αn−r	NOUN
ejpam-6492	192	20	+	+	CCONJ
ejpam-6492	192	21	βn−r)2	βn−r)2	PUNCT
ejpam-6492	193	1	=	=	PUNCT
ejpam-6492	194	1	[	[	PUNCT
ejpam-6492	194	2	α2n+2r	α2n+2r	NOUN
ejpam-6492	194	3	+	+	CCONJ
ejpam-6492	194	4	β2n+2r	β2n+2r	NOUN
ejpam-6492	194	5	+	+	CCONJ
ejpam-6492	194	6	2	2	NUM
ejpam-6492	194	7	(	(	PUNCT
ejpam-6492	194	8	αβ)n+r	αβ)n+r	NOUN
ejpam-6492	194	9	]	]	PUNCT
ejpam-6492	194	10	−	−	PROPN
ejpam-6492	194	11	(	(	PUNCT
ejpam-6492	194	12	αβ)2r	αβ)2r	X
ejpam-6492	194	13	[	[	PUNCT
ejpam-6492	194	14	α2n−2r	α2n−2r	NOUN
ejpam-6492	194	15	+	+	NOUN
ejpam-6492	194	16	β2n−2r	β2n−2r	NOUN
ejpam-6492	194	17	+	+	CCONJ
ejpam-6492	194	18	2	2	NUM
ejpam-6492	194	19	(	(	PUNCT
ejpam-6492	194	20	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	194	21	]	]	X
ejpam-6492	194	22	=	=	PUNCT
ejpam-6492	194	23	α2n+2r	α2n+2r	PROPN
ejpam-6492	194	24	+	+	CCONJ
ejpam-6492	194	25	β2n+2r	β2n+2r	NOUN
ejpam-6492	194	26	−	−	PROPN
ejpam-6492	194	27	α2nβ2r	α2nβ2r	X
ejpam-6492	194	28	−	−	NOUN
ejpam-6492	194	29	α2rβ2n	α2rβ2n	NUM
ejpam-6492	194	30	=	=	SYM
ejpam-6492	194	31	α2n	α2n	PROPN
ejpam-6492	194	32	(	(	PUNCT
ejpam-6492	194	33	α2r	α2r	PROPN
ejpam-6492	194	34	−	−	NOUN
ejpam-6492	194	35	β2r	β2r	PUNCT
ejpam-6492	194	36	)	)	PUNCT
ejpam-6492	195	1	+	+	CCONJ
ejpam-6492	195	2	β2n	β2n	PUNCT
ejpam-6492	195	3	(	(	PUNCT
ejpam-6492	195	4	β2r	β2r	PUNCT
ejpam-6492	195	5	−	−	PROPN
ejpam-6492	195	6	α2r	α2r	PROPN
ejpam-6492	195	7	)	)	PUNCT
ejpam-6492	196	1	=	=	SYM
ejpam-6492	196	2	α2n	α2n	PROPN
ejpam-6492	196	3	(	(	PUNCT
ejpam-6492	196	4	α2r	α2r	PROPN
ejpam-6492	196	5	−	−	NOUN
ejpam-6492	196	6	β2r	β2r	PUNCT
ejpam-6492	196	7	)	)	PUNCT
ejpam-6492	196	8	−	−	PROPN
ejpam-6492	197	1	β2n	β2n	INTJ
ejpam-6492	197	2	(	(	PUNCT
ejpam-6492	197	3	α2r	α2r	PROPN
ejpam-6492	197	4	−	−	NOUN
ejpam-6492	197	5	β2r	β2r	NUM
ejpam-6492	197	6	)	)	PUNCT
ejpam-6492	197	7	=	=	SYM
ejpam-6492	197	8	(	(	PUNCT
ejpam-6492	197	9	α2n	α2n	PROPN
ejpam-6492	197	10	−	−	PROPN
ejpam-6492	197	11	β2n	β2n	PUNCT
ejpam-6492	197	12	)	)	PUNCT
ejpam-6492	197	13	(	(	PUNCT
ejpam-6492	197	14	α2r	α2r	PROPN
ejpam-6492	197	15	−	−	NOUN
ejpam-6492	197	16	β2r	β2r	NUM
ejpam-6492	197	17	)	)	PUNCT
ejpam-6492	197	18	=	=	SYM
ejpam-6492	197	19	(	(	PUNCT
ejpam-6492	197	20	α2n	α2n	PROPN
ejpam-6492	197	21	−	−	PROPN
ejpam-6492	197	22	β2n	β2n	PUNCT
ejpam-6492	197	23	)	)	PUNCT
ejpam-6492	198	1	α	α	INTJ
ejpam-6492	198	2	−	−	NOUN
ejpam-6492	199	1	β	β	X
ejpam-6492	199	2	(	(	PUNCT
ejpam-6492	199	3	α2r	α2r	PROPN
ejpam-6492	199	4	−	−	NOUN
ejpam-6492	199	5	β2r	β2r	NUM
ejpam-6492	199	6	)	)	PUNCT
ejpam-6492	200	1	α	α	PRON
ejpam-6492	200	2	−	−	NOUN
ejpam-6492	201	1	β	β	X
ejpam-6492	201	2	(	(	PUNCT
ejpam-6492	201	3	α	α	NOUN
ejpam-6492	201	4	−	−	PROPN
ejpam-6492	201	5	β)2	β)2	NOUN
ejpam-6492	201	6	=	=	NOUN
ejpam-6492	201	7	∆u2nu2r	∆u2nu2r	NOUN
ejpam-6492	201	8	.	.	PUNCT
ejpam-6492	202	1	the	the	DET
ejpam-6492	202	2	following	following	ADJ
ejpam-6492	202	3	result	result	NOUN
ejpam-6492	202	4	is	be	AUX
ejpam-6492	202	5	a	a	DET
ejpam-6492	202	6	direct	direct	ADJ
ejpam-6492	202	7	consequence	consequence	NOUN
ejpam-6492	202	8	of	of	ADP
ejpam-6492	202	9	theorem	theorem	ADJ
ejpam-6492	202	10	5	5	NUM
ejpam-6492	202	11	and	and	CCONJ
ejpam-6492	202	12	theorem	theorem	VERB
ejpam-6492	202	13	6	6	NUM
ejpam-6492	202	14	.	.	PUNCT
ejpam-6492	202	15	corollary	corollary	ADJ
ejpam-6492	202	16	2	2	NUM
ejpam-6492	202	17	.	.	PUNCT
ejpam-6492	203	1	for	for	ADP
ejpam-6492	203	2	all	all	DET
ejpam-6492	203	3	n	n	NOUN
ejpam-6492	203	4	,	,	PUNCT
ejpam-6492	203	5	r	r	NOUN
ejpam-6492	203	6	∈	∈	PROPN
ejpam-6492	203	7	z	z	X
ejpam-6492	203	8	,	,	PUNCT
ejpam-6492	203	9	it	it	PRON
ejpam-6492	203	10	follows	follow	VERB
ejpam-6492	203	11	that	that	SCONJ
ejpam-6492	203	12	v	v	ADP
ejpam-6492	203	13	2	2	NUM
ejpam-6492	203	14	n+r	n+r	NUM
ejpam-6492	203	15	−	−	NOUN
ejpam-6492	203	16	q2rv	q2rv	NOUN
ejpam-6492	203	17	2	2	NUM
ejpam-6492	203	18	n−r	n−r	NOUN
ejpam-6492	203	19	=	=	SYM
ejpam-6492	203	20	∆	∆	X
ejpam-6492	203	21	[	[	PUNCT
ejpam-6492	203	22	u2	u2	PROPN
ejpam-6492	203	23	n+r	n+r	PROPN
ejpam-6492	204	1	−	−	ADP
ejpam-6492	204	2	q2ru2	q2ru2	NOUN
ejpam-6492	204	3	n−r	n−r	NOUN
ejpam-6492	204	4	]	]	PUNCT
ejpam-6492	204	5	.	.	PUNCT
ejpam-6492	205	1	b.	b.	PROPN
ejpam-6492	205	2	demirtürk	demirtürk	PROPN
ejpam-6492	205	3	,	,	PUNCT
ejpam-6492	205	4	n.	n.	NOUN
ejpam-6492	205	5	topal	topal	PROPN
ejpam-6492	205	6	/	/	SYM
ejpam-6492	205	7	eur	eur	PROPN
ejpam-6492	205	8	.	.	PUNCT
ejpam-6492	206	1	j.	j.	PROPN
ejpam-6492	206	2	pure	pure	PROPN
ejpam-6492	206	3	appl	appl	PROPN
ejpam-6492	206	4	.	.	PROPN
ejpam-6492	206	5	math	math	PROPN
ejpam-6492	206	6	,	,	PUNCT
ejpam-6492	206	7	18	18	NUM
ejpam-6492	206	8	(	(	PUNCT
ejpam-6492	206	9	3	3	NUM
ejpam-6492	206	10	)	)	PUNCT
ejpam-6492	206	11	(	(	PUNCT
ejpam-6492	206	12	2025	2025	NUM
ejpam-6492	206	13	)	)	PUNCT
ejpam-6492	206	14	,	,	PUNCT
ejpam-6492	206	15	6492	6492	NUM
ejpam-6492	206	16	10	10	NUM
ejpam-6492	206	17	of	of	ADP
ejpam-6492	206	18	22	22	NUM
ejpam-6492	206	19	3	3	NUM
ejpam-6492	206	20	.	.	PUNCT
ejpam-6492	206	21	generalized	generalize	VERB
ejpam-6492	206	22	fibonacci	fibonacci	PROPN
ejpam-6492	206	23	and	and	CCONJ
ejpam-6492	206	24	lucas	lucas	PROPN
ejpam-6492	206	25	quaternions	quaternions	PROPN
ejpam-6492	206	26	quaternions	quaternion	NOUN
ejpam-6492	206	27	are	be	AUX
ejpam-6492	206	28	four	four	NUM
ejpam-6492	206	29	-	-	PUNCT
ejpam-6492	206	30	dimensional	dimensional	ADJ
ejpam-6492	206	31	hypercomplex	hypercomplex	NOUN
ejpam-6492	206	32	numbers	number	NOUN
ejpam-6492	206	33	introduced	introduce	VERB
ejpam-6492	206	34	by	by	ADP
ejpam-6492	206	35	william	william	PROPN
ejpam-6492	206	36	rowan	rowan	PROPN
ejpam-6492	206	37	hamilton	hamilton	PROPN
ejpam-6492	206	38	in	in	ADP
ejpam-6492	206	39	the	the	DET
ejpam-6492	206	40	19th	19th	ADJ
ejpam-6492	206	41	century	century	NOUN
ejpam-6492	206	42	and	and	CCONJ
ejpam-6492	206	43	used	use	VERB
ejpam-6492	206	44	to	to	PART
ejpam-6492	206	45	represent	represent	VERB
ejpam-6492	206	46	transformations	transformation	NOUN
ejpam-6492	206	47	in	in	ADP
ejpam-6492	206	48	three	three	NUM
ejpam-6492	206	49	-	-	PUNCT
ejpam-6492	206	50	dimensional	dimensional	ADJ
ejpam-6492	206	51	space	space	NOUN
ejpam-6492	206	52	[	[	X
ejpam-6492	206	53	28	28	NUM
ejpam-6492	206	54	]	]	PUNCT
ejpam-6492	206	55	.	.	PUNCT
ejpam-6492	207	1	quaternions	quaternion	NOUN
ejpam-6492	207	2	,	,	PUNCT
ejpam-6492	207	3	whose	whose	DET
ejpam-6492	207	4	coefficients	coefficient	NOUN
ejpam-6492	207	5	are	be	AUX
ejpam-6492	207	6	fibonacci	fibonacci	NOUN
ejpam-6492	207	7	and	and	CCONJ
ejpam-6492	207	8	lucas	lucas	PROPN
ejpam-6492	207	9	numbers	number	NOUN
ejpam-6492	207	10	,	,	PUNCT
ejpam-6492	207	11	have	have	AUX
ejpam-6492	207	12	also	also	ADV
ejpam-6492	207	13	become	become	VERB
ejpam-6492	207	14	an	an	DET
ejpam-6492	207	15	interesting	interesting	ADJ
ejpam-6492	207	16	sequence	sequence	NOUN
ejpam-6492	207	17	with	with	ADP
ejpam-6492	207	18	many	many	ADJ
ejpam-6492	207	19	generalizations	generalization	NOUN
ejpam-6492	207	20	over	over	ADP
ejpam-6492	207	21	time	time	NOUN
ejpam-6492	207	22	.	.	PUNCT
ejpam-6492	208	1	let	let	VERB
ejpam-6492	208	2	us	we	PRON
ejpam-6492	208	3	start	start	VERB
ejpam-6492	208	4	by	by	ADP
ejpam-6492	208	5	giving	give	VERB
ejpam-6492	208	6	the	the	DET
ejpam-6492	208	7	fundamental	fundamental	ADJ
ejpam-6492	208	8	definition	definition	NOUN
ejpam-6492	208	9	of	of	ADP
ejpam-6492	208	10	a	a	DET
ejpam-6492	208	11	quaternion	quaternion	NOUN
ejpam-6492	208	12	.	.	PUNCT
ejpam-6492	209	1	definition	definition	NOUN
ejpam-6492	209	2	5	5	NUM
ejpam-6492	209	3	.	.	PUNCT
ejpam-6492	210	1	let	let	VERB
ejpam-6492	210	2	a	a	DET
ejpam-6492	210	3	,	,	PUNCT
ejpam-6492	210	4	b	b	NOUN
ejpam-6492	210	5	,	,	PUNCT
ejpam-6492	210	6	c	c	NOUN
ejpam-6492	210	7	,	,	PUNCT
ejpam-6492	210	8	d	d	PROPN
ejpam-6492	210	9	∈	∈	PROPN
ejpam-6492	210	10	r	r	NOUN
ejpam-6492	210	11	and	and	CCONJ
ejpam-6492	210	12	i2	i2	PROPN
ejpam-6492	210	13	=	=	SYM
ejpam-6492	210	14	−1	−1	PROPN
ejpam-6492	210	15	,	,	PUNCT
ejpam-6492	210	16	j2	j2	PROPN
ejpam-6492	210	17	=	=	SYM
ejpam-6492	210	18	−1	−1	PROPN
ejpam-6492	210	19	,	,	PUNCT
ejpam-6492	210	20	k2	k2	NOUN
ejpam-6492	210	21	=	=	SYM
ejpam-6492	210	22	−1	−1	NOUN
ejpam-6492	210	23	,	,	PUNCT
ejpam-6492	210	24	with	with	SCONJ
ejpam-6492	210	25	the	the	DET
ejpam-6492	210	26	multiplication	multiplication	NOUN
ejpam-6492	210	27	rules	rule	VERB
ejpam-6492	210	28	ij	ij	INTJ
ejpam-6492	210	29	=	=	SYM
ejpam-6492	210	30	k	k	PROPN
ejpam-6492	210	31	=	=	SYM
ejpam-6492	210	32	−ki	−ki	PROPN
ejpam-6492	210	33	,	,	PUNCT
ejpam-6492	210	34	jk	jk	X
ejpam-6492	211	1	=	=	PUNCT
ejpam-6492	211	2	i	i	PRON
ejpam-6492	211	3	=	=	SYM
ejpam-6492	211	4	−kj	−kj	PROPN
ejpam-6492	211	5	,	,	PUNCT
ejpam-6492	211	6	and	and	CCONJ
ejpam-6492	211	7	ki	ki	PROPN
ejpam-6492	212	1	=	=	SYM
ejpam-6492	212	2	j	j	PROPN
ejpam-6492	212	3	=	=	SYM
ejpam-6492	212	4	−ik	−ik	PROPN
ejpam-6492	212	5	.	.	PUNCT
ejpam-6492	213	1	a	a	DET
ejpam-6492	213	2	hypercomplex	hypercomplex	ADJ
ejpam-6492	213	3	number	number	NOUN
ejpam-6492	213	4	of	of	ADP
ejpam-6492	213	5	the	the	DET
ejpam-6492	213	6	form	form	NOUN
ejpam-6492	213	7	q	q	NOUN
ejpam-6492	213	8	=	=	PUNCT
ejpam-6492	213	9	a	a	DET
ejpam-6492	213	10	+	+	NOUN
ejpam-6492	213	11	bi	bi	NOUN
ejpam-6492	213	12	+	+	CCONJ
ejpam-6492	213	13	cj	cj	NOUN
ejpam-6492	214	1	+	+	CCONJ
ejpam-6492	214	2	dk	dk	PROPN
ejpam-6492	214	3	is	be	AUX
ejpam-6492	214	4	called	call	VERB
ejpam-6492	214	5	a	a	DET
ejpam-6492	214	6	quaternion	quaternion	NOUN
ejpam-6492	214	7	[	[	X
ejpam-6492	214	8	28	28	NUM
ejpam-6492	214	9	]	]	PUNCT
ejpam-6492	214	10	.	.	PUNCT
ejpam-6492	215	1	the	the	DET
ejpam-6492	215	2	elements	element	NOUN
ejpam-6492	215	3	1	1	NUM
ejpam-6492	215	4	,	,	PUNCT
ejpam-6492	215	5	i	i	PRON
ejpam-6492	215	6	,	,	PUNCT
ejpam-6492	215	7	j	j	PROPN
ejpam-6492	215	8	,	,	PUNCT
ejpam-6492	215	9	and	and	CCONJ
ejpam-6492	215	10	k	k	PROPN
ejpam-6492	215	11	are	be	AUX
ejpam-6492	215	12	called	call	VERB
ejpam-6492	215	13	the	the	DET
ejpam-6492	215	14	basis	basis	NOUN
ejpam-6492	215	15	or	or	CCONJ
ejpam-6492	215	16	characteristic	characteristic	ADJ
ejpam-6492	215	17	elements	element	NOUN
ejpam-6492	215	18	of	of	ADP
ejpam-6492	215	19	the	the	DET
ejpam-6492	215	20	quaternion	quaternion	NOUN
ejpam-6492	215	21	.	.	PUNCT
ejpam-6492	216	1	definition	definition	NOUN
ejpam-6492	216	2	6	6	NUM
ejpam-6492	216	3	.	.	PUNCT
ejpam-6492	217	1	let	let	VERB
ejpam-6492	217	2	a1	a1	NOUN
ejpam-6492	217	3	,	,	PUNCT
ejpam-6492	217	4	a2	a2	PROPN
ejpam-6492	217	5	,	,	PUNCT
ejpam-6492	217	6	a3	a3	NOUN
ejpam-6492	217	7	,	,	PUNCT
ejpam-6492	217	8	a4	a4	PROPN
ejpam-6492	217	9	,	,	PUNCT
ejpam-6492	217	10	b1	b1	NOUN
ejpam-6492	217	11	,	,	PUNCT
ejpam-6492	217	12	b2	b2	NOUN
ejpam-6492	217	13	,	,	PUNCT
ejpam-6492	217	14	b3	b3	NOUN
ejpam-6492	217	15	,	,	PUNCT
ejpam-6492	217	16	b4	b4	PROPN
ejpam-6492	217	17	∈	∈	PROPN
ejpam-6492	217	18	r.	r.	PROPN
ejpam-6492	217	19	for	for	ADP
ejpam-6492	217	20	two	two	NUM
ejpam-6492	217	21	quaternions	quaternion	NOUN
ejpam-6492	217	22	a	a	DET
ejpam-6492	217	23	=	=	NOUN
ejpam-6492	217	24	a1	a1	NOUN
ejpam-6492	217	25	+	+	CCONJ
ejpam-6492	217	26	a2i	a2i	NOUN
ejpam-6492	217	27	+	+	CCONJ
ejpam-6492	217	28	a3j	a3j	PROPN
ejpam-6492	217	29	+	+	CCONJ
ejpam-6492	217	30	a4k	a4k	PROPN
ejpam-6492	217	31	,	,	PUNCT
ejpam-6492	217	32	b	b	X
ejpam-6492	217	33	=	=	SYM
ejpam-6492	217	34	b1	b1	PROPN
ejpam-6492	217	35	+	+	CCONJ
ejpam-6492	217	36	b2i	b2i	PROPN
ejpam-6492	217	37	+	+	CCONJ
ejpam-6492	217	38	b3j	b3j	PROPN
ejpam-6492	217	39	+	+	CCONJ
ejpam-6492	217	40	b4k	b4k	PROPN
ejpam-6492	217	41	,	,	PUNCT
ejpam-6492	217	42	the	the	DET
ejpam-6492	217	43	addition	addition	NOUN
ejpam-6492	217	44	,	,	PUNCT
ejpam-6492	217	45	subtraction	subtraction	NOUN
ejpam-6492	217	46	,	,	PUNCT
ejpam-6492	217	47	and	and	CCONJ
ejpam-6492	217	48	multiplication	multiplication	NOUN
ejpam-6492	217	49	operations	operation	NOUN
ejpam-6492	217	50	in	in	ADP
ejpam-6492	217	51	the	the	DET
ejpam-6492	217	52	quaternion	quaternion	NOUN
ejpam-6492	217	53	algebra	algebra	NOUN
ejpam-6492	217	54	are	be	AUX
ejpam-6492	217	55	defined	define	VERB
ejpam-6492	217	56	as	as	SCONJ
ejpam-6492	217	57	follows	follow	VERB
ejpam-6492	217	58	:	:	PUNCT
ejpam-6492	217	59	a	a	DET
ejpam-6492	217	60	+	+	NUM
ejpam-6492	217	61	b	b	NOUN
ejpam-6492	217	62	=	=	SYM
ejpam-6492	217	63	(	(	PUNCT
ejpam-6492	217	64	a1	a1	NOUN
ejpam-6492	217	65	+	+	X
ejpam-6492	217	66	a2i	a2i	NOUN
ejpam-6492	217	67	+	+	CCONJ
ejpam-6492	217	68	a3j	a3j	PROPN
ejpam-6492	217	69	+	+	X
ejpam-6492	217	70	a4k	a4k	NOUN
ejpam-6492	217	71	)	)	PUNCT
ejpam-6492	218	1	+	+	CCONJ
ejpam-6492	218	2	(	(	PUNCT
ejpam-6492	218	3	b1	b1	NOUN
ejpam-6492	218	4	+	+	CCONJ
ejpam-6492	218	5	b2i	b2i	PROPN
ejpam-6492	218	6	+	+	CCONJ
ejpam-6492	218	7	b3j	b3j	PROPN
ejpam-6492	218	8	+	+	CCONJ
ejpam-6492	218	9	b4k	b4k	NOUN
ejpam-6492	218	10	)	)	PUNCT
ejpam-6492	218	11	=	=	PUNCT
ejpam-6492	218	12	(	(	PUNCT
ejpam-6492	218	13	a1	a1	NOUN
ejpam-6492	218	14	+	+	CCONJ
ejpam-6492	218	15	b1	b1	NOUN
ejpam-6492	218	16	)	)	PUNCT
ejpam-6492	218	17	+	+	CCONJ
ejpam-6492	218	18	(	(	PUNCT
ejpam-6492	218	19	a2	a2	PROPN
ejpam-6492	218	20	+	+	CCONJ
ejpam-6492	218	21	b2)i	b2)i	NOUN
ejpam-6492	218	22	+	+	CCONJ
ejpam-6492	218	23	(	(	PUNCT
ejpam-6492	218	24	a3	a3	NOUN
ejpam-6492	218	25	+	+	CCONJ
ejpam-6492	218	26	b3)j	b3)j	NOUN
ejpam-6492	218	27	+	+	CCONJ
ejpam-6492	218	28	(	(	PUNCT
ejpam-6492	218	29	a4	a4	NOUN
ejpam-6492	218	30	+	+	CCONJ
ejpam-6492	218	31	b4)k	b4)k	VERB
ejpam-6492	218	32	a	a	DET
ejpam-6492	218	33	−	−	PROPN
ejpam-6492	218	34	b	b	NOUN
ejpam-6492	218	35	=	=	PUNCT
ejpam-6492	218	36	(	(	PUNCT
ejpam-6492	218	37	a1	a1	NOUN
ejpam-6492	218	38	+	+	X
ejpam-6492	218	39	a2i	a2i	NOUN
ejpam-6492	218	40	+	+	CCONJ
ejpam-6492	218	41	a3j	a3j	PROPN
ejpam-6492	218	42	+	+	X
ejpam-6492	218	43	a4k	a4k	NOUN
ejpam-6492	218	44	)	)	PUNCT
ejpam-6492	218	45	−	−	PROPN
ejpam-6492	218	46	(	(	PUNCT
ejpam-6492	218	47	b1	b1	NOUN
ejpam-6492	218	48	+	+	CCONJ
ejpam-6492	218	49	b2i	b2i	PROPN
ejpam-6492	218	50	+	+	CCONJ
ejpam-6492	218	51	b3j	b3j	PROPN
ejpam-6492	218	52	+	+	CCONJ
ejpam-6492	218	53	b4k	b4k	NOUN
ejpam-6492	218	54	)	)	PUNCT
ejpam-6492	218	55	=	=	PUNCT
ejpam-6492	219	1	(	(	PUNCT
ejpam-6492	219	2	a1	a1	NOUN
ejpam-6492	219	3	−	−	PROPN
ejpam-6492	219	4	b1	b1	NOUN
ejpam-6492	219	5	)	)	PUNCT
ejpam-6492	220	1	+	+	CCONJ
ejpam-6492	220	2	(	(	PUNCT
ejpam-6492	220	3	a2	a2	PROPN
ejpam-6492	220	4	−	−	PROPN
ejpam-6492	220	5	b2)i	b2)i	NOUN
ejpam-6492	220	6	+	+	CCONJ
ejpam-6492	220	7	(	(	PUNCT
ejpam-6492	220	8	a3	a3	NOUN
ejpam-6492	220	9	−	−	PROPN
ejpam-6492	220	10	b3)j	b3)j	NOUN
ejpam-6492	220	11	+	+	CCONJ
ejpam-6492	220	12	(	(	PUNCT
ejpam-6492	220	13	a4	a4	NOUN
ejpam-6492	220	14	−	−	NOUN
ejpam-6492	220	15	b4)k	b4)k	ADJ
ejpam-6492	220	16	ab	ab	PROPN
ejpam-6492	220	17	=	=	PUNCT
ejpam-6492	220	18	(	(	PUNCT
ejpam-6492	220	19	a1	a1	NOUN
ejpam-6492	220	20	+	+	X
ejpam-6492	220	21	a2i	a2i	NOUN
ejpam-6492	220	22	+	+	CCONJ
ejpam-6492	220	23	a3j	a3j	PROPN
ejpam-6492	220	24	+	+	NUM
ejpam-6492	220	25	a4k)(b1	a4k)(b1	NOUN
ejpam-6492	220	26	+	+	CCONJ
ejpam-6492	220	27	b2i	b2i	VERB
ejpam-6492	220	28	+	+	CCONJ
ejpam-6492	220	29	b3j	b3j	PROPN
ejpam-6492	220	30	+	+	CCONJ
ejpam-6492	220	31	b4k	b4k	NOUN
ejpam-6492	220	32	)	)	PUNCT
ejpam-6492	220	33	=	=	SYM
ejpam-6492	220	34	a1(b1	a1(b1	ADJ
ejpam-6492	220	35	+	+	CCONJ
ejpam-6492	220	36	b2i	b2i	ADJ
ejpam-6492	220	37	+	+	CCONJ
ejpam-6492	220	38	b3j	b3j	PROPN
ejpam-6492	220	39	+	+	CCONJ
ejpam-6492	220	40	b4k	b4k	NOUN
ejpam-6492	220	41	)	)	PUNCT
ejpam-6492	220	42	+	+	CCONJ
ejpam-6492	220	43	a2i(b1	a2i(b1	VERB
ejpam-6492	220	44	+	+	CCONJ
ejpam-6492	220	45	b2i	b2i	ADJ
ejpam-6492	220	46	+	+	CCONJ
ejpam-6492	220	47	b3j	b3j	PROPN
ejpam-6492	220	48	+	+	CCONJ
ejpam-6492	220	49	b4k	b4k	NOUN
ejpam-6492	220	50	)	)	PUNCT
ejpam-6492	221	1	+	+	CCONJ
ejpam-6492	221	2	a3j(b1	a3j(b1	NOUN
ejpam-6492	221	3	+	+	CCONJ
ejpam-6492	221	4	b2i	b2i	ADJ
ejpam-6492	221	5	+	+	CCONJ
ejpam-6492	221	6	b3j	b3j	PROPN
ejpam-6492	221	7	+	+	CCONJ
ejpam-6492	221	8	b4k	b4k	NOUN
ejpam-6492	221	9	)	)	PUNCT
ejpam-6492	221	10	+	+	CCONJ
ejpam-6492	221	11	a4k(b1	a4k(b1	VERB
ejpam-6492	221	12	+	+	CCONJ
ejpam-6492	221	13	b2i	b2i	ADJ
ejpam-6492	221	14	+	+	CCONJ
ejpam-6492	221	15	b3j	b3j	PROPN
ejpam-6492	221	16	+	+	CCONJ
ejpam-6492	221	17	b4k	b4k	NOUN
ejpam-6492	221	18	)	)	PUNCT
ejpam-6492	221	19	=	=	SYM
ejpam-6492	221	20	(	(	PUNCT
ejpam-6492	221	21	a1b1	a1b1	PUNCT
ejpam-6492	221	22	+	+	NUM
ejpam-6492	221	23	a1b2i	a1b2i	NUM
ejpam-6492	221	24	+	+	NUM
ejpam-6492	221	25	a1b3j	a1b3j	SYM
ejpam-6492	221	26	+	+	NUM
ejpam-6492	221	27	a1b4k	a1b4k	NUM
ejpam-6492	221	28	)	)	PUNCT
ejpam-6492	221	29	+	+	CCONJ
ejpam-6492	221	30	(	(	PUNCT
ejpam-6492	221	31	a2b1i	a2b1i	X
ejpam-6492	221	32	+	+	CCONJ
ejpam-6492	221	33	a2b2ii	a2b2ii	PUNCT
ejpam-6492	221	34	+	+	CCONJ
ejpam-6492	221	35	a2b3ij	a2b3ij	X
ejpam-6492	221	36	+	+	NUM
ejpam-6492	221	37	a2b4ik	a2b4ik	ADJ
ejpam-6492	221	38	)	)	PUNCT
ejpam-6492	222	1	+	+	CCONJ
ejpam-6492	222	2	(	(	PUNCT
ejpam-6492	222	3	a3b1j	a3b1j	X
ejpam-6492	222	4	+	+	CCONJ
ejpam-6492	222	5	a3b2ji	a3b2ji	NOUN
ejpam-6492	222	6	+	+	ADJ
ejpam-6492	222	7	a3b3jj	a3b3jj	PUNCT
ejpam-6492	222	8	+	+	CCONJ
ejpam-6492	222	9	a3b4jk	a3b4jk	PROPN
ejpam-6492	222	10	)	)	PUNCT
ejpam-6492	222	11	+	+	CCONJ
ejpam-6492	222	12	(	(	PUNCT
ejpam-6492	222	13	a4b1k	a4b1k	SYM
ejpam-6492	222	14	+	+	CCONJ
ejpam-6492	222	15	a4b2ki	a4b2ki	X
ejpam-6492	222	16	+	+	CCONJ
ejpam-6492	222	17	a4b3kj	a4b3kj	X
ejpam-6492	222	18	+	+	X
ejpam-6492	222	19	a4b4kk	a4b4kk	X
ejpam-6492	222	20	)	)	PUNCT
ejpam-6492	222	21	=	=	SYM
ejpam-6492	223	1	(	(	PUNCT
ejpam-6492	223	2	a1b1	a1b1	INTJ
ejpam-6492	223	3	−	−	NOUN
ejpam-6492	223	4	a2b2	a2b2	ADP
ejpam-6492	223	5	−	−	NOUN
ejpam-6492	223	6	a3b3	a3b3	NOUN
ejpam-6492	223	7	−	−	NOUN
ejpam-6492	223	8	a4b4	a4b4	SYM
ejpam-6492	223	9	)	)	PUNCT
ejpam-6492	224	1	+	+	CCONJ
ejpam-6492	224	2	i(a1b2	i(a1b2	PROPN
ejpam-6492	224	3	+	+	CCONJ
ejpam-6492	224	4	a2b1	a2b1	X
ejpam-6492	224	5	+	+	CCONJ
ejpam-6492	224	6	a3b4	a3b4	PRON
ejpam-6492	224	7	−	−	NOUN
ejpam-6492	224	8	a4b3	a4b3	NOUN
ejpam-6492	224	9	)	)	PUNCT
ejpam-6492	225	1	+	+	NUM
ejpam-6492	225	2	j(a1b3	j(a1b3	NOUN
ejpam-6492	225	3	+	+	CCONJ
ejpam-6492	225	4	a3b1	a3b1	ADP
ejpam-6492	225	5	−	−	NOUN
ejpam-6492	225	6	a2b4	a2b4	NOUN
ejpam-6492	225	7	+	+	NUM
ejpam-6492	225	8	a4b2	a4b2	X
ejpam-6492	225	9	)	)	PUNCT
ejpam-6492	225	10	+	+	CCONJ
ejpam-6492	225	11	k(a1b4	k(a1b4	PROPN
ejpam-6492	225	12	+	+	CCONJ
ejpam-6492	225	13	a4b1	a4b1	ADJ
ejpam-6492	225	14	+	+	CCONJ
ejpam-6492	225	15	a2b3	a2b3	ADJ
ejpam-6492	225	16	−	−	NOUN
ejpam-6492	225	17	a3b2	a3b2	PUNCT
ejpam-6492	225	18	)	)	PUNCT
ejpam-6492	225	19	.	.	PUNCT
ejpam-6492	226	1	the	the	DET
ejpam-6492	226	2	quaternion	quaternion	NOUN
ejpam-6492	226	3	q̄	q̄	NOUN
ejpam-6492	226	4	=	=	NOUN
ejpam-6492	226	5	a	a	DET
ejpam-6492	226	6	−	−	PROPN
ejpam-6492	226	7	bi	bi	NOUN
ejpam-6492	226	8	−	−	PROPN
ejpam-6492	226	9	cj	cj	NOUN
ejpam-6492	227	1	−	−	NOUN
ejpam-6492	228	1	dk	dk	PROPN
ejpam-6492	228	2	is	be	AUX
ejpam-6492	228	3	called	call	VERB
ejpam-6492	228	4	the	the	DET
ejpam-6492	228	5	conjugate	conjugate	NOUN
ejpam-6492	228	6	of	of	ADP
ejpam-6492	228	7	the	the	DET
ejpam-6492	228	8	quaternion	quaternion	NOUN
ejpam-6492	228	9	q	q	NOUN
ejpam-6492	228	10	=	=	PUNCT
ejpam-6492	229	1	a	a	DET
ejpam-6492	229	2	+	+	NOUN
ejpam-6492	229	3	bi	bi	NOUN
ejpam-6492	229	4	+	+	CCONJ
ejpam-6492	229	5	cj	cj	NOUN
ejpam-6492	230	1	+	+	X
ejpam-6492	230	2	dk	dk	PROPN
ejpam-6492	230	3	.	.	PUNCT
ejpam-6492	231	1	the	the	DET
ejpam-6492	231	2	norm	norm	NOUN
ejpam-6492	231	3	of	of	ADP
ejpam-6492	231	4	q	q	PROPN
ejpam-6492	231	5	is	be	AUX
ejpam-6492	231	6	defined	define	VERB
ejpam-6492	231	7	by	by	ADP
ejpam-6492	231	8	n(q	n(q	PROPN
ejpam-6492	231	9	)	)	PUNCT
ejpam-6492	232	1	=	=	SYM
ejpam-6492	232	2	∥q∥	∥q∥	NOUN
ejpam-6492	232	3	=	=	SYM
ejpam-6492	232	4	√	√	ADP
ejpam-6492	232	5	qq̄	qq̄	NOUN
ejpam-6492	232	6	=	=	SYM
ejpam-6492	232	7	√	√	NOUN
ejpam-6492	232	8	a2	a2	PROPN
ejpam-6492	232	9	+	+	CCONJ
ejpam-6492	232	10	b2	b2	NOUN
ejpam-6492	232	11	+	+	CCONJ
ejpam-6492	232	12	c2	c2	PROPN
ejpam-6492	232	13	+	+	CCONJ
ejpam-6492	232	14	d2	d2	PROPN
ejpam-6492	232	15	,	,	PUNCT
ejpam-6492	232	16	[	[	X
ejpam-6492	232	17	29–31	29–31	NOUN
ejpam-6492	232	18	]	]	PUNCT
ejpam-6492	232	19	.	.	PUNCT
ejpam-6492	233	1	b.	b.	PROPN
ejpam-6492	233	2	demirtürk	demirtürk	PROPN
ejpam-6492	233	3	,	,	PUNCT
ejpam-6492	233	4	n.	n.	NOUN
ejpam-6492	233	5	topal	topal	PROPN
ejpam-6492	233	6	/	/	SYM
ejpam-6492	233	7	eur	eur	PROPN
ejpam-6492	233	8	.	.	PUNCT
ejpam-6492	234	1	j.	j.	PROPN
ejpam-6492	234	2	pure	pure	PROPN
ejpam-6492	234	3	appl	appl	PROPN
ejpam-6492	234	4	.	.	PROPN
ejpam-6492	234	5	math	math	PROPN
ejpam-6492	234	6	,	,	PUNCT
ejpam-6492	234	7	18	18	NUM
ejpam-6492	234	8	(	(	PUNCT
ejpam-6492	234	9	3	3	NUM
ejpam-6492	234	10	)	)	PUNCT
ejpam-6492	234	11	(	(	PUNCT
ejpam-6492	234	12	2025	2025	NUM
ejpam-6492	234	13	)	)	PUNCT
ejpam-6492	234	14	,	,	PUNCT
ejpam-6492	234	15	6492	6492	NUM
ejpam-6492	234	16	11	11	NUM
ejpam-6492	234	17	of	of	ADP
ejpam-6492	234	18	22	22	NUM
ejpam-6492	234	19	horadam	horadam	NOUN
ejpam-6492	234	20	,	,	PUNCT
ejpam-6492	234	21	in	in	ADP
ejpam-6492	234	22	[	[	PUNCT
ejpam-6492	234	23	32	32	NUM
ejpam-6492	234	24	]	]	PUNCT
ejpam-6492	234	25	,	,	PUNCT
ejpam-6492	234	26	defined	define	VERB
ejpam-6492	234	27	fibonacci	fibonacci	NOUN
ejpam-6492	234	28	and	and	CCONJ
ejpam-6492	234	29	lucas	lucas	PROPN
ejpam-6492	234	30	quaternions	quaternion	NOUN
ejpam-6492	234	31	as	as	ADP
ejpam-6492	234	32	qn	qn	NOUN
ejpam-6492	234	33	=	=	PROPN
ejpam-6492	234	34	fn	fn	PROPN
ejpam-6492	234	35	+	+	NUM
ejpam-6492	234	36	fn+1i	fn+1i	NOUN
ejpam-6492	234	37	+	+	CCONJ
ejpam-6492	234	38	fn+2j	fn+2j	NOUN
ejpam-6492	234	39	+	+	CCONJ
ejpam-6492	234	40	fn+3k	fn+3k	NOUN
ejpam-6492	234	41	,	,	PUNCT
ejpam-6492	234	42	kn	kn	PROPN
ejpam-6492	234	43	=	=	PUNCT
ejpam-6492	234	44	ln	ln	PROPN
ejpam-6492	235	1	+	+	NUM
ejpam-6492	235	2	ln+1i	ln+1i	ADJ
ejpam-6492	235	3	+	+	CCONJ
ejpam-6492	235	4	ln+2j	ln+2j	ADJ
ejpam-6492	235	5	+	+	CCONJ
ejpam-6492	235	6	ln+3k	ln+3k	NOUN
ejpam-6492	235	7	,	,	PUNCT
ejpam-6492	235	8	for	for	ADP
ejpam-6492	235	9	all	all	DET
ejpam-6492	235	10	n	n	PRON
ejpam-6492	235	11	∈	∈	PROPN
ejpam-6492	235	12	z	z	NOUN
ejpam-6492	235	13	,	,	PUNCT
ejpam-6492	235	14	where	where	SCONJ
ejpam-6492	235	15	fn	fn	NOUN
ejpam-6492	235	16	and	and	CCONJ
ejpam-6492	235	17	ln	ln	NOUN
ejpam-6492	235	18	are	be	AUX
ejpam-6492	235	19	fibonacci	fibonacci	NOUN
ejpam-6492	235	20	and	and	CCONJ
ejpam-6492	235	21	lucas	lucas	PROPN
ejpam-6492	235	22	numbers	number	NOUN
ejpam-6492	235	23	,	,	PUNCT
ejpam-6492	235	24	respectively	respectively	ADV
ejpam-6492	235	25	.	.	PUNCT
ejpam-6492	236	1	in	in	ADP
ejpam-6492	236	2	this	this	DET
ejpam-6492	236	3	section	section	NOUN
ejpam-6492	236	4	,	,	PUNCT
ejpam-6492	236	5	we	we	PRON
ejpam-6492	236	6	define	define	VERB
ejpam-6492	236	7	the	the	DET
ejpam-6492	236	8	(	(	PUNCT
ejpam-6492	236	9	p	p	X
ejpam-6492	236	10	,	,	PUNCT
ejpam-6492	236	11	q	q	ADJ
ejpam-6492	236	12	)	)	PUNCT
ejpam-6492	236	13	generalizations	generalization	NOUN
ejpam-6492	236	14	of	of	ADP
ejpam-6492	236	15	the	the	DET
ejpam-6492	236	16	fibonacci	fibonacci	NOUN
ejpam-6492	236	17	and	and	CCONJ
ejpam-6492	236	18	lucas	lucas	PROPN
ejpam-6492	236	19	quaternions	quaternion	NOUN
ejpam-6492	236	20	and	and	CCONJ
ejpam-6492	236	21	examine	examine	VERB
ejpam-6492	236	22	their	their	PRON
ejpam-6492	236	23	properties	property	NOUN
ejpam-6492	236	24	.	.	PUNCT
ejpam-6492	237	1	definition	definition	NOUN
ejpam-6492	237	2	7	7	NUM
ejpam-6492	237	3	.	.	PUNCT
ejpam-6492	238	1	let	let	VERB
ejpam-6492	238	2	un	un	PROPN
ejpam-6492	238	3	be	be	AUX
ejpam-6492	238	4	the	the	DET
ejpam-6492	238	5	n	n	ADV
ejpam-6492	238	6	-	-	PUNCT
ejpam-6492	238	7	th	th	X
ejpam-6492	238	8	generalized	generalize	VERB
ejpam-6492	238	9	fibonacci	fibonacci	NOUN
ejpam-6492	238	10	number	number	NOUN
ejpam-6492	238	11	.	.	PUNCT
ejpam-6492	239	1	for	for	ADP
ejpam-6492	239	2	all	all	DET
ejpam-6492	239	3	n	n	PRON
ejpam-6492	239	4	∈	∈	PROPN
ejpam-6492	239	5	n	n	CCONJ
ejpam-6492	239	6	,	,	PUNCT
ejpam-6492	239	7	quaternions	quaternion	NOUN
ejpam-6492	239	8	of	of	ADP
ejpam-6492	239	9	the	the	DET
ejpam-6492	239	10	form	form	NOUN
ejpam-6492	239	11	qn	qn	NOUN
ejpam-6492	239	12	=	=	PROPN
ejpam-6492	239	13	un	un	PROPN
ejpam-6492	239	14	+	+	CCONJ
ejpam-6492	239	15	un+1i	un+1i	PROPN
ejpam-6492	239	16	+	+	CCONJ
ejpam-6492	239	17	un+2j	un+2j	NOUN
ejpam-6492	239	18	+	+	CCONJ
ejpam-6492	239	19	un+3k	un+3k	PROPN
ejpam-6492	239	20	are	be	AUX
ejpam-6492	239	21	called	call	VERB
ejpam-6492	239	22	generalized	generalized	ADJ
ejpam-6492	239	23	fibonacci	fibonacci	NOUN
ejpam-6492	239	24	quaternions	quaternion	NOUN
ejpam-6492	239	25	.	.	PUNCT
ejpam-6492	240	1	let	let	VERB
ejpam-6492	240	2	vn	vn	PART
ejpam-6492	240	3	be	be	AUX
ejpam-6492	240	4	the	the	DET
ejpam-6492	240	5	n	n	ADV
ejpam-6492	240	6	-	-	PUNCT
ejpam-6492	240	7	th	th	X
ejpam-6492	240	8	generalized	generalize	VERB
ejpam-6492	240	9	lucas	lucas	NOUN
ejpam-6492	240	10	number	number	NOUN
ejpam-6492	240	11	.	.	PUNCT
ejpam-6492	241	1	for	for	ADP
ejpam-6492	241	2	all	all	DET
ejpam-6492	241	3	n	n	PRON
ejpam-6492	241	4	∈	∈	PROPN
ejpam-6492	241	5	n	n	CCONJ
ejpam-6492	241	6	,	,	PUNCT
ejpam-6492	241	7	quaternions	quaternion	NOUN
ejpam-6492	241	8	of	of	ADP
ejpam-6492	241	9	the	the	DET
ejpam-6492	241	10	form	form	NOUN
ejpam-6492	241	11	kn	kn	PROPN
ejpam-6492	241	12	=	=	PUNCT
ejpam-6492	241	13	vn	vn	PROPN
ejpam-6492	241	14	+	+	NUM
ejpam-6492	241	15	vn+1i	vn+1i	NOUN
ejpam-6492	241	16	+	+	CCONJ
ejpam-6492	241	17	vn+2j	vn+2j	PROPN
ejpam-6492	241	18	+	+	CCONJ
ejpam-6492	241	19	vn+3k	vn+3k	NOUN
ejpam-6492	241	20	are	be	AUX
ejpam-6492	241	21	called	call	VERB
ejpam-6492	241	22	generalized	generalized	ADJ
ejpam-6492	241	23	lucas	lucas	NOUN
ejpam-6492	241	24	quaternions	quaternion	NOUN
ejpam-6492	241	25	[	[	X
ejpam-6492	241	26	28	28	NUM
ejpam-6492	241	27	,	,	PUNCT
ejpam-6492	241	28	29	29	NUM
ejpam-6492	241	29	,	,	PUNCT
ejpam-6492	241	30	31	31	NUM
ejpam-6492	241	31	,	,	PUNCT
ejpam-6492	241	32	33	33	NUM
ejpam-6492	241	33	]	]	PUNCT
ejpam-6492	241	34	.	.	PUNCT
ejpam-6492	242	1	binet	binet	PROPN
ejpam-6492	242	2	fomulas	fomula	NOUN
ejpam-6492	242	3	for	for	ADP
ejpam-6492	242	4	generalized	generalized	ADJ
ejpam-6492	242	5	fibonacci	fibonacci	NOUN
ejpam-6492	242	6	and	and	CCONJ
ejpam-6492	242	7	lucas	lucas	PROPN
ejpam-6492	242	8	quaternions	quaternion	NOUN
ejpam-6492	242	9	will	will	AUX
ejpam-6492	242	10	be	be	AUX
ejpam-6492	242	11	given	give	VERB
ejpam-6492	242	12	in	in	ADP
ejpam-6492	242	13	theorem	theorem	ADJ
ejpam-6492	242	14	7	7	NUM
ejpam-6492	242	15	.	.	PUNCT
ejpam-6492	242	16	theorem	theorem	VERB
ejpam-6492	242	17	7	7	NUM
ejpam-6492	242	18	.	.	PUNCT
ejpam-6492	243	1	(	(	PUNCT
ejpam-6492	243	2	[	[	X
ejpam-6492	243	3	33	33	NUM
ejpam-6492	243	4	]	]	PUNCT
ejpam-6492	243	5	)	)	PUNCT
ejpam-6492	243	6	let	let	VERB
ejpam-6492	243	7	α̂	α̂	NOUN
ejpam-6492	243	8	=	=	SYM
ejpam-6492	243	9	1	1	NUM
ejpam-6492	243	10	+	+	CCONJ
ejpam-6492	243	11	αi	αi	VERB
ejpam-6492	244	1	+	+	CCONJ
ejpam-6492	244	2	α2j	α2j	X
ejpam-6492	244	3	+	+	CCONJ
ejpam-6492	244	4	α3k	α3k	X
ejpam-6492	244	5	and	and	CCONJ
ejpam-6492	244	6	β̂	β̂	PUNCT
ejpam-6492	244	7	=	=	SYM
ejpam-6492	244	8	1	1	NUM
ejpam-6492	244	9	+	+	CCONJ
ejpam-6492	244	10	βi	βi	PROPN
ejpam-6492	244	11	+	+	NUM
ejpam-6492	244	12	β2j	β2j	PROPN
ejpam-6492	244	13	+	+	CCONJ
ejpam-6492	244	14	β3k	β3k	NOUN
ejpam-6492	244	15	,	,	PUNCT
ejpam-6492	244	16	then	then	ADV
ejpam-6492	244	17	the	the	DET
ejpam-6492	244	18	generalized	generalized	ADJ
ejpam-6492	244	19	fibonacci	fibonacci	NOUN
ejpam-6492	244	20	and	and	CCONJ
ejpam-6492	244	21	lucas	lucas	PROPN
ejpam-6492	244	22	quaternions	quaternion	NOUN
ejpam-6492	244	23	are	be	AUX
ejpam-6492	244	24	given	give	VERB
ejpam-6492	244	25	by	by	ADP
ejpam-6492	244	26	qn	qn	NOUN
ejpam-6492	244	27	=	=	PROPN
ejpam-6492	244	28	αnα̂	αnα̂	PROPN
ejpam-6492	244	29	−	−	PROPN
ejpam-6492	244	30	βnβ̂	βnβ̂	PROPN
ejpam-6492	245	1	α	α	INTJ
ejpam-6492	245	2	−	−	NOUN
ejpam-6492	246	1	β	β	X
ejpam-6492	246	2	and	and	CCONJ
ejpam-6492	246	3	kn	kn	PROPN
ejpam-6492	246	4	=	=	PUNCT
ejpam-6492	246	5	αnα̂	αnα̂	PROPN
ejpam-6492	246	6	+	+	PROPN
ejpam-6492	246	7	βnβ̂	βnβ̂	PROPN
ejpam-6492	246	8	,	,	PUNCT
ejpam-6492	246	9	for	for	ADP
ejpam-6492	246	10	all	all	DET
ejpam-6492	246	11	n	n	PRON
ejpam-6492	246	12	∈	∈	NOUN
ejpam-6492	246	13	n.	n.	NOUN
ejpam-6492	246	14	proof	proof	NOUN
ejpam-6492	246	15	.	.	PUNCT
ejpam-6492	247	1	using	use	VERB
ejpam-6492	247	2	(	(	PUNCT
ejpam-6492	247	3	6	6	NUM
ejpam-6492	247	4	)	)	PUNCT
ejpam-6492	247	5	in	in	ADP
ejpam-6492	247	6	the	the	DET
ejpam-6492	247	7	expression	expression	NOUN
ejpam-6492	247	8	qn	qn	NOUN
ejpam-6492	247	9	=	=	PROPN
ejpam-6492	247	10	un	un	PROPN
ejpam-6492	247	11	+	+	CCONJ
ejpam-6492	247	12	un+1i	un+1i	PROPN
ejpam-6492	247	13	+	+	CCONJ
ejpam-6492	247	14	un+2j	un+2j	NOUN
ejpam-6492	247	15	+	+	CCONJ
ejpam-6492	247	16	un+3k	un+3k	ADJ
ejpam-6492	247	17	,	,	PUNCT
ejpam-6492	247	18	we	we	PRON
ejpam-6492	247	19	obtain	obtain	VERB
ejpam-6492	247	20	qn	qn	NOUN
ejpam-6492	247	21	=	=	NOUN
ejpam-6492	247	22	αn	αn	NOUN
ejpam-6492	248	1	−	−	NOUN
ejpam-6492	248	2	βn	βn	NOUN
ejpam-6492	248	3	α	α	NOUN
ejpam-6492	248	4	−	−	X
ejpam-6492	248	5	β	β	X
ejpam-6492	249	1	+	+	NOUN
ejpam-6492	249	2	αn+1	αn+1	NUM
ejpam-6492	249	3	−	−	NOUN
ejpam-6492	249	4	βn+1	βn+1	PUNCT
ejpam-6492	250	1	α	α	NOUN
ejpam-6492	250	2	−	−	NOUN
ejpam-6492	251	1	β	β	NOUN
ejpam-6492	251	2	i	i	PRON
ejpam-6492	252	1	+	+	PROPN
ejpam-6492	252	2	αn+2	αn+2	NUM
ejpam-6492	252	3	−	−	PROPN
ejpam-6492	252	4	βn+2	βn+2	NUM
ejpam-6492	253	1	α	α	NOUN
ejpam-6492	253	2	−	−	NOUN
ejpam-6492	253	3	β	β	SYM
ejpam-6492	253	4	j	j	PROPN
ejpam-6492	254	1	+	+	CCONJ
ejpam-6492	254	2	αn+3	αn+3	ADV
ejpam-6492	254	3	−	−	VERB
ejpam-6492	254	4	βn+3	βn+3	NOUN
ejpam-6492	255	1	α	α	NOUN
ejpam-6492	255	2	−	−	X
ejpam-6492	256	1	β	β	X
ejpam-6492	256	2	k	k	X
ejpam-6492	256	3	=	=	PUNCT
ejpam-6492	256	4	(	(	PUNCT
ejpam-6492	256	5	αn	αn	NOUN
ejpam-6492	256	6	+	+	CCONJ
ejpam-6492	256	7	αn+1i	αn+1i	ADJ
ejpam-6492	256	8	+	+	CCONJ
ejpam-6492	256	9	αn+2j	αn+2j	ADJ
ejpam-6492	256	10	+	+	CCONJ
ejpam-6492	256	11	αn+3k	αn+3k	NOUN
ejpam-6492	256	12	)	)	PUNCT
ejpam-6492	256	13	−	−	PROPN
ejpam-6492	256	14	(	(	PUNCT
ejpam-6492	256	15	βn	βn	NOUN
ejpam-6492	256	16	+	+	CCONJ
ejpam-6492	256	17	βn+1i	βn+1i	NOUN
ejpam-6492	256	18	+	+	NUM
ejpam-6492	256	19	βn+2j	βn+2j	NOUN
ejpam-6492	256	20	+	+	CCONJ
ejpam-6492	256	21	βn+3k	βn+3k	X
ejpam-6492	256	22	)	)	PUNCT
ejpam-6492	257	1	α	α	NOUN
ejpam-6492	257	2	−	−	NOUN
ejpam-6492	257	3	β	β	X
ejpam-6492	257	4	=	=	PUNCT
ejpam-6492	257	5	αn(1	αn(1	PROPN
ejpam-6492	257	6	+	+	NUM
ejpam-6492	257	7	αi	αi	VERB
ejpam-6492	257	8	+	+	CCONJ
ejpam-6492	257	9	α2j	α2j	PROPN
ejpam-6492	257	10	+	+	CCONJ
ejpam-6492	257	11	α3k	α3k	X
ejpam-6492	257	12	)	)	PUNCT
ejpam-6492	258	1	−	−	NOUN
ejpam-6492	258	2	βn(1	βn(1	NOUN
ejpam-6492	258	3	+	+	CCONJ
ejpam-6492	258	4	βi	βi	PROPN
ejpam-6492	258	5	+	+	NUM
ejpam-6492	258	6	β2j	β2j	PRON
ejpam-6492	258	7	+	+	CCONJ
ejpam-6492	258	8	β3k	β3k	X
ejpam-6492	258	9	)	)	PUNCT
ejpam-6492	258	10	α	α	NOUN
ejpam-6492	258	11	−	−	NOUN
ejpam-6492	259	1	β	β	NOUN
ejpam-6492	259	2	=	=	PUNCT
ejpam-6492	259	3	αnα̂	αnα̂	PROPN
ejpam-6492	259	4	−	−	PROPN
ejpam-6492	259	5	βnβ̂	βnβ̂	PROPN
ejpam-6492	260	1	α	α	INTJ
ejpam-6492	260	2	−	−	NOUN
ejpam-6492	261	1	β	β	X
ejpam-6492	261	2	.	.	PUNCT
ejpam-6492	262	1	similarly	similarly	ADV
ejpam-6492	262	2	,	,	PUNCT
ejpam-6492	262	3	using	use	VERB
ejpam-6492	262	4	(	(	PUNCT
ejpam-6492	262	5	6	6	NUM
ejpam-6492	262	6	)	)	PUNCT
ejpam-6492	262	7	in	in	ADP
ejpam-6492	262	8	the	the	DET
ejpam-6492	262	9	equation	equation	NOUN
ejpam-6492	262	10	kn	kn	NOUN
ejpam-6492	263	1	=	=	PUNCT
ejpam-6492	263	2	vn	vn	PROPN
ejpam-6492	263	3	+	+	NUM
ejpam-6492	263	4	vn+1i	vn+1i	NOUN
ejpam-6492	263	5	+	+	CCONJ
ejpam-6492	263	6	vn+2j	vn+2j	NOUN
ejpam-6492	263	7	+	+	CCONJ
ejpam-6492	263	8	vn+3k	vn+3k	NOUN
ejpam-6492	263	9	,	,	PUNCT
ejpam-6492	263	10	we	we	PRON
ejpam-6492	263	11	get	get	VERB
ejpam-6492	263	12	kn	kn	NOUN
ejpam-6492	263	13	=	=	PUNCT
ejpam-6492	263	14	(	(	PUNCT
ejpam-6492	263	15	αn	αn	NOUN
ejpam-6492	263	16	+	+	CCONJ
ejpam-6492	263	17	βn	βn	ADJ
ejpam-6492	263	18	)	)	PUNCT
ejpam-6492	264	1	+	+	CCONJ
ejpam-6492	264	2	(	(	PUNCT
ejpam-6492	264	3	αn+1	αn+1	NUM
ejpam-6492	264	4	+	+	CCONJ
ejpam-6492	264	5	βn+1)i	βn+1)i	ADJ
ejpam-6492	264	6	+	+	CCONJ
ejpam-6492	264	7	(	(	PUNCT
ejpam-6492	264	8	αn+2	αn+2	NUM
ejpam-6492	264	9	+	+	CCONJ
ejpam-6492	264	10	βn+2)j	βn+2)j	PUNCT
ejpam-6492	264	11	+	+	X
ejpam-6492	264	12	(	(	PUNCT
ejpam-6492	264	13	αn+3	αn+3	NUM
ejpam-6492	264	14	+	+	CCONJ
ejpam-6492	264	15	βn+3)k	βn+3)k	NOUN
ejpam-6492	264	16	=	=	SYM
ejpam-6492	264	17	αn(1	αn(1	PROPN
ejpam-6492	264	18	+	+	NUM
ejpam-6492	264	19	αi	αi	VERB
ejpam-6492	264	20	+	+	CCONJ
ejpam-6492	264	21	α2j	α2j	PROPN
ejpam-6492	264	22	+	+	CCONJ
ejpam-6492	264	23	α3k	α3k	X
ejpam-6492	264	24	)	)	PUNCT
ejpam-6492	265	1	+	+	NUM
ejpam-6492	265	2	βn(1	βn(1	X
ejpam-6492	265	3	+	+	CCONJ
ejpam-6492	265	4	βi	βi	PROPN
ejpam-6492	265	5	+	+	NUM
ejpam-6492	265	6	β2j	β2j	PROPN
ejpam-6492	265	7	+	+	NUM
ejpam-6492	265	8	β3k	β3k	X
ejpam-6492	265	9	)	)	PUNCT
ejpam-6492	265	10	b.	b.	PROPN
ejpam-6492	265	11	demirtürk	demirtürk	PROPN
ejpam-6492	265	12	,	,	PUNCT
ejpam-6492	265	13	n.	n.	NOUN
ejpam-6492	265	14	topal	topal	PROPN
ejpam-6492	265	15	/	/	SYM
ejpam-6492	265	16	eur	eur	PROPN
ejpam-6492	265	17	.	.	PUNCT
ejpam-6492	266	1	j.	j.	PROPN
ejpam-6492	266	2	pure	pure	PROPN
ejpam-6492	266	3	appl	appl	PROPN
ejpam-6492	266	4	.	.	PROPN
ejpam-6492	266	5	math	math	PROPN
ejpam-6492	266	6	,	,	PUNCT
ejpam-6492	266	7	18	18	NUM
ejpam-6492	266	8	(	(	PUNCT
ejpam-6492	266	9	3	3	NUM
ejpam-6492	266	10	)	)	PUNCT
ejpam-6492	266	11	(	(	PUNCT
ejpam-6492	266	12	2025	2025	NUM
ejpam-6492	266	13	)	)	PUNCT
ejpam-6492	266	14	,	,	PUNCT
ejpam-6492	266	15	6492	6492	NUM
ejpam-6492	266	16	12	12	NUM
ejpam-6492	266	17	of	of	ADP
ejpam-6492	266	18	22	22	NUM
ejpam-6492	266	19	=	=	SYM
ejpam-6492	266	20	αnα̂	αnα̂	PROPN
ejpam-6492	266	21	+	+	PROPN
ejpam-6492	266	22	βnβ̂.	βnβ̂.	NOUN
ejpam-6492	266	23	the	the	DET
ejpam-6492	266	24	generalized	generalize	VERB
ejpam-6492	266	25	negative	negative	ADJ
ejpam-6492	266	26	-	-	PUNCT
ejpam-6492	266	27	indexed	index	VERB
ejpam-6492	266	28	fibonacci	fibonacci	NOUN
ejpam-6492	266	29	and	and	CCONJ
ejpam-6492	266	30	lucas	lucas	PROPN
ejpam-6492	266	31	quaternions	quaternion	NOUN
ejpam-6492	266	32	were	be	AUX
ejpam-6492	266	33	defined	define	VERB
ejpam-6492	266	34	by	by	ADP
ejpam-6492	266	35	iakin	iakin	NOUN
ejpam-6492	266	36	in	in	ADP
ejpam-6492	266	37	[	[	X
ejpam-6492	266	38	31	31	NUM
ejpam-6492	266	39	]	]	PUNCT
ejpam-6492	266	40	as	as	ADP
ejpam-6492	266	41	:	:	PUNCT
ejpam-6492	266	42	q−n	q−n	PROPN
ejpam-6492	266	43	=	=	PUNCT
ejpam-6492	266	44	u−n	u−n	PROPN
ejpam-6492	266	45	+	+	NOUN
ejpam-6492	266	46	u−n+1i+u−n+2j	u−n+1i+u−n+2j	ADJ
ejpam-6492	266	47	+	+	ADJ
ejpam-6492	266	48	u−n+3k	u−n+3k	PROPN
ejpam-6492	266	49	and	and	CCONJ
ejpam-6492	266	50	k−n	k−n	PROPN
ejpam-6492	266	51	=	=	PUNCT
ejpam-6492	266	52	v−n	v−n	NOUN
ejpam-6492	266	53	+	+	CCONJ
ejpam-6492	266	54	v−n+1i+v−n+2j	v−n+1i+v−n+2j	PROPN
ejpam-6492	266	55	+	+	ADJ
ejpam-6492	266	56	v−n+3k	v−n+3k	NOUN
ejpam-6492	266	57	for	for	ADP
ejpam-6492	266	58	all	all	DET
ejpam-6492	266	59	n	n	PRON
ejpam-6492	266	60	∈	∈	PROPN
ejpam-6492	266	61	n.	n.	NOUN
ejpam-6492	266	62	for	for	ADP
ejpam-6492	266	63	example	example	NOUN
ejpam-6492	266	64	,	,	PUNCT
ejpam-6492	266	65	q−1	q−1	PROPN
ejpam-6492	266	66	=	=	PUNCT
ejpam-6492	266	67	u−1+u0i+u1j+u2k	u−1+u0i+u1j+u2k	NUM
ejpam-6492	266	68	=	=	SYM
ejpam-6492	266	69	1	1	NUM
ejpam-6492	266	70	q	q	NOUN
ejpam-6492	266	71	+0i+j+pk	+0i+j+pk	NOUN
ejpam-6492	266	72	,	,	PUNCT
ejpam-6492	266	73	k−1	k−1	PROPN
ejpam-6492	266	74	=	=	PUNCT
ejpam-6492	266	75	v−1+v0i+v1j+v2k	v−1+v0i+v1j+v2k	PROPN
ejpam-6492	266	76	=	=	SYM
ejpam-6492	266	77	−p	−p	NOUN
ejpam-6492	266	78	q	q	NOUN
ejpam-6492	267	1	+2i+pj+(p2	+2i+pj+(p2	PROPN
ejpam-6492	267	2	+	+	PROPN
ejpam-6492	267	3	2q)k	2q)k	NUM
ejpam-6492	267	4	.	.	PUNCT
ejpam-6492	268	1	according	accord	VERB
ejpam-6492	268	2	to	to	ADP
ejpam-6492	268	3	iyer	iyer	PROPN
ejpam-6492	268	4	[	[	X
ejpam-6492	268	5	34	34	NUM
ejpam-6492	268	6	]	]	PUNCT
ejpam-6492	268	7	,	,	PUNCT
ejpam-6492	268	8	the	the	DET
ejpam-6492	268	9	following	follow	VERB
ejpam-6492	268	10	properties	property	NOUN
ejpam-6492	268	11	involving	involve	VERB
ejpam-6492	268	12	the	the	DET
ejpam-6492	268	13	fibonacci	fibonacci	NOUN
ejpam-6492	268	14	and	and	CCONJ
ejpam-6492	268	15	lucas	lucas	PROPN
ejpam-6492	268	16	quaternions	quaternions	PROPN
ejpam-6492	268	17	hold	hold	VERB
ejpam-6492	268	18	.	.	PUNCT
ejpam-6492	269	1	here	here	ADV
ejpam-6492	269	2	,	,	PUNCT
ejpam-6492	269	3	qn	qn	PROPN
ejpam-6492	269	4	and	and	CCONJ
ejpam-6492	269	5	kn	kn	PROPN
ejpam-6492	269	6	are	be	AUX
ejpam-6492	269	7	the	the	DET
ejpam-6492	269	8	fibonacci	fibonacci	NOUN
ejpam-6492	269	9	and	and	CCONJ
ejpam-6492	269	10	lucas	lucas	PROPN
ejpam-6492	269	11	quaternions	quaternion	NOUN
ejpam-6492	269	12	,	,	PUNCT
ejpam-6492	269	13	not	not	PART
ejpam-6492	269	14	their	their	PRON
ejpam-6492	269	15	generalizations	generalization	NOUN
ejpam-6492	269	16	.	.	PUNCT
ejpam-6492	270	1	qn	qn	INTJ
ejpam-6492	270	2	−	−	PROPN
ejpam-6492	270	3	iqn+1	iqn+1	PROPN
ejpam-6492	270	4	−	−	PROPN
ejpam-6492	270	5	jqn+2	jqn+2	PROPN
ejpam-6492	270	6	−	−	PROPN
ejpam-6492	270	7	kqn+3	kqn+3	NOUN
ejpam-6492	270	8	=	=	SYM
ejpam-6492	270	9	ln+3	ln+3	PROPN
ejpam-6492	270	10	,	,	PUNCT
ejpam-6492	270	11	q2	q2	PROPN
ejpam-6492	270	12	n−1	n−1	PROPN
ejpam-6492	271	1	+	+	CCONJ
ejpam-6492	271	2	q2	q2	PROPN
ejpam-6492	271	3	n	n	NOUN
ejpam-6492	271	4	=	=	SYM
ejpam-6492	271	5	2q2n−1	2q2n−1	NUM
ejpam-6492	272	1	−	−	PROPN
ejpam-6492	272	2	3l2n+2	3l2n+2	NUM
ejpam-6492	272	3	,	,	PUNCT
ejpam-6492	272	4	q2	q2	NOUN
ejpam-6492	272	5	n+1	n+1	PROPN
ejpam-6492	272	6	−	−	PROPN
ejpam-6492	272	7	q2	q2	PROPN
ejpam-6492	272	8	n−1	n−1	PROPN
ejpam-6492	272	9	=	=	PUNCT
ejpam-6492	272	10	qnkn	qnkn	NOUN
ejpam-6492	272	11	=	=	PUNCT
ejpam-6492	272	12	(	(	PUNCT
ejpam-6492	272	13	2q2n	2q2n	NUM
ejpam-6492	272	14	−	−	PROPN
ejpam-6492	272	15	3l2n+3	3l2n+3	NUM
ejpam-6492	272	16	)	)	PUNCT
ejpam-6492	273	1	+	+	CCONJ
ejpam-6492	273	2	2(−1)n+1(q0	2(−1)n+1(q0	NUM
ejpam-6492	273	3	−	−	NOUN
ejpam-6492	273	4	3k	3k	NUM
ejpam-6492	273	5	)	)	PUNCT
ejpam-6492	273	6	,	,	PUNCT
ejpam-6492	273	7	qn−2qn−1	qn−2qn−1	PROPN
ejpam-6492	273	8	+	+	PROPN
ejpam-6492	273	9	qnqn+1	qnqn+1	PROPN
ejpam-6492	273	10	=	=	SYM
ejpam-6492	274	1	6fnqn−1	6fnqn−1	NUM
ejpam-6492	274	2	−	−	NUM
ejpam-6492	274	3	9f2n+2	9f2n+2	NUM
ejpam-6492	275	1	+	+	CCONJ
ejpam-6492	275	2	2(−1)n+1(q−1	2(−1)n+1(q−1	NUM
ejpam-6492	275	3	−	−	NOUN
ejpam-6492	275	4	3k	3k	NUM
ejpam-6492	275	5	)	)	PUNCT
ejpam-6492	275	6	,	,	PUNCT
ejpam-6492	275	7	qn−1qn+3	qn−1qn+3	NOUN
ejpam-6492	275	8	−	−	PROPN
ejpam-6492	275	9	q2	q2	NOUN
ejpam-6492	275	10	n+1	n+1	PROPN
ejpam-6492	275	11	=	=	PUNCT
ejpam-6492	275	12	(	(	PUNCT
ejpam-6492	275	13	−1)n[2	−1)n[2	X
ejpam-6492	275	14	+	+	NUM
ejpam-6492	275	15	4i	4i	NOUN
ejpam-6492	276	1	+	+	CCONJ
ejpam-6492	276	2	3j	3j	NUM
ejpam-6492	276	3	+	+	X
ejpam-6492	276	4	k	k	X
ejpam-6492	276	5	]	]	X
ejpam-6492	276	6	,	,	PUNCT
ejpam-6492	276	7	qn−1qn+1	qn−1qn+1	PROPN
ejpam-6492	276	8	−	−	PROPN
ejpam-6492	276	9	qn−2qn+2	qn−2qn+2	ADV
ejpam-6492	276	10	=	=	PUNCT
ejpam-6492	277	1	(	(	PUNCT
ejpam-6492	277	2	−1)n[2k0	−1)n[2k0	PROPN
ejpam-6492	277	3	−	−	PROPN
ejpam-6492	277	4	k	k	X
ejpam-6492	277	5	]	]	X
ejpam-6492	278	1	+	+	CCONJ
ejpam-6492	278	2	4(−1)n+1[q0	4(−1)n+1[q0	ADJ
ejpam-6492	278	3	−	−	PROPN
ejpam-6492	278	4	2k	2k	NUM
ejpam-6492	278	5	]	]	PUNCT
ejpam-6492	278	6	,	,	PUNCT
ejpam-6492	278	7	qn−3qn−2	qn−3qn−2	PROPN
ejpam-6492	278	8	+	+	PROPN
ejpam-6492	278	9	qnqn+1	qnqn+1	NOUN
ejpam-6492	278	10	=	=	SYM
ejpam-6492	278	11	4q2n−2	4q2n−2	NUM
ejpam-6492	279	1	−	−	PROPN
ejpam-6492	279	2	6l2n+1	6l2n+1	NUM
ejpam-6492	279	3	,	,	PUNCT
ejpam-6492	279	4	q2	q2	PROPN
ejpam-6492	279	5	n−1	n−1	PROPN
ejpam-6492	279	6	+	+	CCONJ
ejpam-6492	279	7	q2	q2	NOUN
ejpam-6492	279	8	n+1	n+1	PROPN
ejpam-6492	279	9	=	=	SYM
ejpam-6492	279	10	6fn+1qn−1	6fn+1qn−1	NUM
ejpam-6492	279	11	−	−	NOUN
ejpam-6492	279	12	9f2n+3	9f2n+3	NUM
ejpam-6492	279	13	+	+	CCONJ
ejpam-6492	279	14	2(−1)nq−2	2(−1)nq−2	NUM
ejpam-6492	279	15	,	,	PUNCT
ejpam-6492	279	16	qn+r	qn+r	PROPN
ejpam-6492	279	17	+	+	CCONJ
ejpam-6492	279	18	(	(	PUNCT
ejpam-6492	279	19	−1)rqn−r	−1)rqn−r	PROPN
ejpam-6492	279	20	=	=	SYM
ejpam-6492	279	21	qnlr	qnlr	NOUN
ejpam-6492	279	22	,	,	PUNCT
ejpam-6492	279	23	qn+1−rqn+1+r	qn+1−rqn+1+r	VERB
ejpam-6492	279	24	−	−	PROPN
ejpam-6492	279	25	q2	q2	NOUN
ejpam-6492	279	26	n+1	n+1	PROPN
ejpam-6492	280	1	=	=	SYM
ejpam-6492	281	1	(	(	PUNCT
ejpam-6492	281	2	−1)n−r[f	−1)n−r[f	NOUN
ejpam-6492	281	3	2	2	NUM
ejpam-6492	281	4	r	r	NOUN
ejpam-6492	281	5	k0	k0	NOUN
ejpam-6492	281	6	+	+	CCONJ
ejpam-6492	281	7	f2r(q0	f2r(q0	NOUN
ejpam-6492	281	8	−	−	NOUN
ejpam-6492	281	9	3r	3r	NUM
ejpam-6492	281	10	)	)	PUNCT
ejpam-6492	281	11	]	]	PUNCT
ejpam-6492	281	12	,	,	PUNCT
ejpam-6492	281	13	qn+rln+r	qn+rln+r	PROPN
ejpam-6492	281	14	=	=	PUNCT
ejpam-6492	281	15	q2n+2r	q2n+2r	ADP
ejpam-6492	281	16	+	+	PROPN
ejpam-6492	281	17	(	(	PUNCT
ejpam-6492	281	18	−1)n+rq0	−1)n+rq0	NUM
ejpam-6492	281	19	,	,	PUNCT
ejpam-6492	281	20	qn−rln−r	qn−rln−r	PROPN
ejpam-6492	281	21	=	=	X
ejpam-6492	281	22	q2n−2r	q2n−2r	ADV
ejpam-6492	281	23	+	+	CCONJ
ejpam-6492	281	24	(	(	PUNCT
ejpam-6492	281	25	−1)n+rq0	−1)n+rq0	NUM
ejpam-6492	281	26	,	,	PUNCT
ejpam-6492	281	27	qn+rln+r	qn+rln+r	PROPN
ejpam-6492	281	28	+	+	CCONJ
ejpam-6492	281	29	qn−rln−r	qn−rln−r	PROPN
ejpam-6492	281	30	=	=	X
ejpam-6492	281	31	q2nl2r	q2nl2r	NOUN
ejpam-6492	281	32	+	+	CCONJ
ejpam-6492	281	33	2(−1)n+rq0	2(−1)n+rq0	NUM
ejpam-6492	281	34	,	,	PUNCT
ejpam-6492	281	35	qn+rln+r	qn+rln+r	NOUN
ejpam-6492	281	36	−	−	PROPN
ejpam-6492	281	37	qn−rln−r	qn−rln−r	PROPN
ejpam-6492	281	38	=	=	SYM
ejpam-6492	281	39	f2rk2n	f2rk2n	ADP
ejpam-6492	281	40	,	,	PUNCT
ejpam-6492	281	41	qn+rln−r	qn+rln−r	NOUN
ejpam-6492	281	42	=	=	SYM
ejpam-6492	281	43	q2n	q2n	PROPN
ejpam-6492	281	44	+	+	CCONJ
ejpam-6492	281	45	(	(	PUNCT
ejpam-6492	281	46	−1)n−rq2r	−1)n−rq2r	INTJ
ejpam-6492	281	47	.	.	PUNCT
ejpam-6492	282	1	before	before	ADP
ejpam-6492	282	2	presenting	present	VERB
ejpam-6492	282	3	the	the	DET
ejpam-6492	282	4	catalan	catalan	NOUN
ejpam-6492	282	5	’s	’s	PART
ejpam-6492	282	6	identity	identity	NOUN
ejpam-6492	282	7	for	for	ADP
ejpam-6492	282	8	generalized	generalized	ADJ
ejpam-6492	282	9	fibonacci	fibonacci	NOUN
ejpam-6492	282	10	and	and	CCONJ
ejpam-6492	282	11	lucas	lucas	PROPN
ejpam-6492	282	12	quaternions	quaternion	NOUN
ejpam-6492	282	13	,	,	PUNCT
ejpam-6492	282	14	we	we	PRON
ejpam-6492	282	15	will	will	AUX
ejpam-6492	282	16	first	first	ADV
ejpam-6492	282	17	prove	prove	VERB
ejpam-6492	282	18	some	some	DET
ejpam-6492	282	19	preliminary	preliminary	ADJ
ejpam-6492	282	20	lemmas	lemma	NOUN
ejpam-6492	282	21	.	.	PUNCT
ejpam-6492	283	1	lemma	lemma	PROPN
ejpam-6492	283	2	1	1	X
ejpam-6492	283	3	.	.	PUNCT
ejpam-6492	284	1	let	let	VERB
ejpam-6492	284	2	α̂	α̂	NOUN
ejpam-6492	284	3	=	=	SYM
ejpam-6492	284	4	1	1	NUM
ejpam-6492	284	5	+	+	CCONJ
ejpam-6492	284	6	αi	αi	VERB
ejpam-6492	285	1	+	+	CCONJ
ejpam-6492	285	2	α2j	α2j	PROPN
ejpam-6492	285	3	+	+	CCONJ
ejpam-6492	285	4	α3k	α3k	ADP
ejpam-6492	285	5	,	,	PUNCT
ejpam-6492	285	6	β̂	β̂	PUNCT
ejpam-6492	285	7	=	=	SYM
ejpam-6492	285	8	1	1	NUM
ejpam-6492	285	9	+	+	CCONJ
ejpam-6492	285	10	βi	βi	PROPN
ejpam-6492	285	11	+	+	NUM
ejpam-6492	285	12	β2j	β2j	PROPN
ejpam-6492	285	13	+	+	CCONJ
ejpam-6492	285	14	β3k	β3k	PROPN
ejpam-6492	285	15	,	,	PUNCT
ejpam-6492	285	16	and	and	CCONJ
ejpam-6492	285	17	define	define	VERB
ejpam-6492	285	18	a	a	DET
ejpam-6492	285	19	=	=	SYM
ejpam-6492	285	20	k0	k0	PROPN
ejpam-6492	285	21	−	−	PROPN
ejpam-6492	285	22	(	(	PUNCT
ejpam-6492	285	23	1	1	NUM
ejpam-6492	285	24	−	−	NUM
ejpam-6492	285	25	q)(1	q)(1	PROPN
ejpam-6492	285	26	+	+	X
ejpam-6492	285	27	q2	q2	NOUN
ejpam-6492	285	28	)	)	PUNCT
ejpam-6492	285	29	,	,	PUNCT
ejpam-6492	285	30	b	b	X
ejpam-6492	285	31	=	=	SYM
ejpam-6492	285	32	(	(	PUNCT
ejpam-6492	285	33	−q)i	−q)i	NOUN
ejpam-6492	285	34	+	+	CCONJ
ejpam-6492	285	35	(	(	PUNCT
ejpam-6492	285	36	−p)j	−p)j	NOUN
ejpam-6492	286	1	+	+	CCONJ
ejpam-6492	286	2	k	k	NOUN
ejpam-6492	286	3	,	,	PUNCT
ejpam-6492	286	4	then	then	ADV
ejpam-6492	286	5	the	the	DET
ejpam-6492	286	6	following	follow	VERB
ejpam-6492	286	7	identities	identity	NOUN
ejpam-6492	286	8	hold	hold	VERB
ejpam-6492	286	9	α̂β̂	α̂β̂	X
ejpam-6492	286	10	=	=	PUNCT
ejpam-6492	286	11	a	a	DET
ejpam-6492	286	12	+	+	NUM
ejpam-6492	286	13	qb	qb	PROPN
ejpam-6492	286	14	√	√	ADJ
ejpam-6492	286	15	∆	∆	PROPN
ejpam-6492	286	16	,	,	PUNCT
ejpam-6492	286	17	(	(	PUNCT
ejpam-6492	286	18	10	10	NUM
ejpam-6492	286	19	)	)	PUNCT
ejpam-6492	286	20	β̂α̂	β̂α̂	PUNCT
ejpam-6492	287	1	=	=	PUNCT
ejpam-6492	287	2	a	a	DET
ejpam-6492	287	3	−	−	PROPN
ejpam-6492	287	4	qb	qb	PROPN
ejpam-6492	287	5	√	√	PROPN
ejpam-6492	287	6	∆.	∆.	NOUN
ejpam-6492	287	7	(	(	PUNCT
ejpam-6492	287	8	11	11	NUM
ejpam-6492	287	9	)	)	PUNCT
ejpam-6492	287	10	b.	b.	NOUN
ejpam-6492	287	11	demirtürk	demirtürk	PROPN
ejpam-6492	287	12	,	,	PUNCT
ejpam-6492	287	13	n.	n.	NOUN
ejpam-6492	287	14	topal	topal	PROPN
ejpam-6492	287	15	/	/	SYM
ejpam-6492	287	16	eur	eur	PROPN
ejpam-6492	287	17	.	.	PUNCT
ejpam-6492	288	1	j.	j.	PROPN
ejpam-6492	288	2	pure	pure	PROPN
ejpam-6492	288	3	appl	appl	PROPN
ejpam-6492	288	4	.	.	PROPN
ejpam-6492	288	5	math	math	PROPN
ejpam-6492	288	6	,	,	PUNCT
ejpam-6492	288	7	18	18	NUM
ejpam-6492	288	8	(	(	PUNCT
ejpam-6492	288	9	3	3	NUM
ejpam-6492	288	10	)	)	PUNCT
ejpam-6492	288	11	(	(	PUNCT
ejpam-6492	288	12	2025	2025	NUM
ejpam-6492	288	13	)	)	PUNCT
ejpam-6492	288	14	,	,	PUNCT
ejpam-6492	288	15	6492	6492	NUM
ejpam-6492	288	16	13	13	NUM
ejpam-6492	288	17	of	of	ADP
ejpam-6492	288	18	22	22	NUM
ejpam-6492	288	19	proof	proof	NOUN
ejpam-6492	288	20	.	.	PUNCT
ejpam-6492	289	1	using	use	VERB
ejpam-6492	289	2	quaternion	quaternion	NOUN
ejpam-6492	289	3	multiplication	multiplication	NOUN
ejpam-6492	289	4	and	and	CCONJ
ejpam-6492	289	5	the	the	DET
ejpam-6492	289	6	identity	identity	NOUN
ejpam-6492	289	7	αβ	αβ	NOUN
ejpam-6492	289	8	=	=	SYM
ejpam-6492	289	9	−q	−q	NOUN
ejpam-6492	289	10	,	,	PUNCT
ejpam-6492	289	11	we	we	PRON
ejpam-6492	289	12	compute	compute	VERB
ejpam-6492	289	13	α̂β̂	α̂β̂	X
ejpam-6492	289	14	=	=	PUNCT
ejpam-6492	289	15	(	(	PUNCT
ejpam-6492	289	16	1	1	NUM
ejpam-6492	289	17	+	+	NUM
ejpam-6492	289	18	αi	αi	VERB
ejpam-6492	290	1	+	+	CCONJ
ejpam-6492	290	2	α2j	α2j	NUM
ejpam-6492	290	3	+	+	CCONJ
ejpam-6492	290	4	α3k)(1	α3k)(1	NUM
ejpam-6492	290	5	+	+	PRON
ejpam-6492	290	6	βi	βi	PROPN
ejpam-6492	290	7	+	+	NUM
ejpam-6492	290	8	β2j	β2j	PRON
ejpam-6492	290	9	+	+	CCONJ
ejpam-6492	290	10	β3k	β3k	PUNCT
ejpam-6492	290	11	)	)	PUNCT
ejpam-6492	291	1	=	=	PUNCT
ejpam-6492	292	1	[	[	X
ejpam-6492	292	2	2+(α+β)i+(α2+β2)j+(α3+β3)k]−[1+αβ+α2β2+α3β3]−αβ(α−β)[αβi−(α+β)j+k	2+(α+β)i+(α2+β2)j+(α3+β3)k]−[1+αβ+α2β2+α3β3]−αβ(α−β)[αβi−(α+β)j+k	X
ejpam-6492	292	3	]	]	X
ejpam-6492	292	4	=	=	PUNCT
ejpam-6492	292	5	k0	k0	PROPN
ejpam-6492	292	6	−	−	PROPN
ejpam-6492	292	7	(	(	PUNCT
ejpam-6492	292	8	1	1	NUM
ejpam-6492	292	9	−	−	NUM
ejpam-6492	292	10	q)(1	q)(1	PROPN
ejpam-6492	292	11	+	+	X
ejpam-6492	292	12	q2	q2	NOUN
ejpam-6492	292	13	)	)	PUNCT
ejpam-6492	293	1	+	+	CCONJ
ejpam-6492	293	2	q	q	PUNCT
ejpam-6492	293	3	√	√	NUM
ejpam-6492	293	4	∆[(−q)i	∆[(−q)i	NOUN
ejpam-6492	293	5	+	+	CCONJ
ejpam-6492	293	6	(	(	PUNCT
ejpam-6492	293	7	−p)j	−p)j	NOUN
ejpam-6492	293	8	+	+	CCONJ
ejpam-6492	293	9	k	k	X
ejpam-6492	293	10	]	]	X
ejpam-6492	293	11	=	=	PUNCT
ejpam-6492	293	12	a	a	DET
ejpam-6492	293	13	+	+	NUM
ejpam-6492	293	14	qb	qb	PROPN
ejpam-6492	293	15	√	√	ADJ
ejpam-6492	293	16	∆	∆	PROPN
ejpam-6492	293	17	,	,	PUNCT
ejpam-6492	293	18	which	which	PRON
ejpam-6492	293	19	completes	complete	VERB
ejpam-6492	293	20	the	the	DET
ejpam-6492	293	21	proof	proof	NOUN
ejpam-6492	293	22	.	.	PUNCT
ejpam-6492	294	1	similarly	similarly	ADV
ejpam-6492	294	2	,	,	PUNCT
ejpam-6492	294	3	it	it	PRON
ejpam-6492	294	4	can	can	AUX
ejpam-6492	294	5	be	be	AUX
ejpam-6492	294	6	shown	show	VERB
ejpam-6492	294	7	that	that	SCONJ
ejpam-6492	294	8	β̂α̂	β̂α̂	PUNCT
ejpam-6492	294	9	=	=	PUNCT
ejpam-6492	294	10	a	a	DET
ejpam-6492	294	11	−	−	PROPN
ejpam-6492	294	12	qb	qb	PROPN
ejpam-6492	294	13	√	√	PROPN
ejpam-6492	294	14	∆.	∆.	X
ejpam-6492	294	15	lemma	lemma	PROPN
ejpam-6492	294	16	2	2	X
ejpam-6492	294	17	.	.	PUNCT
ejpam-6492	295	1	let	let	VERB
ejpam-6492	295	2	α̂	α̂	NOUN
ejpam-6492	295	3	=	=	SYM
ejpam-6492	295	4	1	1	NUM
ejpam-6492	295	5	+	+	CCONJ
ejpam-6492	295	6	αi	αi	VERB
ejpam-6492	296	1	+	+	CCONJ
ejpam-6492	296	2	α2j	α2j	PROPN
ejpam-6492	296	3	+	+	CCONJ
ejpam-6492	296	4	α3k	α3k	ADP
ejpam-6492	296	5	,	,	PUNCT
ejpam-6492	296	6	β̂	β̂	PUNCT
ejpam-6492	296	7	=	=	SYM
ejpam-6492	296	8	1	1	NUM
ejpam-6492	296	9	+	+	CCONJ
ejpam-6492	296	10	βi	βi	PROPN
ejpam-6492	296	11	+	+	NUM
ejpam-6492	296	12	β2j	β2j	PROPN
ejpam-6492	296	13	+	+	CCONJ
ejpam-6492	296	14	β3k	β3k	PROPN
ejpam-6492	296	15	,	,	PUNCT
ejpam-6492	296	16	and	and	CCONJ
ejpam-6492	296	17	r	r	NOUN
ejpam-6492	296	18	∈	∈	PROPN
ejpam-6492	296	19	n.	n.	NOUN
ejpam-6492	296	20	then	then	ADV
ejpam-6492	296	21	the	the	DET
ejpam-6492	296	22	following	follow	VERB
ejpam-6492	296	23	identities	identity	NOUN
ejpam-6492	296	24	hold	hold	VERB
ejpam-6492	296	25	:	:	PUNCT
ejpam-6492	296	26	α̂β̂βr	α̂β̂βr	NOUN
ejpam-6492	296	27	−	−	PROPN
ejpam-6492	296	28	β̂α̂αr	β̂α̂αr	NOUN
ejpam-6492	296	29	α	α	NOUN
ejpam-6492	296	30	−	−	NOUN
ejpam-6492	296	31	β	β	NOUN
ejpam-6492	296	32	=	=	SYM
ejpam-6492	296	33	qbvr	qbvr	NOUN
ejpam-6492	296	34	−	−	PROPN
ejpam-6492	296	35	aur	aur	PROPN
ejpam-6492	296	36	,	,	PUNCT
ejpam-6492	296	37	(	(	PUNCT
ejpam-6492	296	38	12	12	NUM
ejpam-6492	296	39	)	)	PUNCT
ejpam-6492	296	40	α̂β̂βr	α̂β̂βr	NOUN
ejpam-6492	296	41	+	+	NOUN
ejpam-6492	296	42	β̂α̂αr	β̂α̂αr	NOUN
ejpam-6492	296	43	=	=	SYM
ejpam-6492	296	44	avr	avr	PROPN
ejpam-6492	296	45	−	−	PROPN
ejpam-6492	296	46	q∆bur	q∆bur	PROPN
ejpam-6492	296	47	,	,	PUNCT
ejpam-6492	296	48	(	(	PUNCT
ejpam-6492	296	49	13	13	NUM
ejpam-6492	296	50	)	)	PUNCT
ejpam-6492	296	51	where	where	SCONJ
ejpam-6492	296	52	a	a	DET
ejpam-6492	296	53	=	=	X
ejpam-6492	296	54	k0	k0	PROPN
ejpam-6492	296	55	−	−	PROPN
ejpam-6492	296	56	(	(	PUNCT
ejpam-6492	296	57	1	1	NUM
ejpam-6492	296	58	−	−	NUM
ejpam-6492	296	59	q)(1	q)(1	PROPN
ejpam-6492	296	60	+	+	X
ejpam-6492	296	61	q2	q2	NOUN
ejpam-6492	296	62	)	)	PUNCT
ejpam-6492	296	63	and	and	CCONJ
ejpam-6492	296	64	b	b	X
ejpam-6492	296	65	=	=	SYM
ejpam-6492	296	66	(	(	PUNCT
ejpam-6492	296	67	−q)i	−q)i	NOUN
ejpam-6492	296	68	+	+	CCONJ
ejpam-6492	296	69	(	(	PUNCT
ejpam-6492	296	70	−p)j	−p)j	NOUN
ejpam-6492	296	71	+	+	CCONJ
ejpam-6492	296	72	k.	k.	PROPN
ejpam-6492	296	73	proof	proof	NOUN
ejpam-6492	296	74	.	.	PUNCT
ejpam-6492	297	1	using	use	VERB
ejpam-6492	297	2	equations	equation	NOUN
ejpam-6492	297	3	(	(	PUNCT
ejpam-6492	297	4	10	10	NUM
ejpam-6492	297	5	)	)	PUNCT
ejpam-6492	297	6	and	and	CCONJ
ejpam-6492	297	7	(	(	PUNCT
ejpam-6492	297	8	11	11	NUM
ejpam-6492	297	9	)	)	PUNCT
ejpam-6492	297	10	,	,	PUNCT
ejpam-6492	297	11	we	we	PRON
ejpam-6492	297	12	get	get	VERB
ejpam-6492	297	13	α̂β̂βr	α̂β̂βr	NOUN
ejpam-6492	297	14	−	−	NOUN
ejpam-6492	297	15	β̂α̂αr	β̂α̂αr	NOUN
ejpam-6492	298	1	α	α	NOUN
ejpam-6492	298	2	−	−	NOUN
ejpam-6492	298	3	β	β	X
ejpam-6492	298	4	=	=	SYM
ejpam-6492	298	5	(	(	PUNCT
ejpam-6492	298	6	a	a	DET
ejpam-6492	298	7	+	+	X
ejpam-6492	298	8	qb	qb	PROPN
ejpam-6492	298	9	√	√	PROPN
ejpam-6492	298	10	∆)βr	∆)βr	PROPN
ejpam-6492	298	11	−	−	PROPN
ejpam-6492	298	12	(	(	PUNCT
ejpam-6492	298	13	a	a	DET
ejpam-6492	298	14	−	−	PROPN
ejpam-6492	298	15	qb	qb	PROPN
ejpam-6492	298	16	√	√	PROPN
ejpam-6492	298	17	∆)αr	∆)αr	PROPN
ejpam-6492	298	18	α	α	NOUN
ejpam-6492	298	19	−	−	NOUN
ejpam-6492	298	20	β	β	X
ejpam-6492	298	21	=	=	PUNCT
ejpam-6492	298	22	qb	qb	PROPN
ejpam-6492	298	23	√	√	NUM
ejpam-6492	298	24	∆(αr	∆(αr	NOUN
ejpam-6492	298	25	+	+	CCONJ
ejpam-6492	298	26	βr	βr	X
ejpam-6492	298	27	)	)	PUNCT
ejpam-6492	298	28	−	−	NOUN
ejpam-6492	298	29	a(αr	a(αr	ADV
ejpam-6492	298	30	−	−	PROPN
ejpam-6492	298	31	βr	βr	NUM
ejpam-6492	298	32	)	)	PUNCT
ejpam-6492	298	33	α	α	NOUN
ejpam-6492	298	34	−	−	NOUN
ejpam-6492	298	35	β	β	NOUN
ejpam-6492	298	36	=	=	SYM
ejpam-6492	298	37	qbvr	qbvr	NOUN
ejpam-6492	298	38	−	−	PROPN
ejpam-6492	298	39	aur	aur	PROPN
ejpam-6492	298	40	,	,	PUNCT
ejpam-6492	298	41	which	which	PRON
ejpam-6492	298	42	proves	prove	VERB
ejpam-6492	298	43	the	the	DET
ejpam-6492	298	44	equation	equation	NOUN
ejpam-6492	298	45	(	(	PUNCT
ejpam-6492	298	46	12	12	NUM
ejpam-6492	298	47	)	)	PUNCT
ejpam-6492	298	48	.	.	PUNCT
ejpam-6492	299	1	now	now	ADV
ejpam-6492	299	2	considering	consider	VERB
ejpam-6492	299	3	the	the	DET
ejpam-6492	299	4	equations	equation	NOUN
ejpam-6492	299	5	(	(	PUNCT
ejpam-6492	299	6	10	10	NUM
ejpam-6492	299	7	)	)	PUNCT
ejpam-6492	299	8	and	and	CCONJ
ejpam-6492	299	9	(	(	PUNCT
ejpam-6492	299	10	11	11	X
ejpam-6492	299	11	)	)	PUNCT
ejpam-6492	299	12	we	we	PRON
ejpam-6492	299	13	have	have	VERB
ejpam-6492	299	14	α̂β̂βr	α̂β̂βr	NOUN
ejpam-6492	299	15	+	+	NOUN
ejpam-6492	299	16	β̂α̂αr	β̂α̂αr	NOUN
ejpam-6492	299	17	=	=	SYM
ejpam-6492	299	18	(	(	PUNCT
ejpam-6492	299	19	a	a	DET
ejpam-6492	299	20	+	+	X
ejpam-6492	299	21	qb	qb	PROPN
ejpam-6492	299	22	√	√	NOUN
ejpam-6492	299	23	∆)βr	∆)βr	PROPN
ejpam-6492	299	24	+	+	CCONJ
ejpam-6492	299	25	(	(	PUNCT
ejpam-6492	299	26	a	a	DET
ejpam-6492	299	27	−	−	PROPN
ejpam-6492	299	28	qb	qb	PROPN
ejpam-6492	299	29	√	√	NOUN
ejpam-6492	299	30	∆)αr	∆)αr	NOUN
ejpam-6492	299	31	=	=	SYM
ejpam-6492	299	32	a(αr	a(αr	PROPN
ejpam-6492	299	33	+	+	NUM
ejpam-6492	299	34	βr	βr	X
ejpam-6492	299	35	)	)	PUNCT
ejpam-6492	300	1	−	−	PROPN
ejpam-6492	301	1	qb	qb	PROPN
ejpam-6492	301	2	√	√	NUM
ejpam-6492	301	3	∆(αr	∆(αr	NOUN
ejpam-6492	301	4	−	−	NUM
ejpam-6492	301	5	βr	βr	NOUN
ejpam-6492	301	6	)	)	PUNCT
ejpam-6492	301	7	=	=	SYM
ejpam-6492	301	8	avr	avr	PROPN
ejpam-6492	301	9	−	−	PROPN
ejpam-6492	301	10	q∆bur	q∆bur	PROPN
ejpam-6492	301	11	,	,	PUNCT
ejpam-6492	301	12	which	which	PRON
ejpam-6492	301	13	proves	prove	VERB
ejpam-6492	301	14	the	the	DET
ejpam-6492	301	15	equation	equation	NOUN
ejpam-6492	301	16	(	(	PUNCT
ejpam-6492	301	17	13	13	NUM
ejpam-6492	301	18	)	)	PUNCT
ejpam-6492	301	19	.	.	PUNCT
ejpam-6492	302	1	lemma	lemma	PROPN
ejpam-6492	302	2	3	3	X
ejpam-6492	302	3	.	.	PUNCT
ejpam-6492	303	1	let	let	VERB
ejpam-6492	303	2	α̂	α̂	NOUN
ejpam-6492	303	3	=	=	SYM
ejpam-6492	303	4	1	1	NUM
ejpam-6492	303	5	+	+	CCONJ
ejpam-6492	303	6	αi	αi	VERB
ejpam-6492	304	1	+	+	CCONJ
ejpam-6492	304	2	α2j	α2j	PROPN
ejpam-6492	304	3	+	+	CCONJ
ejpam-6492	304	4	α3k	α3k	ADP
ejpam-6492	304	5	,	,	PUNCT
ejpam-6492	304	6	β̂	β̂	PUNCT
ejpam-6492	304	7	=	=	SYM
ejpam-6492	304	8	1	1	NUM
ejpam-6492	304	9	+	+	CCONJ
ejpam-6492	304	10	βi	βi	PROPN
ejpam-6492	304	11	+	+	NUM
ejpam-6492	304	12	β2j	β2j	PROPN
ejpam-6492	304	13	+	+	CCONJ
ejpam-6492	304	14	β3k	β3k	PROPN
ejpam-6492	304	15	.	.	PUNCT
ejpam-6492	305	1	then	then	ADV
ejpam-6492	305	2	it	it	PRON
ejpam-6492	305	3	follows	follow	VERB
ejpam-6492	305	4	that	that	PRON
ejpam-6492	305	5	α̂β	α̂β	NUM
ejpam-6492	305	6	−	−	NOUN
ejpam-6492	305	7	β̂α	β̂α	PUNCT
ejpam-6492	306	1	=	=	PUNCT
ejpam-6492	306	2	(	(	PUNCT
ejpam-6492	306	3	−q	−q	ADJ
ejpam-6492	306	4	)	)	PUNCT
ejpam-6492	306	5	√	√	NUM
ejpam-6492	306	6	∆q−r	∆q−r	NOUN
ejpam-6492	306	7	,	,	PUNCT
ejpam-6492	306	8	(	(	PUNCT
ejpam-6492	306	9	14	14	NUM
ejpam-6492	306	10	)	)	PUNCT
ejpam-6492	306	11	α̂βr	α̂βr	NOUN
ejpam-6492	306	12	−	−	NOUN
ejpam-6492	306	13	β̂αr	β̂αr	NOUN
ejpam-6492	306	14	=	=	PUNCT
ejpam-6492	306	15	√	√	ADP
ejpam-6492	306	16	∆(−q)rq−r	∆(−q)rq−r	PROPN
ejpam-6492	306	17	,	,	PUNCT
ejpam-6492	306	18	(	(	PUNCT
ejpam-6492	306	19	15	15	NUM
ejpam-6492	306	20	)	)	PUNCT
ejpam-6492	306	21	for	for	ADP
ejpam-6492	306	22	all	all	DET
ejpam-6492	306	23	r	r	NOUN
ejpam-6492	306	24	∈	∈	NOUN
ejpam-6492	306	25	z.	z.	NOUN
ejpam-6492	306	26	proof	proof	NOUN
ejpam-6492	306	27	.	.	PUNCT
ejpam-6492	307	1	using	use	VERB
ejpam-6492	307	2	the	the	DET
ejpam-6492	307	3	binet	binet	NOUN
ejpam-6492	307	4	formula	formula	NOUN
ejpam-6492	307	5	,	,	PUNCT
ejpam-6492	307	6	we	we	PRON
ejpam-6492	307	7	achive	achive	VERB
ejpam-6492	307	8	that	that	PRON
ejpam-6492	307	9	α̂β	α̂β	NUM
ejpam-6492	307	10	−	−	NOUN
ejpam-6492	307	11	β̂α	β̂α	PUNCT
ejpam-6492	308	1	=	=	PUNCT
ejpam-6492	308	2	αβ	αβ	INTJ
ejpam-6492	308	3	(	(	PUNCT
ejpam-6492	308	4	α̂	α̂	NUM
ejpam-6492	308	5	α	α	PROPN
ejpam-6492	308	6	−	−	PROPN
ejpam-6492	308	7	β̂	β̂	PUNCT
ejpam-6492	308	8	β	β	NOUN
ejpam-6492	308	9	)	)	PUNCT
ejpam-6492	309	1	=	=	SYM
ejpam-6492	309	2	(	(	PUNCT
ejpam-6492	309	3	−q	−q	ADJ
ejpam-6492	309	4	)	)	PUNCT
ejpam-6492	309	5	√	√	NUM
ejpam-6492	309	6	∆q−r	∆q−r	NOUN
ejpam-6492	309	7	,	,	PUNCT
ejpam-6492	309	8	α̂βr	α̂βr	NOUN
ejpam-6492	309	9	−	−	NOUN
ejpam-6492	309	10	β̂αr	β̂αr	NOUN
ejpam-6492	309	11	=	=	SYM
ejpam-6492	309	12	(	(	PUNCT
ejpam-6492	309	13	αβ)r	αβ)r	PROPN
ejpam-6492	309	14	(	(	PUNCT
ejpam-6492	309	15	α̂	α̂	NUM
ejpam-6492	309	16	αr	αr	ADP
ejpam-6492	309	17	−	−	PROPN
ejpam-6492	309	18	β̂	β̂	SYM
ejpam-6492	309	19	βr	βr	NOUN
ejpam-6492	309	20	)	)	PUNCT
ejpam-6492	310	1	=	=	PUNCT
ejpam-6492	311	1	√	√	ADP
ejpam-6492	311	2	∆(−q)rq−r	∆(−q)rq−r	PROPN
ejpam-6492	311	3	.	.	PUNCT
ejpam-6492	312	1	now	now	ADV
ejpam-6492	312	2	we	we	PRON
ejpam-6492	312	3	can	can	AUX
ejpam-6492	312	4	prove	prove	VERB
ejpam-6492	312	5	the	the	DET
ejpam-6492	312	6	generalized	generalized	ADJ
ejpam-6492	312	7	product	product	NOUN
ejpam-6492	312	8	differences	difference	NOUN
ejpam-6492	312	9	of	of	ADP
ejpam-6492	312	10	fibonacci	fibonacci	NOUN
ejpam-6492	312	11	and	and	CCONJ
ejpam-6492	312	12	lucas	lucas	PROPN
ejpam-6492	312	13	quaternions	quaternion	NOUN
ejpam-6492	312	14	in	in	ADP
ejpam-6492	312	15	theorem	theorem	ADJ
ejpam-6492	312	16	8	8	NUM
ejpam-6492	312	17	and	and	CCONJ
ejpam-6492	312	18	theorem	theorem	VERB
ejpam-6492	312	19	9	9	NUM
ejpam-6492	312	20	.	.	PUNCT
ejpam-6492	312	21	b.	b.	PROPN
ejpam-6492	312	22	demirtürk	demirtürk	PROPN
ejpam-6492	312	23	,	,	PUNCT
ejpam-6492	312	24	n.	n.	NOUN
ejpam-6492	312	25	topal	topal	PROPN
ejpam-6492	312	26	/	/	SYM
ejpam-6492	312	27	eur	eur	PROPN
ejpam-6492	312	28	.	.	PUNCT
ejpam-6492	313	1	j.	j.	PROPN
ejpam-6492	313	2	pure	pure	PROPN
ejpam-6492	313	3	appl	appl	PROPN
ejpam-6492	313	4	.	.	PROPN
ejpam-6492	313	5	math	math	PROPN
ejpam-6492	313	6	,	,	PUNCT
ejpam-6492	313	7	18	18	NUM
ejpam-6492	313	8	(	(	PUNCT
ejpam-6492	313	9	3	3	NUM
ejpam-6492	313	10	)	)	PUNCT
ejpam-6492	313	11	(	(	PUNCT
ejpam-6492	313	12	2025	2025	NUM
ejpam-6492	313	13	)	)	PUNCT
ejpam-6492	313	14	,	,	PUNCT
ejpam-6492	313	15	6492	6492	NUM
ejpam-6492	313	16	14	14	NUM
ejpam-6492	313	17	of	of	ADP
ejpam-6492	313	18	22	22	NUM
ejpam-6492	313	19	theorem	theorem	NOUN
ejpam-6492	313	20	8	8	NUM
ejpam-6492	313	21	.	.	PUNCT
ejpam-6492	314	1	for	for	ADP
ejpam-6492	314	2	all	all	DET
ejpam-6492	314	3	n	n	NOUN
ejpam-6492	314	4	,	,	PUNCT
ejpam-6492	314	5	r	r	NOUN
ejpam-6492	314	6	∈	∈	PROPN
ejpam-6492	314	7	z	z	X
ejpam-6492	314	8	,	,	PUNCT
ejpam-6492	314	9	it	it	PRON
ejpam-6492	314	10	follows	follow	VERB
ejpam-6492	314	11	that	that	SCONJ
ejpam-6492	314	12	qn−rqn+r	qn−rqn+r	PROPN
ejpam-6492	314	13	−	−	PROPN
ejpam-6492	314	14	q2	q2	NOUN
ejpam-6492	314	15	n	n	NOUN
ejpam-6492	314	16	=	=	SYM
ejpam-6492	314	17	(	(	PUNCT
ejpam-6492	314	18	−q)n−rur[qbvr	−q)n−rur[qbvr	NOUN
ejpam-6492	314	19	−	−	PROPN
ejpam-6492	314	20	aur	aur	NOUN
ejpam-6492	314	21	]	]	X
ejpam-6492	314	22	,	,	PUNCT
ejpam-6492	314	23	where	where	SCONJ
ejpam-6492	314	24	a	a	DET
ejpam-6492	314	25	=	=	X
ejpam-6492	314	26	k0	k0	PROPN
ejpam-6492	314	27	−	−	PROPN
ejpam-6492	314	28	(	(	PUNCT
ejpam-6492	314	29	1	1	NUM
ejpam-6492	314	30	−	−	NUM
ejpam-6492	314	31	q)(1	q)(1	PROPN
ejpam-6492	314	32	+	+	X
ejpam-6492	314	33	q2	q2	NOUN
ejpam-6492	314	34	)	)	PUNCT
ejpam-6492	314	35	and	and	CCONJ
ejpam-6492	314	36	b	b	X
ejpam-6492	314	37	=	=	SYM
ejpam-6492	314	38	(	(	PUNCT
ejpam-6492	314	39	−q)i	−q)i	NOUN
ejpam-6492	314	40	+	+	CCONJ
ejpam-6492	314	41	(	(	PUNCT
ejpam-6492	314	42	−p)j	−p)j	NOUN
ejpam-6492	314	43	+	+	CCONJ
ejpam-6492	314	44	k.	k.	PROPN
ejpam-6492	314	45	proof	proof	NOUN
ejpam-6492	314	46	.	.	PUNCT
ejpam-6492	315	1	using	use	VERB
ejpam-6492	315	2	the	the	DET
ejpam-6492	315	3	binet	binet	NOUN
ejpam-6492	315	4	formula	formula	NOUN
ejpam-6492	315	5	and	and	CCONJ
ejpam-6492	315	6	(	(	PUNCT
ejpam-6492	315	7	12	12	NUM
ejpam-6492	315	8	)	)	PUNCT
ejpam-6492	315	9	,	,	PUNCT
ejpam-6492	315	10	we	we	PRON
ejpam-6492	315	11	obtain	obtain	VERB
ejpam-6492	315	12	qn−rqn+r	qn−rqn+r	PROPN
ejpam-6492	315	13	−	−	PROPN
ejpam-6492	315	14	q2	q2	NOUN
ejpam-6492	315	15	n	n	NOUN
ejpam-6492	315	16	=	=	PUNCT
ejpam-6492	315	17	(	(	PUNCT
ejpam-6492	315	18	α̂αn−r	α̂αn−r	PROPN
ejpam-6492	315	19	−	−	NOUN
ejpam-6492	315	20	β̂βn−r	β̂βn−r	NOUN
ejpam-6492	316	1	α	α	NOUN
ejpam-6492	316	2	−	−	NOUN
ejpam-6492	316	3	β	β	NOUN
ejpam-6492	316	4	)	)	PUNCT
ejpam-6492	316	5	(	(	PUNCT
ejpam-6492	316	6	α̂αn+r	α̂αn+r	PROPN
ejpam-6492	317	1	−	−	PROPN
ejpam-6492	317	2	β̂βn+r	β̂βn+r	PROPN
ejpam-6492	318	1	α	α	NOUN
ejpam-6492	318	2	−	−	NOUN
ejpam-6492	318	3	β	β	NOUN
ejpam-6492	318	4	)	)	PUNCT
ejpam-6492	318	5	−	−	PROPN
ejpam-6492	319	1	(	(	PUNCT
ejpam-6492	319	2	α̂αn	α̂αn	NOUN
ejpam-6492	319	3	−	−	PROPN
ejpam-6492	319	4	β̂βn	β̂βn	ADV
ejpam-6492	319	5	α	α	NOUN
ejpam-6492	319	6	−	−	NOUN
ejpam-6492	319	7	β	β	X
ejpam-6492	319	8	)	)	PUNCT
ejpam-6492	319	9	2	2	NUM
ejpam-6492	319	10	=	=	SYM
ejpam-6492	319	11	α̂2α2n	α̂2α2n	PROPN
ejpam-6492	319	12	+	+	CCONJ
ejpam-6492	319	13	β̂2β2n	β̂2β2n	NOUN
ejpam-6492	319	14	−	−	ADP
ejpam-6492	319	15	α̂β̂αn−rβn+r	α̂β̂αn−rβn+r	NOUN
ejpam-6492	319	16	−	−	NOUN
ejpam-6492	319	17	β̂α̂αn+rβn−r	β̂α̂αn+rβn−r	PUNCT
ejpam-6492	319	18	(	(	PUNCT
ejpam-6492	319	19	α	α	NOUN
ejpam-6492	319	20	−	−	NOUN
ejpam-6492	319	21	β)2	β)2	ADV
ejpam-6492	319	22	−	−	PROPN
ejpam-6492	319	23	α̂2α2n	α̂2α2n	PROPN
ejpam-6492	319	24	+	+	CCONJ
ejpam-6492	319	25	β̂2β2n	β̂2β2n	NUM
ejpam-6492	319	26	−	−	PROPN
ejpam-6492	319	27	2α̂β̂αnβn	2α̂β̂αnβn	NUM
ejpam-6492	319	28	(	(	PUNCT
ejpam-6492	319	29	α	α	NOUN
ejpam-6492	319	30	−	−	NOUN
ejpam-6492	320	1	β)2	β)2	ADV
ejpam-6492	320	2	=	=	NOUN
ejpam-6492	320	3	−α̂β̂αn−rβn+r	−α̂β̂αn−rβn+r	NOUN
ejpam-6492	320	4	−	−	NOUN
ejpam-6492	320	5	β̂α̂αn+rβn−r	β̂α̂αn+rβn−r	PUNCT
ejpam-6492	321	1	+	+	CCONJ
ejpam-6492	321	2	2α̂β̂αnβn	2α̂β̂αnβn	NUM
ejpam-6492	321	3	(	(	PUNCT
ejpam-6492	321	4	α	α	NOUN
ejpam-6492	321	5	−	−	NOUN
ejpam-6492	321	6	β)2	β)2	NOUN
ejpam-6492	321	7	=	=	X
ejpam-6492	321	8	(	(	PUNCT
ejpam-6492	321	9	αβ)n	αβ)n	PROPN
ejpam-6492	321	10	(	(	PUNCT
ejpam-6492	321	11	α	α	NOUN
ejpam-6492	321	12	−	−	PROPN
ejpam-6492	321	13	β)2	β)2	NOUN
ejpam-6492	321	14	[	[	PUNCT
ejpam-6492	321	15	α̂β̂	α̂β̂	X
ejpam-6492	321	16	(	(	PUNCT
ejpam-6492	321	17	αr	αr	INTJ
ejpam-6492	321	18	−	−	PROPN
ejpam-6492	321	19	βr	βr	INTJ
ejpam-6492	321	20	αr	αr	NUM
ejpam-6492	321	21	)	)	PUNCT
ejpam-6492	322	1	+	+	CCONJ
ejpam-6492	322	2	β̂α̂	β̂α̂	PUNCT
ejpam-6492	322	3	(	(	PUNCT
ejpam-6492	322	4	βr	βr	INTJ
ejpam-6492	322	5	−	−	ADP
ejpam-6492	322	6	αr	αr	NUM
ejpam-6492	322	7	βr	βr	NUM
ejpam-6492	322	8	)	)	PUNCT
ejpam-6492	322	9	]	]	PUNCT
ejpam-6492	323	1	=	=	PUNCT
ejpam-6492	323	2	(	(	PUNCT
ejpam-6492	323	3	αβ)n(αr	αβ)n(αr	NOUN
ejpam-6492	323	4	−	−	NOUN
ejpam-6492	323	5	βr	βr	NOUN
ejpam-6492	323	6	)	)	PUNCT
ejpam-6492	323	7	(	(	PUNCT
ejpam-6492	323	8	α	α	NOUN
ejpam-6492	323	9	−	−	PROPN
ejpam-6492	323	10	β)2	β)2	NOUN
ejpam-6492	323	11	(	(	PUNCT
ejpam-6492	323	12	α̂β̂	α̂β̂	X
ejpam-6492	323	13	αr	αr	ADP
ejpam-6492	323	14	−	−	PROPN
ejpam-6492	323	15	β̂α̂	β̂α̂	PUNCT
ejpam-6492	323	16	βr	βr	NOUN
ejpam-6492	323	17	)	)	PUNCT
ejpam-6492	323	18	=	=	PRON
ejpam-6492	324	1	(	(	PUNCT
ejpam-6492	324	2	αβ)n(αr	αβ)n(αr	NOUN
ejpam-6492	324	3	−	−	NOUN
ejpam-6492	324	4	βr	βr	NOUN
ejpam-6492	324	5	)	)	PUNCT
ejpam-6492	324	6	(	(	PUNCT
ejpam-6492	324	7	α	α	NOUN
ejpam-6492	324	8	−	−	PROPN
ejpam-6492	324	9	β)2	β)2	NOUN
ejpam-6492	324	10	(	(	PUNCT
ejpam-6492	324	11	α̂β̂βr	α̂β̂βr	NOUN
ejpam-6492	324	12	−	−	NOUN
ejpam-6492	324	13	β̂α̂βr	β̂α̂βr	NOUN
ejpam-6492	324	14	(	(	PUNCT
ejpam-6492	324	15	αβ)r	αβ)r	NUM
ejpam-6492	324	16	)	)	PUNCT
ejpam-6492	324	17	=	=	SYM
ejpam-6492	325	1	(	(	PUNCT
ejpam-6492	325	2	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	325	3	·	·	PUNCT
ejpam-6492	325	4	(	(	PUNCT
ejpam-6492	325	5	αr	αr	INTJ
ejpam-6492	325	6	−	−	NOUN
ejpam-6492	325	7	βr	βr	NOUN
ejpam-6492	325	8	)	)	PUNCT
ejpam-6492	325	9	(	(	PUNCT
ejpam-6492	325	10	α	α	NOUN
ejpam-6492	325	11	−	−	NOUN
ejpam-6492	325	12	β	β	NOUN
ejpam-6492	325	13	)	)	PUNCT
ejpam-6492	325	14	·	·	PUNCT
ejpam-6492	325	15	α̂β̂βr	α̂β̂βr	NOUN
ejpam-6492	325	16	−	−	NOUN
ejpam-6492	325	17	β̂α̂βr	β̂α̂βr	NOUN
ejpam-6492	325	18	(	(	PUNCT
ejpam-6492	325	19	α	α	NOUN
ejpam-6492	325	20	−	−	VERB
ejpam-6492	325	21	β	β	NOUN
ejpam-6492	325	22	)	)	PUNCT
ejpam-6492	325	23	=	=	SYM
ejpam-6492	325	24	(	(	PUNCT
ejpam-6492	325	25	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	325	26	·	·	PUNCT
ejpam-6492	325	27	αr	αr	X
ejpam-6492	325	28	−	−	NOUN
ejpam-6492	325	29	βr	βr	ADP
ejpam-6492	326	1	α	α	PRON
ejpam-6492	326	2	−	−	NOUN
ejpam-6492	326	3	β	β	X
ejpam-6492	326	4	·	·	PUNCT
ejpam-6492	326	5	(	(	PUNCT
ejpam-6492	326	6	α̂β̂βr	α̂β̂βr	NOUN
ejpam-6492	326	7	−	−	PROPN
ejpam-6492	326	8	β̂α̂βr	β̂α̂βr	NOUN
ejpam-6492	327	1	α	α	INTJ
ejpam-6492	327	2	−	−	NOUN
ejpam-6492	327	3	β	β	NOUN
ejpam-6492	327	4	)	)	PUNCT
ejpam-6492	328	1	=	=	SYM
ejpam-6492	328	2	(	(	PUNCT
ejpam-6492	328	3	αβ)n−rur	αβ)n−rur	X
ejpam-6492	328	4	·	·	PUNCT
ejpam-6492	328	5	(	(	PUNCT
ejpam-6492	328	6	qbvr	qbvr	NOUN
ejpam-6492	328	7	−	−	PROPN
ejpam-6492	328	8	aur	aur	NOUN
ejpam-6492	328	9	)	)	PUNCT
ejpam-6492	328	10	=	=	PRON
ejpam-6492	329	1	(	(	PUNCT
ejpam-6492	329	2	−q)n−rur[qbvr	−q)n−rur[qbvr	NOUN
ejpam-6492	329	3	−	−	PROPN
ejpam-6492	330	1	aur	aur	NOUN
ejpam-6492	330	2	]	]	X
ejpam-6492	330	3	,	,	PUNCT
ejpam-6492	330	4	as	as	SCONJ
ejpam-6492	330	5	desired	desire	VERB
ejpam-6492	330	6	.	.	PUNCT
ejpam-6492	331	1	theorem	theorem	VERB
ejpam-6492	331	2	9	9	NUM
ejpam-6492	331	3	.	.	PUNCT
ejpam-6492	332	1	for	for	ADP
ejpam-6492	332	2	all	all	DET
ejpam-6492	332	3	n	n	NOUN
ejpam-6492	332	4	,	,	PUNCT
ejpam-6492	332	5	r	r	NOUN
ejpam-6492	332	6	∈	∈	PROPN
ejpam-6492	332	7	z	z	X
ejpam-6492	332	8	,	,	PUNCT
ejpam-6492	332	9	it	it	PRON
ejpam-6492	332	10	follows	follow	VERB
ejpam-6492	332	11	that	that	SCONJ
ejpam-6492	332	12	kn−rkn+r	kn−rkn+r	PROPN
ejpam-6492	332	13	−	−	PROPN
ejpam-6492	332	14	k2	k2	PROPN
ejpam-6492	332	15	n	n	PROPN
ejpam-6492	332	16	=	=	SYM
ejpam-6492	332	17	−(−q)n−rur∆[qbvr	−(−q)n−rur∆[qbvr	PROPN
ejpam-6492	332	18	−	−	PROPN
ejpam-6492	332	19	aur	aur	PROPN
ejpam-6492	332	20	]	]	X
ejpam-6492	332	21	,	,	PUNCT
ejpam-6492	332	22	where	where	SCONJ
ejpam-6492	332	23	a	a	DET
ejpam-6492	332	24	=	=	X
ejpam-6492	332	25	k0	k0	PROPN
ejpam-6492	332	26	−	−	PROPN
ejpam-6492	332	27	(	(	PUNCT
ejpam-6492	332	28	1	1	NUM
ejpam-6492	332	29	−	−	NUM
ejpam-6492	332	30	q)(1	q)(1	PROPN
ejpam-6492	332	31	+	+	X
ejpam-6492	332	32	q2	q2	NOUN
ejpam-6492	332	33	)	)	PUNCT
ejpam-6492	332	34	and	and	CCONJ
ejpam-6492	332	35	b	b	X
ejpam-6492	332	36	=	=	SYM
ejpam-6492	332	37	(	(	PUNCT
ejpam-6492	332	38	−q)i	−q)i	NOUN
ejpam-6492	332	39	+	+	CCONJ
ejpam-6492	332	40	(	(	PUNCT
ejpam-6492	332	41	−p)j	−p)j	NOUN
ejpam-6492	332	42	+	+	CCONJ
ejpam-6492	332	43	k.	k.	PROPN
ejpam-6492	332	44	proof	proof	NOUN
ejpam-6492	332	45	.	.	PUNCT
ejpam-6492	333	1	using	use	VERB
ejpam-6492	333	2	the	the	DET
ejpam-6492	333	3	binet	binet	NOUN
ejpam-6492	333	4	formula	formula	NOUN
ejpam-6492	333	5	and	and	CCONJ
ejpam-6492	333	6	equation	equation	NOUN
ejpam-6492	333	7	(	(	PUNCT
ejpam-6492	333	8	14	14	NUM
ejpam-6492	333	9	)	)	PUNCT
ejpam-6492	333	10	,	,	PUNCT
ejpam-6492	333	11	we	we	PRON
ejpam-6492	333	12	have	have	VERB
ejpam-6492	333	13	:	:	PUNCT
ejpam-6492	334	1	kn−rkn+r	kn−rkn+r	PROPN
ejpam-6492	334	2	−	−	PROPN
ejpam-6492	334	3	k2	k2	PROPN
ejpam-6492	334	4	n	n	NOUN
ejpam-6492	334	5	=	=	PUNCT
ejpam-6492	334	6	(	(	PUNCT
ejpam-6492	334	7	α̂αn−r	α̂αn−r	NOUN
ejpam-6492	334	8	+	+	NOUN
ejpam-6492	334	9	β̂βn−r)(α̂αn+r	β̂βn−r)(α̂αn+r	PUNCT
ejpam-6492	334	10	+	+	SYM
ejpam-6492	334	11	β̂βn+r	β̂βn+r	NUM
ejpam-6492	334	12	)	)	PUNCT
ejpam-6492	334	13	−	−	PROPN
ejpam-6492	334	14	(	(	PUNCT
ejpam-6492	334	15	α̂αn	α̂αn	NOUN
ejpam-6492	334	16	+	+	CCONJ
ejpam-6492	334	17	β̂βn)2	β̂βn)2	X
ejpam-6492	334	18	=	=	SYM
ejpam-6492	334	19	α̂2α2n	α̂2α2n	PROPN
ejpam-6492	334	20	+	+	CCONJ
ejpam-6492	334	21	β̂2β2n	β̂2β2n	PUNCT
ejpam-6492	334	22	+	+	CCONJ
ejpam-6492	334	23	α̂β̂αn−rβn+r	α̂β̂αn−rβn+r	NOUN
ejpam-6492	334	24	+	+	CCONJ
ejpam-6492	334	25	β̂α̂αn+rβn−r	β̂α̂αn+rβn−r	NOUN
ejpam-6492	334	26	−	−	PROPN
ejpam-6492	334	27	(	(	PUNCT
ejpam-6492	334	28	α̂2α2n	α̂2α2n	PROPN
ejpam-6492	334	29	+	+	CCONJ
ejpam-6492	334	30	β̂2β2n	β̂2β2n	PUNCT
ejpam-6492	334	31	+	+	X
ejpam-6492	334	32	2α̂β̂αnβn	2α̂β̂αnβn	NUM
ejpam-6492	334	33	)	)	PUNCT
ejpam-6492	335	1	=	=	NOUN
ejpam-6492	335	2	−α̂β̂αn−rβn+r	−α̂β̂αn−rβn+r	NOUN
ejpam-6492	335	3	−	−	NOUN
ejpam-6492	335	4	β̂α̂αn+rβn−r	β̂α̂αn+rβn−r	PUNCT
ejpam-6492	336	1	+	+	CCONJ
ejpam-6492	336	2	2α̂β̂αnβn	2α̂β̂αnβn	NUM
ejpam-6492	336	3	=	=	SYM
ejpam-6492	336	4	−(−q)n−rur∆[qbvr	−(−q)n−rur∆[qbvr	PROPN
ejpam-6492	336	5	−	−	PROPN
ejpam-6492	336	6	aur	aur	NOUN
ejpam-6492	336	7	]	]	X
ejpam-6492	336	8	.	.	PUNCT
ejpam-6492	337	1	as	as	ADP
ejpam-6492	337	2	a	a	DET
ejpam-6492	337	3	consequence	consequence	NOUN
ejpam-6492	337	4	of	of	ADP
ejpam-6492	337	5	theorem	theorem	ADJ
ejpam-6492	337	6	8	8	NUM
ejpam-6492	337	7	and	and	CCONJ
ejpam-6492	337	8	theorem	theorem	VERB
ejpam-6492	337	9	9	9	NUM
ejpam-6492	337	10	,	,	PUNCT
ejpam-6492	337	11	we	we	PRON
ejpam-6492	337	12	can	can	AUX
ejpam-6492	337	13	give	give	VERB
ejpam-6492	337	14	corollary	corollary	ADJ
ejpam-6492	337	15	3	3	NUM
ejpam-6492	337	16	.	.	PUNCT
ejpam-6492	337	17	b.	b.	PROPN
ejpam-6492	337	18	demirtürk	demirtürk	PROPN
ejpam-6492	337	19	,	,	PUNCT
ejpam-6492	337	20	n.	n.	NOUN
ejpam-6492	337	21	topal	topal	PROPN
ejpam-6492	337	22	/	/	SYM
ejpam-6492	337	23	eur	eur	PROPN
ejpam-6492	337	24	.	.	PUNCT
ejpam-6492	338	1	j.	j.	PROPN
ejpam-6492	338	2	pure	pure	PROPN
ejpam-6492	338	3	appl	appl	PROPN
ejpam-6492	338	4	.	.	PROPN
ejpam-6492	338	5	math	math	PROPN
ejpam-6492	338	6	,	,	PUNCT
ejpam-6492	338	7	18	18	NUM
ejpam-6492	338	8	(	(	PUNCT
ejpam-6492	338	9	3	3	NUM
ejpam-6492	338	10	)	)	PUNCT
ejpam-6492	338	11	(	(	PUNCT
ejpam-6492	338	12	2025	2025	NUM
ejpam-6492	338	13	)	)	PUNCT
ejpam-6492	338	14	,	,	PUNCT
ejpam-6492	338	15	6492	6492	NUM
ejpam-6492	338	16	15	15	NUM
ejpam-6492	338	17	of	of	ADP
ejpam-6492	338	18	22	22	NUM
ejpam-6492	338	19	corollary	corollary	ADJ
ejpam-6492	338	20	3	3	NUM
ejpam-6492	338	21	.	.	PUNCT
ejpam-6492	339	1	for	for	ADP
ejpam-6492	339	2	all	all	DET
ejpam-6492	339	3	n	n	NOUN
ejpam-6492	339	4	,	,	PUNCT
ejpam-6492	339	5	r	r	NOUN
ejpam-6492	339	6	∈	∈	PROPN
ejpam-6492	339	7	z	z	X
ejpam-6492	339	8	,	,	PUNCT
ejpam-6492	339	9	it	it	PRON
ejpam-6492	339	10	follows	follow	VERB
ejpam-6492	339	11	that	that	SCONJ
ejpam-6492	339	12	kn−rkn+r	kn−rkn+r	PROPN
ejpam-6492	339	13	−	−	PROPN
ejpam-6492	339	14	k2	k2	PROPN
ejpam-6492	339	15	n	n	NOUN
ejpam-6492	339	16	=	=	SYM
ejpam-6492	339	17	−∆[qn−rqn+r	−∆[qn−rqn+r	PROPN
ejpam-6492	339	18	−	−	PROPN
ejpam-6492	339	19	q2	q2	NOUN
ejpam-6492	339	20	n	n	CCONJ
ejpam-6492	339	21	]	]	PUNCT
ejpam-6492	339	22	.	.	PUNCT
ejpam-6492	340	1	theorem	theorem	ADJ
ejpam-6492	340	2	10	10	NUM
ejpam-6492	340	3	.	.	PUNCT
ejpam-6492	341	1	for	for	ADP
ejpam-6492	341	2	all	all	DET
ejpam-6492	341	3	n	n	NOUN
ejpam-6492	341	4	,	,	PUNCT
ejpam-6492	341	5	r	r	NOUN
ejpam-6492	341	6	∈	∈	PROPN
ejpam-6492	341	7	z	z	X
ejpam-6492	341	8	,	,	PUNCT
ejpam-6492	341	9	it	it	PRON
ejpam-6492	341	10	follows	follow	VERB
ejpam-6492	341	11	that	that	SCONJ
ejpam-6492	341	12	q2	q2	PROPN
ejpam-6492	341	13	n+r	n+r	PROPN
ejpam-6492	341	14	−	−	PROPN
ejpam-6492	341	15	q2rq2	q2rq2	NOUN
ejpam-6492	341	16	n−r	n−r	NOUN
ejpam-6492	341	17	=	=	PUNCT
ejpam-6492	341	18	u2r	u2r	PUNCT
ejpam-6492	342	1	[	[	PUNCT
ejpam-6492	342	2	2q2n	2q2n	NOUN
ejpam-6492	342	3	−	−	PROPN
ejpam-6492	342	4	(	(	PUNCT
ejpam-6492	342	5	u2n	u2n	PROPN
ejpam-6492	342	6	+	+	CCONJ
ejpam-6492	342	7	u2n+2	u2n+2	PROPN
ejpam-6492	343	1	+	+	CCONJ
ejpam-6492	343	2	u2n+4	u2n+4	X
ejpam-6492	343	3	+	+	CCONJ
ejpam-6492	343	4	u2n+6	u2n+6	ADJ
ejpam-6492	343	5	)	)	PUNCT
ejpam-6492	343	6	]	]	PUNCT
ejpam-6492	343	7	.	.	PUNCT
ejpam-6492	344	1	proof	proof	NOUN
ejpam-6492	344	2	.	.	PUNCT
ejpam-6492	345	1	q2	q2	PROPN
ejpam-6492	345	2	n+r	n+r	PROPN
ejpam-6492	345	3	−	−	PROPN
ejpam-6492	345	4	q2rq2	q2rq2	NOUN
ejpam-6492	345	5	n−r	n−r	NOUN
ejpam-6492	345	6	=	=	SYM
ejpam-6492	345	7	(	(	PUNCT
ejpam-6492	345	8	α̂αn+r	α̂αn+r	PROPN
ejpam-6492	345	9	−	−	PROPN
ejpam-6492	346	1	β̂βn+r	β̂βn+r	PROPN
ejpam-6492	347	1	α	α	NOUN
ejpam-6492	347	2	−	−	NOUN
ejpam-6492	347	3	β	β	NOUN
ejpam-6492	347	4	)	)	PUNCT
ejpam-6492	347	5	2	2	NUM
ejpam-6492	347	6	−	−	PROPN
ejpam-6492	347	7	(	(	PUNCT
ejpam-6492	347	8	−q)2r	−q)2r	INTJ
ejpam-6492	347	9	(	(	PUNCT
ejpam-6492	347	10	α̂αn−r	α̂αn−r	PROPN
ejpam-6492	347	11	−	−	NOUN
ejpam-6492	347	12	β̂βn−r	β̂βn−r	NOUN
ejpam-6492	347	13	α	α	NOUN
ejpam-6492	347	14	−	−	NOUN
ejpam-6492	347	15	β	β	NOUN
ejpam-6492	347	16	)	)	PUNCT
ejpam-6492	347	17	2	2	NUM
ejpam-6492	347	18	=	=	SYM
ejpam-6492	347	19	α̂α̂α2n+2r	α̂α̂α2n+2r	NUM
ejpam-6492	347	20	+	+	CCONJ
ejpam-6492	347	21	β̂β̂β2n+2r	β̂β̂β2n+2r	NUM
ejpam-6492	347	22	−	−	PROPN
ejpam-6492	347	23	(	(	PUNCT
ejpam-6492	347	24	αβ)n+r	αβ)n+r	PROPN
ejpam-6492	347	25	(	(	PUNCT
ejpam-6492	347	26	α̂β̂	α̂β̂	X
ejpam-6492	347	27	+	+	CCONJ
ejpam-6492	347	28	β̂α̂	β̂α̂	PUNCT
ejpam-6492	347	29	)	)	PUNCT
ejpam-6492	347	30	(	(	PUNCT
ejpam-6492	347	31	α	α	NOUN
ejpam-6492	347	32	−	−	PROPN
ejpam-6492	347	33	β)2	β)2	ADV
ejpam-6492	347	34	−(αβ)2r	−(αβ)2r	PROPN
ejpam-6492	347	35	α̂α̂α2n−2r	α̂α̂α2n−2r	PROPN
ejpam-6492	347	36	+	+	CCONJ
ejpam-6492	347	37	β̂β̂β2n−2r	β̂β̂β2n−2r	PROPN
ejpam-6492	347	38	−	−	PROPN
ejpam-6492	347	39	(	(	PUNCT
ejpam-6492	347	40	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	347	41	(	(	PUNCT
ejpam-6492	347	42	α̂β̂	α̂β̂	X
ejpam-6492	347	43	+	+	CCONJ
ejpam-6492	347	44	β̂α̂	β̂α̂	PUNCT
ejpam-6492	347	45	)	)	PUNCT
ejpam-6492	348	1	(	(	PUNCT
ejpam-6492	348	2	α	α	NOUN
ejpam-6492	348	3	−	−	NOUN
ejpam-6492	348	4	β)2	β)2	NOUN
ejpam-6492	348	5	=	=	X
ejpam-6492	348	6	[	[	PUNCT
ejpam-6492	348	7	α̂α̂α2n+2r	α̂α̂α2n+2r	NUM
ejpam-6492	348	8	+	+	CCONJ
ejpam-6492	348	9	β̂β̂β2n+2r	β̂β̂β2n+2r	NUM
ejpam-6492	348	10	−	−	PROPN
ejpam-6492	348	11	(	(	PUNCT
ejpam-6492	348	12	αβ)n+r	αβ)n+r	PROPN
ejpam-6492	348	13	(	(	PUNCT
ejpam-6492	348	14	α̂β̂	α̂β̂	X
ejpam-6492	348	15	+	+	CCONJ
ejpam-6492	348	16	β̂α̂	β̂α̂	PUNCT
ejpam-6492	348	17	)	)	PUNCT
ejpam-6492	348	18	]	]	PUNCT
ejpam-6492	349	1	−	−	PROPN
ejpam-6492	349	2	(	(	PUNCT
ejpam-6492	349	3	αβ)2r	αβ)2r	X
ejpam-6492	349	4	[	[	PUNCT
ejpam-6492	349	5	α̂α̂α2n−2r	α̂α̂α2n−2r	PROPN
ejpam-6492	349	6	+	+	CCONJ
ejpam-6492	349	7	β̂β̂β2n−2r	β̂β̂β2n−2r	PROPN
ejpam-6492	349	8	−	−	PROPN
ejpam-6492	349	9	(	(	PUNCT
ejpam-6492	349	10	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	349	11	(	(	PUNCT
ejpam-6492	349	12	α̂β̂	α̂β̂	X
ejpam-6492	349	13	+	+	CCONJ
ejpam-6492	349	14	β̂α̂	β̂α̂	PUNCT
ejpam-6492	349	15	)	)	PUNCT
ejpam-6492	349	16	]	]	PUNCT
ejpam-6492	350	1	(	(	PUNCT
ejpam-6492	350	2	α	α	NOUN
ejpam-6492	350	3	−	−	NOUN
ejpam-6492	350	4	β)2	β)2	NOUN
ejpam-6492	350	5	=	=	NOUN
ejpam-6492	350	6	α̂α̂α2n+2r	α̂α̂α2n+2r	NUM
ejpam-6492	350	7	+	+	CCONJ
ejpam-6492	350	8	β̂β̂β2n+2r	β̂β̂β2n+2r	NUM
ejpam-6492	350	9	−	−	NUM
ejpam-6492	350	10	α̂α̂α2nβ2r	α̂α̂α2nβ2r	ADJ
ejpam-6492	350	11	−	−	PROPN
ejpam-6492	350	12	β̂β̂α2rβ2n	β̂β̂α2rβ2n	NOUN
ejpam-6492	350	13	(	(	PUNCT
ejpam-6492	350	14	α	α	NOUN
ejpam-6492	350	15	−	−	NOUN
ejpam-6492	350	16	β)2	β)2	NOUN
ejpam-6492	350	17	=	=	PROPN
ejpam-6492	350	18	α̂α̂	α̂α̂	PROPN
ejpam-6492	350	19	(	(	PUNCT
ejpam-6492	350	20	α2n+2r	α2n+2r	NOUN
ejpam-6492	350	21	−	−	PROPN
ejpam-6492	350	22	α2nβ2r	α2nβ2r	NUM
ejpam-6492	350	23	)	)	PUNCT
ejpam-6492	350	24	+	+	NUM
ejpam-6492	350	25	β̂β̂	β̂β̂	PUNCT
ejpam-6492	351	1	(	(	PUNCT
ejpam-6492	351	2	β2n+2r	β2n+2r	NOUN
ejpam-6492	351	3	−	−	PROPN
ejpam-6492	351	4	α2rβ2n	α2rβ2n	NUM
ejpam-6492	351	5	)	)	PUNCT
ejpam-6492	351	6	(	(	PUNCT
ejpam-6492	351	7	α	α	NOUN
ejpam-6492	351	8	−	−	NOUN
ejpam-6492	351	9	β)2	β)2	NOUN
ejpam-6492	351	10	=	=	PROPN
ejpam-6492	351	11	α̂α̂α2n	α̂α̂α2n	PROPN
ejpam-6492	351	12	(	(	PUNCT
ejpam-6492	351	13	α2r	α2r	PROPN
ejpam-6492	351	14	−	−	NOUN
ejpam-6492	351	15	β2r	β2r	PUNCT
ejpam-6492	351	16	)	)	PUNCT
ejpam-6492	352	1	+	+	CCONJ
ejpam-6492	352	2	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	352	3	(	(	PUNCT
ejpam-6492	352	4	β2r	β2r	PUNCT
ejpam-6492	352	5	−	−	PROPN
ejpam-6492	352	6	α2r	α2r	PROPN
ejpam-6492	352	7	)	)	PUNCT
ejpam-6492	352	8	(	(	PUNCT
ejpam-6492	352	9	α	α	NOUN
ejpam-6492	352	10	−	−	NOUN
ejpam-6492	352	11	β)2	β)2	NOUN
ejpam-6492	352	12	=	=	PROPN
ejpam-6492	352	13	α̂α̂α2n	α̂α̂α2n	PROPN
ejpam-6492	352	14	(	(	PUNCT
ejpam-6492	352	15	α2r	α2r	PROPN
ejpam-6492	352	16	−	−	NOUN
ejpam-6492	352	17	β2r	β2r	NUM
ejpam-6492	352	18	)	)	PUNCT
ejpam-6492	352	19	−	−	PROPN
ejpam-6492	353	1	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	353	2	(	(	PUNCT
ejpam-6492	353	3	α2r	α2r	PROPN
ejpam-6492	353	4	−	−	NOUN
ejpam-6492	353	5	β2r	β2r	NUM
ejpam-6492	353	6	)	)	PUNCT
ejpam-6492	353	7	(	(	PUNCT
ejpam-6492	353	8	α	α	NOUN
ejpam-6492	353	9	−	−	NOUN
ejpam-6492	353	10	β)2	β)2	NOUN
ejpam-6492	353	11	=	=	X
ejpam-6492	353	12	α2r	α2r	PROPN
ejpam-6492	353	13	−	−	ADP
ejpam-6492	353	14	β2r	β2r	PUNCT
ejpam-6492	353	15	α	α	NOUN
ejpam-6492	353	16	−	−	NOUN
ejpam-6492	353	17	β	β	X
ejpam-6492	353	18	α̂α̂α2n	α̂α̂α2n	ADJ
ejpam-6492	353	19	−	−	PUNCT
ejpam-6492	353	20	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	353	21	α	α	NOUN
ejpam-6492	353	22	−	−	NOUN
ejpam-6492	353	23	β	β	X
ejpam-6492	353	24	=	=	PUNCT
ejpam-6492	353	25	u2r	u2r	PROPN
ejpam-6492	353	26	α̂α̂α2n	α̂α̂α2n	ADJ
ejpam-6492	353	27	−	−	PROPN
ejpam-6492	353	28	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	353	29	α	α	NOUN
ejpam-6492	353	30	−	−	NOUN
ejpam-6492	353	31	β	β	X
ejpam-6492	353	32	=	=	PUNCT
ejpam-6492	353	33	u2r	u2r	PROPN
ejpam-6492	353	34	α2n	α2n	PROPN
ejpam-6492	353	35	(	(	PUNCT
ejpam-6492	353	36	2α̂	2α̂	NUM
ejpam-6492	353	37	−	−	NOUN
ejpam-6492	353	38	1	1	NUM
ejpam-6492	353	39	−	−	NOUN
ejpam-6492	353	40	α2	α2	ADJ
ejpam-6492	353	41	−	−	NOUN
ejpam-6492	353	42	α4	α4	NOUN
ejpam-6492	353	43	−	−	NOUN
ejpam-6492	353	44	α6)−	α6)−	NOUN
ejpam-6492	353	45	β2n	β2n	PUNCT
ejpam-6492	353	46	(	(	PUNCT
ejpam-6492	353	47	2β̂	2β̂	NUM
ejpam-6492	353	48	−	−	NOUN
ejpam-6492	353	49	1	1	NUM
ejpam-6492	353	50	−	−	PROPN
ejpam-6492	353	51	β2	β2	NOUN
ejpam-6492	353	52	−	−	PROPN
ejpam-6492	353	53	β4	β4	PROPN
ejpam-6492	353	54	−	−	PROPN
ejpam-6492	353	55	β6	β6	PROPN
ejpam-6492	353	56	)	)	PUNCT
ejpam-6492	354	1	α	α	NOUN
ejpam-6492	354	2	−	−	NOUN
ejpam-6492	355	1	β	β	X
ejpam-6492	355	2	=	=	PUNCT
ejpam-6492	355	3	u2r	u2r	PROPN
ejpam-6492	355	4	2	2	NUM
ejpam-6492	355	5	(	(	PUNCT
ejpam-6492	355	6	α̂α2n	α̂α2n	ADJ
ejpam-6492	355	7	−	−	NOUN
ejpam-6492	355	8	β̂β2n	β̂β2n	NOUN
ejpam-6492	355	9	)	)	PUNCT
ejpam-6492	355	10	−	−	PROPN
ejpam-6492	356	1	(	(	PUNCT
ejpam-6492	356	2	α2n	α2n	PROPN
ejpam-6492	356	3	−	−	PROPN
ejpam-6492	356	4	β2n	β2n	PUNCT
ejpam-6492	356	5	)	)	PUNCT
ejpam-6492	356	6	−	−	PROPN
ejpam-6492	357	1	(	(	PUNCT
ejpam-6492	357	2	α2n+2	α2n+2	PRON
ejpam-6492	357	3	−	−	VERB
ejpam-6492	357	4	β2n+2)−	β2n+2)−	VERB
ejpam-6492	357	5	(	(	PUNCT
ejpam-6492	357	6	α2n+4	α2n+4	X
ejpam-6492	357	7	−	−	PROPN
ejpam-6492	357	8	β2n+4)−	β2n+4)−	VERB
ejpam-6492	357	9	(	(	PUNCT
ejpam-6492	357	10	α2n+6	α2n+6	NOUN
ejpam-6492	357	11	−	−	NOUN
ejpam-6492	357	12	β2n+6	β2n+6	NOUN
ejpam-6492	357	13	)	)	PUNCT
ejpam-6492	358	1	α	α	NOUN
ejpam-6492	358	2	−	−	NOUN
ejpam-6492	359	1	β	β	X
ejpam-6492	359	2	=	=	PUNCT
ejpam-6492	359	3	u2r	u2r	X
ejpam-6492	359	4	[	[	PUNCT
ejpam-6492	359	5	2	2	NUM
ejpam-6492	359	6	α̂α2n	α̂α2n	NOUN
ejpam-6492	359	7	−	−	ADP
ejpam-6492	359	8	β̂β2n	β̂β2n	NOUN
ejpam-6492	359	9	α	α	NOUN
ejpam-6492	359	10	−	−	NOUN
ejpam-6492	359	11	β	β	NOUN
ejpam-6492	359	12	−	−	PROPN
ejpam-6492	359	13	α2n	α2n	PROPN
ejpam-6492	360	1	−	−	PROPN
ejpam-6492	360	2	β2n	β2n	PUNCT
ejpam-6492	361	1	α	α	NOUN
ejpam-6492	361	2	−	−	NOUN
ejpam-6492	362	1	β	β	X
ejpam-6492	362	2	−	−	PROPN
ejpam-6492	363	1	α2n+2	α2n+2	PRON
ejpam-6492	363	2	−	−	PROPN
ejpam-6492	364	1	β2n+2	β2n+2	ADV
ejpam-6492	365	1	α	α	INTJ
ejpam-6492	365	2	−	−	NOUN
ejpam-6492	366	1	β	β	X
ejpam-6492	367	1	−	−	PROPN
ejpam-6492	368	1	α2n+4	α2n+4	PRON
ejpam-6492	368	2	−	−	PROPN
ejpam-6492	368	3	β2n+4	β2n+4	INTJ
ejpam-6492	369	1	α	α	X
ejpam-6492	369	2	−	−	VERB
ejpam-6492	369	3	β	β	NOUN
ejpam-6492	369	4	−	−	NOUN
ejpam-6492	369	5	α2n+6	α2n+6	NOUN
ejpam-6492	369	6	−	−	NOUN
ejpam-6492	369	7	β2n+6	β2n+6	NOUN
ejpam-6492	370	1	α	α	NOUN
ejpam-6492	370	2	−	−	NOUN
ejpam-6492	370	3	β	β	X
ejpam-6492	370	4	]	]	X
ejpam-6492	371	1	=	=	PUNCT
ejpam-6492	371	2	u2r	u2r	PUNCT
ejpam-6492	371	3	[	[	X
ejpam-6492	371	4	2q2n	2q2n	NOUN
ejpam-6492	371	5	−	−	PROPN
ejpam-6492	371	6	(	(	PUNCT
ejpam-6492	371	7	u2n	u2n	PROPN
ejpam-6492	371	8	+	+	CCONJ
ejpam-6492	371	9	u2n+2	u2n+2	PROPN
ejpam-6492	371	10	+	+	CCONJ
ejpam-6492	371	11	u2n+4	u2n+4	X
ejpam-6492	371	12	+	+	CCONJ
ejpam-6492	371	13	u2n+6	u2n+6	ADJ
ejpam-6492	371	14	)	)	PUNCT
ejpam-6492	371	15	]	]	PUNCT
ejpam-6492	371	16	.	.	PUNCT
ejpam-6492	372	1	b.	b.	PROPN
ejpam-6492	372	2	demirtürk	demirtürk	PROPN
ejpam-6492	372	3	,	,	PUNCT
ejpam-6492	372	4	n.	n.	NOUN
ejpam-6492	372	5	topal	topal	PROPN
ejpam-6492	372	6	/	/	SYM
ejpam-6492	372	7	eur	eur	PROPN
ejpam-6492	372	8	.	.	PUNCT
ejpam-6492	373	1	j.	j.	PROPN
ejpam-6492	373	2	pure	pure	PROPN
ejpam-6492	373	3	appl	appl	PROPN
ejpam-6492	373	4	.	.	PROPN
ejpam-6492	373	5	math	math	PROPN
ejpam-6492	373	6	,	,	PUNCT
ejpam-6492	373	7	18	18	NUM
ejpam-6492	373	8	(	(	PUNCT
ejpam-6492	373	9	3	3	NUM
ejpam-6492	373	10	)	)	PUNCT
ejpam-6492	373	11	(	(	PUNCT
ejpam-6492	373	12	2025	2025	NUM
ejpam-6492	373	13	)	)	PUNCT
ejpam-6492	373	14	,	,	PUNCT
ejpam-6492	373	15	6492	6492	NUM
ejpam-6492	373	16	16	16	NUM
ejpam-6492	373	17	of	of	ADP
ejpam-6492	373	18	22	22	NUM
ejpam-6492	373	19	theorem	theorem	NOUN
ejpam-6492	373	20	11	11	NUM
ejpam-6492	373	21	.	.	PUNCT
ejpam-6492	374	1	for	for	ADP
ejpam-6492	374	2	all	all	DET
ejpam-6492	374	3	n	n	NOUN
ejpam-6492	374	4	,	,	PUNCT
ejpam-6492	374	5	r	r	NOUN
ejpam-6492	374	6	∈	∈	PROPN
ejpam-6492	374	7	z	z	X
ejpam-6492	374	8	,	,	PUNCT
ejpam-6492	374	9	it	it	PRON
ejpam-6492	374	10	follows	follow	VERB
ejpam-6492	374	11	that	that	SCONJ
ejpam-6492	374	12	k2	k2	PROPN
ejpam-6492	374	13	n+r	n+r	PROPN
ejpam-6492	374	14	−	−	PROPN
ejpam-6492	374	15	q2rk2	q2rk2	NOUN
ejpam-6492	374	16	n−r	n−r	NOUN
ejpam-6492	374	17	=	=	SYM
ejpam-6492	374	18	∆u2r	∆u2r	X
ejpam-6492	374	19	[	[	X
ejpam-6492	374	20	2q2n	2q2n	NUM
ejpam-6492	374	21	−	−	PROPN
ejpam-6492	374	22	(	(	PUNCT
ejpam-6492	374	23	u2n	u2n	PROPN
ejpam-6492	374	24	+	+	CCONJ
ejpam-6492	374	25	u2n+2	u2n+2	PROPN
ejpam-6492	375	1	+	+	CCONJ
ejpam-6492	375	2	u2n+4	u2n+4	X
ejpam-6492	375	3	+	+	CCONJ
ejpam-6492	375	4	u2n+6	u2n+6	ADJ
ejpam-6492	375	5	)	)	PUNCT
ejpam-6492	375	6	]	]	PUNCT
ejpam-6492	375	7	.	.	PUNCT
ejpam-6492	376	1	proof	proof	NOUN
ejpam-6492	376	2	.	.	PUNCT
ejpam-6492	377	1	k2	k2	PROPN
ejpam-6492	377	2	n+r	n+r	PROPN
ejpam-6492	377	3	−	−	PROPN
ejpam-6492	377	4	q2rk2	q2rk2	NOUN
ejpam-6492	377	5	n−r	n−r	NOUN
ejpam-6492	377	6	=	=	SYM
ejpam-6492	377	7	(	(	PUNCT
ejpam-6492	377	8	α̂αn+r	α̂αn+r	PROPN
ejpam-6492	377	9	+	+	CCONJ
ejpam-6492	377	10	β̂βn+r	β̂βn+r	NUM
ejpam-6492	377	11	)	)	PUNCT
ejpam-6492	377	12	2	2	NUM
ejpam-6492	377	13	−	−	NOUN
ejpam-6492	377	14	(	(	PUNCT
ejpam-6492	377	15	−q)2r	−q)2r	INTJ
ejpam-6492	377	16	(	(	PUNCT
ejpam-6492	377	17	α̂αn−r	α̂αn−r	PROPN
ejpam-6492	377	18	+	+	CCONJ
ejpam-6492	377	19	β̂βn−r	β̂βn−r	ADJ
ejpam-6492	377	20	)	)	PUNCT
ejpam-6492	377	21	2	2	X
ejpam-6492	377	22	=	=	PUNCT
ejpam-6492	377	23	[	[	PUNCT
ejpam-6492	377	24	α̂α̂α2n+2r	α̂α̂α2n+2r	NUM
ejpam-6492	377	25	+	+	NOUN
ejpam-6492	377	26	β̂β̂β2n+2r	β̂β̂β2n+2r	NUM
ejpam-6492	377	27	+	+	CCONJ
ejpam-6492	377	28	(	(	PUNCT
ejpam-6492	377	29	αβ)n+r	αβ)n+r	PROPN
ejpam-6492	377	30	(	(	PUNCT
ejpam-6492	377	31	α̂β̂	α̂β̂	X
ejpam-6492	377	32	+	+	CCONJ
ejpam-6492	377	33	β̂α̂	β̂α̂	PUNCT
ejpam-6492	377	34	)	)	PUNCT
ejpam-6492	377	35	]	]	PUNCT
ejpam-6492	378	1	−(αβ)2r	−(αβ)2r	PRON
ejpam-6492	378	2	[	[	PUNCT
ejpam-6492	378	3	α̂α̂α2n−2r	α̂α̂α2n−2r	PROPN
ejpam-6492	378	4	+	+	CCONJ
ejpam-6492	378	5	β̂β̂β2n−2r	β̂β̂β2n−2r	PUNCT
ejpam-6492	378	6	+	+	CCONJ
ejpam-6492	378	7	(	(	PUNCT
ejpam-6492	378	8	αβ)n−r	αβ)n−r	PROPN
ejpam-6492	378	9	(	(	PUNCT
ejpam-6492	378	10	α̂β̂	α̂β̂	X
ejpam-6492	378	11	+	+	CCONJ
ejpam-6492	378	12	β̂α̂	β̂α̂	PUNCT
ejpam-6492	378	13	)	)	PUNCT
ejpam-6492	378	14	]	]	PUNCT
ejpam-6492	379	1	=	=	PUNCT
ejpam-6492	379	2	α̂α̂α2n+2r+β̂β̂β2n+2r−α̂α̂α2nβ2r−β̂β̂α2rβ2n	α̂α̂α2n+2r+β̂β̂β2n+2r−α̂α̂α2nβ2r−β̂β̂α2rβ2n	NOUN
ejpam-6492	379	3	=	=	SYM
ejpam-6492	379	4	α̂α̂	α̂α̂	PROPN
ejpam-6492	379	5	(	(	PUNCT
ejpam-6492	379	6	α2n+2r	α2n+2r	NOUN
ejpam-6492	379	7	−	−	PROPN
ejpam-6492	379	8	α2nβ2r	α2nβ2r	NUM
ejpam-6492	379	9	)	)	PUNCT
ejpam-6492	379	10	+	+	X
ejpam-6492	379	11	β̂β̂	β̂β̂	X
ejpam-6492	379	12	(	(	PUNCT
ejpam-6492	379	13	β2n+2r	β2n+2r	NOUN
ejpam-6492	379	14	−	−	NOUN
ejpam-6492	379	15	α2rβ2n	α2rβ2n	NUM
ejpam-6492	379	16	)	)	PUNCT
ejpam-6492	379	17	=	=	SYM
ejpam-6492	379	18	α̂α̂α2n	α̂α̂α2n	ADJ
ejpam-6492	379	19	(	(	PUNCT
ejpam-6492	379	20	α2r	α2r	PROPN
ejpam-6492	379	21	−	−	NOUN
ejpam-6492	379	22	β2r	β2r	PUNCT
ejpam-6492	379	23	)	)	PUNCT
ejpam-6492	380	1	+	+	CCONJ
ejpam-6492	380	2	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	380	3	(	(	PUNCT
ejpam-6492	380	4	β2r	β2r	PUNCT
ejpam-6492	380	5	−	−	PROPN
ejpam-6492	380	6	α2r	α2r	PROPN
ejpam-6492	380	7	)	)	PUNCT
ejpam-6492	380	8	=	=	SYM
ejpam-6492	380	9	α̂α̂α2n	α̂α̂α2n	ADJ
ejpam-6492	380	10	(	(	PUNCT
ejpam-6492	380	11	α2r	α2r	PROPN
ejpam-6492	380	12	−	−	NOUN
ejpam-6492	380	13	β2r	β2r	NUM
ejpam-6492	380	14	)	)	PUNCT
ejpam-6492	380	15	−	−	PROPN
ejpam-6492	381	1	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	381	2	(	(	PUNCT
ejpam-6492	381	3	α2r	α2r	PROPN
ejpam-6492	381	4	−	−	NOUN
ejpam-6492	381	5	β2r	β2r	NUM
ejpam-6492	381	6	)	)	PUNCT
ejpam-6492	381	7	=	=	PUNCT
ejpam-6492	382	1	[	[	PUNCT
ejpam-6492	382	2	α2r	α2r	PROPN
ejpam-6492	382	3	−	−	NOUN
ejpam-6492	382	4	β2r	β2r	SYM
ejpam-6492	382	5	]	]	PUNCT
ejpam-6492	382	6	[	[	PUNCT
ejpam-6492	382	7	α̂α̂α2n	α̂α̂α2n	ADJ
ejpam-6492	382	8	−	−	PROPN
ejpam-6492	382	9	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	382	10	]	]	PUNCT
ejpam-6492	383	1	=	=	PUNCT
ejpam-6492	383	2	[	[	PUNCT
ejpam-6492	383	3	α2r	α2r	PROPN
ejpam-6492	383	4	−	−	NOUN
ejpam-6492	383	5	β2r	β2r	SYM
ejpam-6492	383	6	]	]	PUNCT
ejpam-6492	383	7	[	[	PUNCT
ejpam-6492	383	8	α2n	α2n	PROPN
ejpam-6492	383	9	(	(	PUNCT
ejpam-6492	383	10	2α̂	2α̂	NUM
ejpam-6492	383	11	−	−	NOUN
ejpam-6492	383	12	1	1	NUM
ejpam-6492	383	13	−	−	NOUN
ejpam-6492	383	14	α2	α2	ADJ
ejpam-6492	383	15	−	−	NOUN
ejpam-6492	383	16	α4	α4	NOUN
ejpam-6492	383	17	−	−	PROPN
ejpam-6492	383	18	α6	α6	NOUN
ejpam-6492	383	19	)	)	PUNCT
ejpam-6492	383	20	−	−	PROPN
ejpam-6492	384	1	β2n	β2n	INTJ
ejpam-6492	384	2	(	(	PUNCT
ejpam-6492	384	3	2β̂	2β̂	NUM
ejpam-6492	384	4	−	−	NOUN
ejpam-6492	384	5	1	1	NUM
ejpam-6492	384	6	−	−	PROPN
ejpam-6492	384	7	β2	β2	NOUN
ejpam-6492	384	8	−	−	PROPN
ejpam-6492	384	9	β4	β4	PROPN
ejpam-6492	384	10	−	−	PROPN
ejpam-6492	384	11	β6	β6	PROPN
ejpam-6492	384	12	)	)	PUNCT
ejpam-6492	384	13	]	]	PUNCT
ejpam-6492	385	1	=	=	PUNCT
ejpam-6492	385	2	[	[	PUNCT
ejpam-6492	385	3	α2r	α2r	PROPN
ejpam-6492	385	4	−	−	NOUN
ejpam-6492	385	5	β2r	β2r	SYM
ejpam-6492	385	6	]	]	PUNCT
ejpam-6492	385	7	[	[	PUNCT
ejpam-6492	385	8	2	2	NUM
ejpam-6492	385	9	(	(	PUNCT
ejpam-6492	385	10	α̂α2n	α̂α2n	ADJ
ejpam-6492	385	11	−	−	NOUN
ejpam-6492	385	12	β̂β2n	β̂β2n	NOUN
ejpam-6492	385	13	)	)	PUNCT
ejpam-6492	385	14	−	−	PROPN
ejpam-6492	386	1	(	(	PUNCT
ejpam-6492	386	2	α2n	α2n	PROPN
ejpam-6492	386	3	−	−	PROPN
ejpam-6492	386	4	β2n	β2n	PUNCT
ejpam-6492	386	5	)	)	PUNCT
ejpam-6492	386	6	−	−	PROPN
ejpam-6492	387	1	(	(	PUNCT
ejpam-6492	387	2	α2n+2	α2n+2	PRON
ejpam-6492	387	3	−	−	VERB
ejpam-6492	388	1	β2n+2	β2n+2	PRON
ejpam-6492	388	2	)	)	PUNCT
ejpam-6492	389	1	−	−	PROPN
ejpam-6492	390	1	(	(	PUNCT
ejpam-6492	390	2	α2n+4	α2n+4	NUM
ejpam-6492	390	3	−	−	NOUN
ejpam-6492	390	4	β2n+4	β2n+4	PUNCT
ejpam-6492	390	5	)	)	PUNCT
ejpam-6492	390	6	−	−	PROPN
ejpam-6492	391	1	(	(	PUNCT
ejpam-6492	391	2	α2n+6	α2n+6	NOUN
ejpam-6492	391	3	−	−	NOUN
ejpam-6492	391	4	β2n+6	β2n+6	NOUN
ejpam-6492	391	5	)	)	PUNCT
ejpam-6492	391	6	]	]	PUNCT
ejpam-6492	392	1	=	=	PUNCT
ejpam-6492	392	2	(	(	PUNCT
ejpam-6492	392	3	α	α	NOUN
ejpam-6492	392	4	−	−	NOUN
ejpam-6492	392	5	β)2	β)2	NOUN
ejpam-6492	392	6	[	[	PUNCT
ejpam-6492	392	7	α2r	α2r	PROPN
ejpam-6492	392	8	−	−	ADP
ejpam-6492	392	9	β2r	β2r	PUNCT
ejpam-6492	392	10	α	α	NOUN
ejpam-6492	392	11	−	−	NOUN
ejpam-6492	392	12	β	β	X
ejpam-6492	392	13	]	]	X
ejpam-6492	392	14	[	[	PUNCT
ejpam-6492	392	15	2	2	NUM
ejpam-6492	392	16	α̂α2n	α̂α2n	NOUN
ejpam-6492	392	17	−	−	ADP
ejpam-6492	392	18	β̂β2n	β̂β2n	NOUN
ejpam-6492	392	19	α	α	NOUN
ejpam-6492	392	20	−	−	NOUN
ejpam-6492	392	21	β	β	NOUN
ejpam-6492	392	22	−	−	PROPN
ejpam-6492	392	23	α2n	α2n	PROPN
ejpam-6492	392	24	−	−	PROPN
ejpam-6492	392	25	β2n	β2n	PUNCT
ejpam-6492	392	26	α	α	NOUN
ejpam-6492	392	27	−	−	NOUN
ejpam-6492	392	28	β	β	X
ejpam-6492	392	29	−	−	PROPN
ejpam-6492	393	1	α2n+2	α2n+2	PRON
ejpam-6492	393	2	−	−	PROPN
ejpam-6492	394	1	β2n+2	β2n+2	ADV
ejpam-6492	395	1	α	α	INTJ
ejpam-6492	395	2	−	−	NOUN
ejpam-6492	396	1	β	β	X
ejpam-6492	397	1	−	−	PROPN
ejpam-6492	398	1	α2n+4	α2n+4	PRON
ejpam-6492	398	2	−	−	PROPN
ejpam-6492	398	3	β2n+4	β2n+4	INTJ
ejpam-6492	399	1	α	α	X
ejpam-6492	399	2	−	−	VERB
ejpam-6492	399	3	β	β	NOUN
ejpam-6492	399	4	−	−	NOUN
ejpam-6492	399	5	α2n+6	α2n+6	NOUN
ejpam-6492	399	6	−	−	NOUN
ejpam-6492	399	7	β2n+6	β2n+6	NOUN
ejpam-6492	400	1	α	α	NOUN
ejpam-6492	400	2	−	−	NOUN
ejpam-6492	400	3	β	β	X
ejpam-6492	400	4	]	]	X
ejpam-6492	400	5	=	=	PUNCT
ejpam-6492	400	6	∆u2r	∆u2r	X
ejpam-6492	400	7	[	[	X
ejpam-6492	400	8	2q2n	2q2n	NUM
ejpam-6492	400	9	−	−	PROPN
ejpam-6492	400	10	(	(	PUNCT
ejpam-6492	400	11	u2n	u2n	PROPN
ejpam-6492	400	12	+	+	CCONJ
ejpam-6492	400	13	u2n+2	u2n+2	PROPN
ejpam-6492	401	1	+	+	CCONJ
ejpam-6492	401	2	u2n+4	u2n+4	X
ejpam-6492	401	3	+	+	CCONJ
ejpam-6492	401	4	u2n+6	u2n+6	NOUN
ejpam-6492	401	5	)	)	PUNCT
ejpam-6492	401	6	]	]	PUNCT
ejpam-6492	401	7	as	as	ADP
ejpam-6492	401	8	a	a	DET
ejpam-6492	401	9	consequence	consequence	NOUN
ejpam-6492	401	10	of	of	ADP
ejpam-6492	401	11	theorem	theorem	ADJ
ejpam-6492	401	12	10	10	NUM
ejpam-6492	401	13	and	and	CCONJ
ejpam-6492	401	14	theorem	theorem	VERB
ejpam-6492	401	15	11	11	NUM
ejpam-6492	401	16	,	,	PUNCT
ejpam-6492	401	17	we	we	PRON
ejpam-6492	401	18	can	can	AUX
ejpam-6492	401	19	give	give	VERB
ejpam-6492	401	20	corollary	corollary	ADJ
ejpam-6492	401	21	4	4	NUM
ejpam-6492	401	22	.	.	PUNCT
ejpam-6492	401	23	corollary	corollary	ADJ
ejpam-6492	401	24	4	4	NUM
ejpam-6492	401	25	.	.	PUNCT
ejpam-6492	402	1	for	for	ADP
ejpam-6492	402	2	all	all	DET
ejpam-6492	402	3	n	n	NOUN
ejpam-6492	402	4	,	,	PUNCT
ejpam-6492	402	5	r	r	NOUN
ejpam-6492	402	6	∈	∈	PROPN
ejpam-6492	402	7	z	z	X
ejpam-6492	402	8	,	,	PUNCT
ejpam-6492	402	9	it	it	PRON
ejpam-6492	402	10	follows	follow	VERB
ejpam-6492	402	11	that	that	SCONJ
ejpam-6492	402	12	k2	k2	PROPN
ejpam-6492	402	13	n+r	n+r	PROPN
ejpam-6492	402	14	−	−	PROPN
ejpam-6492	402	15	q2rk2	q2rk2	NOUN
ejpam-6492	402	16	n−r	n−r	NOUN
ejpam-6492	402	17	=	=	SYM
ejpam-6492	402	18	∆	∆	X
ejpam-6492	402	19	[	[	PUNCT
ejpam-6492	402	20	q2	q2	NOUN
ejpam-6492	402	21	n+r	n+r	PROPN
ejpam-6492	402	22	−	−	PROPN
ejpam-6492	402	23	q2rq2	q2rq2	NOUN
ejpam-6492	402	24	n−r	n−r	NOUN
ejpam-6492	402	25	]	]	PUNCT
ejpam-6492	402	26	.	.	PUNCT
ejpam-6492	403	1	theorem	theorem	NOUN
ejpam-6492	403	2	12	12	NUM
ejpam-6492	403	3	.	.	PUNCT
ejpam-6492	404	1	for	for	ADP
ejpam-6492	404	2	all	all	DET
ejpam-6492	404	3	n	n	NOUN
ejpam-6492	404	4	,	,	PUNCT
ejpam-6492	404	5	r	r	NOUN
ejpam-6492	404	6	∈	∈	PROPN
ejpam-6492	404	7	z	z	X
ejpam-6492	404	8	,	,	PUNCT
ejpam-6492	404	9	it	it	PRON
ejpam-6492	404	10	follows	follow	VERB
ejpam-6492	404	11	that	that	SCONJ
ejpam-6492	404	12	k2	k2	PROPN
ejpam-6492	404	13	n	n	CCONJ
ejpam-6492	404	14	−	−	PROPN
ejpam-6492	404	15	∆q2	∆q2	PROPN
ejpam-6492	404	16	n	n	PROPN
ejpam-6492	404	17	=	=	SYM
ejpam-6492	404	18	4	4	NUM
ejpam-6492	404	19	(	(	PUNCT
ejpam-6492	404	20	−q)n	−q)n	NOUN
ejpam-6492	404	21	a	a	X
ejpam-6492	404	22	,	,	PUNCT
ejpam-6492	404	23	where	where	SCONJ
ejpam-6492	404	24	a	a	DET
ejpam-6492	404	25	=	=	X
ejpam-6492	404	26	k0	k0	PROPN
ejpam-6492	404	27	−	−	PROPN
ejpam-6492	404	28	(	(	PUNCT
ejpam-6492	404	29	1	1	NUM
ejpam-6492	404	30	−	−	NUM
ejpam-6492	404	31	q)(1	q)(1	PROPN
ejpam-6492	404	32	+	+	X
ejpam-6492	404	33	q2	q2	NOUN
ejpam-6492	404	34	)	)	PUNCT
ejpam-6492	404	35	and	and	CCONJ
ejpam-6492	404	36	b	b	X
ejpam-6492	404	37	=	=	SYM
ejpam-6492	404	38	(	(	PUNCT
ejpam-6492	404	39	−q)i	−q)i	NOUN
ejpam-6492	404	40	+	+	CCONJ
ejpam-6492	404	41	(	(	PUNCT
ejpam-6492	404	42	−p)j	−p)j	PROPN
ejpam-6492	404	43	+	+	PROPN
ejpam-6492	404	44	k.	k.	PROPN
ejpam-6492	404	45	b.	b.	PROPN
ejpam-6492	404	46	demirtürk	demirtürk	PROPN
ejpam-6492	404	47	,	,	PUNCT
ejpam-6492	404	48	n.	n.	NOUN
ejpam-6492	404	49	topal	topal	PROPN
ejpam-6492	404	50	/	/	SYM
ejpam-6492	404	51	eur	eur	PROPN
ejpam-6492	404	52	.	.	PUNCT
ejpam-6492	405	1	j.	j.	PROPN
ejpam-6492	405	2	pure	pure	PROPN
ejpam-6492	405	3	appl	appl	PROPN
ejpam-6492	405	4	.	.	PROPN
ejpam-6492	405	5	math	math	PROPN
ejpam-6492	405	6	,	,	PUNCT
ejpam-6492	405	7	18	18	NUM
ejpam-6492	405	8	(	(	PUNCT
ejpam-6492	405	9	3	3	NUM
ejpam-6492	405	10	)	)	PUNCT
ejpam-6492	405	11	(	(	PUNCT
ejpam-6492	405	12	2025	2025	NUM
ejpam-6492	405	13	)	)	PUNCT
ejpam-6492	405	14	,	,	PUNCT
ejpam-6492	405	15	6492	6492	NUM
ejpam-6492	405	16	17	17	NUM
ejpam-6492	405	17	of	of	ADP
ejpam-6492	405	18	22	22	NUM
ejpam-6492	405	19	proof	proof	NOUN
ejpam-6492	405	20	.	.	PUNCT
ejpam-6492	406	1	k2	k2	ADJ
ejpam-6492	406	2	n	n	CCONJ
ejpam-6492	406	3	−	−	PROPN
ejpam-6492	406	4	∆q2	∆q2	PROPN
ejpam-6492	406	5	n	n	PROPN
ejpam-6492	406	6	=	=	SYM
ejpam-6492	406	7	(	(	PUNCT
ejpam-6492	406	8	α̂αn	α̂αn	NOUN
ejpam-6492	406	9	+	+	CCONJ
ejpam-6492	406	10	β̂βn	β̂βn	INTJ
ejpam-6492	406	11	)	)	PUNCT
ejpam-6492	406	12	2	2	NUM
ejpam-6492	406	13	−	−	NOUN
ejpam-6492	406	14	∆	∆	PROPN
ejpam-6492	406	15	(	(	PUNCT
ejpam-6492	406	16	α̂αn	α̂αn	NOUN
ejpam-6492	406	17	−	−	PROPN
ejpam-6492	406	18	β̂βn	β̂βn	ADV
ejpam-6492	406	19	α	α	NOUN
ejpam-6492	407	1	−	−	NOUN
ejpam-6492	408	1	β	β	X
ejpam-6492	408	2	)	)	PUNCT
ejpam-6492	408	3	2	2	NUM
ejpam-6492	408	4	=	=	SYM
ejpam-6492	408	5	[	[	PUNCT
ejpam-6492	408	6	α̂α̂α2n	α̂α̂α2n	PROPN
ejpam-6492	408	7	+	+	CCONJ
ejpam-6492	408	8	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	408	9	+	+	CCONJ
ejpam-6492	408	10	(	(	PUNCT
ejpam-6492	408	11	αβ)n	αβ)n	PROPN
ejpam-6492	408	12	(	(	PUNCT
ejpam-6492	408	13	α̂β̂	α̂β̂	X
ejpam-6492	408	14	+	+	CCONJ
ejpam-6492	408	15	β̂α̂	β̂α̂	PUNCT
ejpam-6492	408	16	)	)	PUNCT
ejpam-6492	408	17	]	]	PUNCT
ejpam-6492	409	1	−	−	PROPN
ejpam-6492	409	2	∆	∆	PROPN
ejpam-6492	409	3			PROPN
ejpam-6492	409	4	α̂α̂α2n	α̂α̂α2n	ADJ
ejpam-6492	409	5	+	+	PROPN
ejpam-6492	409	6	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	409	7	−	−	NOUN
ejpam-6492	409	8	(	(	PUNCT
ejpam-6492	409	9	αβ)n	αβ)n	PROPN
ejpam-6492	409	10	(	(	PUNCT
ejpam-6492	409	11	α̂β̂	α̂β̂	X
ejpam-6492	409	12	+	+	CCONJ
ejpam-6492	409	13	β̂α̂	β̂α̂	PUNCT
ejpam-6492	409	14	)	)	PUNCT
ejpam-6492	409	15	(	(	PUNCT
ejpam-6492	409	16	α	α	NOUN
ejpam-6492	409	17	−	−	NOUN
ejpam-6492	409	18	β)2	β)2	ADV
ejpam-6492	409	19			PUNCT
ejpam-6492	409	20	=	=	PUNCT
ejpam-6492	409	21	[	[	PUNCT
ejpam-6492	409	22	α̂α̂α2n	α̂α̂α2n	PROPN
ejpam-6492	409	23	+	+	CCONJ
ejpam-6492	409	24	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	409	25	+	+	CCONJ
ejpam-6492	409	26	(	(	PUNCT
ejpam-6492	409	27	αβ)n	αβ)n	PROPN
ejpam-6492	409	28	(	(	PUNCT
ejpam-6492	409	29	α̂β̂	α̂β̂	X
ejpam-6492	409	30	+	+	CCONJ
ejpam-6492	409	31	β̂α̂	β̂α̂	PUNCT
ejpam-6492	409	32	)	)	PUNCT
ejpam-6492	409	33	]	]	PUNCT
ejpam-6492	410	1	−	−	PROPN
ejpam-6492	410	2	[	[	PUNCT
ejpam-6492	410	3	α̂α̂α2n	α̂α̂α2n	PROPN
ejpam-6492	410	4	+	+	CCONJ
ejpam-6492	410	5	β̂β̂β2n	β̂β̂β2n	NOUN
ejpam-6492	410	6	−	−	NOUN
ejpam-6492	410	7	(	(	PUNCT
ejpam-6492	410	8	αβ)n	αβ)n	PROPN
ejpam-6492	410	9	(	(	PUNCT
ejpam-6492	410	10	α̂β̂	α̂β̂	X
ejpam-6492	410	11	+	+	CCONJ
ejpam-6492	410	12	β̂α̂	β̂α̂	PUNCT
ejpam-6492	410	13	)	)	PUNCT
ejpam-6492	410	14	]	]	PUNCT
ejpam-6492	411	1	=	=	SYM
ejpam-6492	411	2	2	2	X
ejpam-6492	411	3	(	(	PUNCT
ejpam-6492	411	4	αβ)n	αβ)n	PROPN
ejpam-6492	411	5	(	(	PUNCT
ejpam-6492	411	6	2a	2a	NUM
ejpam-6492	411	7	)	)	PUNCT
ejpam-6492	411	8	=	=	SYM
ejpam-6492	411	9	4	4	NUM
ejpam-6492	411	10	(	(	PUNCT
ejpam-6492	411	11	−q)n	−q)n	NOUN
ejpam-6492	411	12	a.	a.	NOUN
ejpam-6492	411	13	theorem	theorem	VERB
ejpam-6492	411	14	13	13	NUM
ejpam-6492	411	15	.	.	PUNCT
ejpam-6492	412	1	for	for	ADP
ejpam-6492	412	2	all	all	DET
ejpam-6492	412	3	m	m	PROPN
ejpam-6492	412	4	,	,	PUNCT
ejpam-6492	412	5	n	n	PROPN
ejpam-6492	412	6	∈	∈	PROPN
ejpam-6492	412	7	z	z	NOUN
ejpam-6492	412	8	,	,	PUNCT
ejpam-6492	412	9	it	it	PRON
ejpam-6492	412	10	follows	follow	VERB
ejpam-6492	412	11	that	that	SCONJ
ejpam-6492	412	12	k2mk2n	k2mk2n	PROPN
ejpam-6492	413	1	−	−	PROPN
ejpam-6492	414	1	∆q2	∆q2	PROPN
ejpam-6492	414	2	m+n	m+n	PROPN
ejpam-6492	414	3	=	=	PUNCT
ejpam-6492	414	4	(	(	PUNCT
ejpam-6492	414	5	−q)2n	−q)2n	NUM
ejpam-6492	414	6	vm−n	vm−n	PROPN
ejpam-6492	414	7	[	[	X
ejpam-6492	414	8	avm−n	avm−n	ADJ
ejpam-6492	414	9	+	+	X
ejpam-6492	414	10	bq∆um−n	bq∆um−n	PROPN
ejpam-6492	414	11	]	]	PUNCT
ejpam-6492	414	12	,	,	PUNCT
ejpam-6492	414	13	where	where	SCONJ
ejpam-6492	414	14	a	a	DET
ejpam-6492	414	15	=	=	X
ejpam-6492	414	16	k0	k0	PROPN
ejpam-6492	414	17	−	−	PROPN
ejpam-6492	414	18	(	(	PUNCT
ejpam-6492	414	19	1	1	NUM
ejpam-6492	414	20	−	−	NUM
ejpam-6492	414	21	q)(1	q)(1	PROPN
ejpam-6492	414	22	+	+	X
ejpam-6492	414	23	q2	q2	NOUN
ejpam-6492	414	24	)	)	PUNCT
ejpam-6492	414	25	and	and	CCONJ
ejpam-6492	414	26	b	b	X
ejpam-6492	414	27	=	=	SYM
ejpam-6492	414	28	(	(	PUNCT
ejpam-6492	414	29	−q)i	−q)i	NOUN
ejpam-6492	414	30	+	+	CCONJ
ejpam-6492	414	31	(	(	PUNCT
ejpam-6492	414	32	−p)j	−p)j	NOUN
ejpam-6492	414	33	+	+	CCONJ
ejpam-6492	414	34	k.	k.	PROPN
ejpam-6492	414	35	proof	proof	NOUN
ejpam-6492	414	36	.	.	PUNCT
ejpam-6492	414	37	k2mk2n−∆q2	k2mk2n−∆q2	PROPN
ejpam-6492	414	38	m+n	m+n	PROPN
ejpam-6492	415	1	=	=	PUNCT
ejpam-6492	415	2	(	(	PUNCT
ejpam-6492	415	3	α̂α2	α̂α2	PROPN
ejpam-6492	415	4	m	m	PROPN
ejpam-6492	415	5	+	+	NOUN
ejpam-6492	415	6	β̂β2	β̂β2	PROPN
ejpam-6492	415	7	m	m	NOUN
ejpam-6492	415	8	)	)	PUNCT
ejpam-6492	415	9	(	(	PUNCT
ejpam-6492	415	10	α̂α2n	α̂α2n	ADP
ejpam-6492	415	11	+	+	CCONJ
ejpam-6492	415	12	β̂β2n	β̂β2n	X
ejpam-6492	415	13	)	)	PUNCT
ejpam-6492	415	14	−∆	−∆	NOUN
ejpam-6492	415	15	(	(	PUNCT
ejpam-6492	415	16	α̂αm+n	α̂αm+n	NOUN
ejpam-6492	415	17	−	−	PROPN
ejpam-6492	415	18	β̂βm+n	β̂βm+n	PROPN
ejpam-6492	416	1	α	α	NOUN
ejpam-6492	416	2	−	−	NOUN
ejpam-6492	416	3	β	β	NOUN
ejpam-6492	416	4	)	)	PUNCT
ejpam-6492	416	5	2	2	NUM
ejpam-6492	416	6	=	=	PUNCT
ejpam-6492	416	7	[	[	PUNCT
ejpam-6492	416	8	α̂α̂α2m+2n	α̂α̂α2m+2n	NUM
ejpam-6492	416	9	+	+	CCONJ
ejpam-6492	416	10	β̂β̂β2m+2n	β̂β̂β2m+2n	PUNCT
ejpam-6492	416	11	+	+	X
ejpam-6492	416	12	α̂β̂α2mβ2n	α̂β̂α2mβ2n	VERB
ejpam-6492	416	13	+	+	CCONJ
ejpam-6492	417	1	β̂α̂α2nβ2	β̂α̂α2nβ2	PROPN
ejpam-6492	417	2	m	m	VERB
ejpam-6492	417	3	]	]	X
ejpam-6492	417	4	−	−	PROPN
ejpam-6492	418	1	[	[	PUNCT
ejpam-6492	418	2	α̂α̂α2m+2n	α̂α̂α2m+2n	X
ejpam-6492	418	3	+	+	CCONJ
ejpam-6492	418	4	β̂β̂β2m+2n	β̂β̂β2m+2n	NUM
ejpam-6492	418	5	−	−	PROPN
ejpam-6492	418	6	(	(	PUNCT
ejpam-6492	418	7	αβ)m+n	αβ)m+n	PROPN
ejpam-6492	418	8	(	(	PUNCT
ejpam-6492	418	9	α̂β̂	α̂β̂	X
ejpam-6492	418	10	+	+	CCONJ
ejpam-6492	418	11	β̂α̂	β̂α̂	PUNCT
ejpam-6492	418	12	)	)	PUNCT
ejpam-6492	418	13	]	]	PUNCT
ejpam-6492	419	1	=	=	SYM
ejpam-6492	419	2	α̂β̂	α̂β̂	NOUN
ejpam-6492	419	3	(	(	PUNCT
ejpam-6492	419	4	α2mβ2n	α2mβ2n	NUM
ejpam-6492	419	5	+	+	NUM
ejpam-6492	419	6	αm+nβm+n	αm+nβm+n	NUM
ejpam-6492	419	7	)	)	PUNCT
ejpam-6492	420	1	+	+	CCONJ
ejpam-6492	420	2	β̂α̂	β̂α̂	PUNCT
ejpam-6492	420	3	(	(	PUNCT
ejpam-6492	420	4	α2nβ2	α2nβ2	NOUN
ejpam-6492	420	5	m	m	VERB
ejpam-6492	420	6	+	+	X
ejpam-6492	420	7	αm+nβm+n	αm+nβm+n	NUM
ejpam-6492	420	8	)	)	PUNCT
ejpam-6492	420	9	=	=	NOUN
ejpam-6492	420	10	αm+nβ2nα̂β̂	αm+nβ2nα̂β̂	NOUN
ejpam-6492	420	11	(	(	PUNCT
ejpam-6492	420	12	αm−n	αm−n	NOUN
ejpam-6492	420	13	+	+	CCONJ
ejpam-6492	420	14	βm−n	βm−n	NOUN
ejpam-6492	420	15	)	)	PUNCT
ejpam-6492	421	1	+	+	CCONJ
ejpam-6492	421	2	α2nβm+nβ̂α̂	α2nβm+nβ̂α̂	PROPN
ejpam-6492	421	3	(	(	PUNCT
ejpam-6492	421	4	αm−n	αm−n	NOUN
ejpam-6492	421	5	+	+	SYM
ejpam-6492	421	6	βm−n	βm−n	NOUN
ejpam-6492	421	7	)	)	PUNCT
ejpam-6492	421	8	=	=	SYM
ejpam-6492	421	9	(	(	PUNCT
ejpam-6492	421	10	αm−n	αm−n	NOUN
ejpam-6492	421	11	+	+	SYM
ejpam-6492	421	12	βm−n	βm−n	NOUN
ejpam-6492	421	13	)	)	PUNCT
ejpam-6492	421	14	[	[	PUNCT
ejpam-6492	421	15	αm+nβ2nα̂β̂	αm+nβ2nα̂β̂	X
ejpam-6492	421	16	+	+	NUM
ejpam-6492	421	17	α2nβm+nβ̂α̂	α2nβm+nβ̂α̂	ADJ
ejpam-6492	421	18	]	]	X
ejpam-6492	422	1	=	=	PUNCT
ejpam-6492	422	2	(	(	PUNCT
ejpam-6492	422	3	αβ)2n	αβ)2n	NOUN
ejpam-6492	422	4	(	(	PUNCT
ejpam-6492	422	5	αm−n	αm−n	NOUN
ejpam-6492	422	6	+	+	CCONJ
ejpam-6492	422	7	βm−n	βm−n	NOUN
ejpam-6492	422	8	)	)	PUNCT
ejpam-6492	422	9	[	[	PUNCT
ejpam-6492	422	10	αm−nα̂β̂	αm−nα̂β̂	NOUN
ejpam-6492	422	11	+	+	CCONJ
ejpam-6492	422	12	βm−nβ̂α̂	βm−nβ̂α̂	X
ejpam-6492	422	13	]	]	X
ejpam-6492	422	14	=	=	PUNCT
ejpam-6492	422	15	(	(	PUNCT
ejpam-6492	422	16	αβ)2n	αβ)2n	NOUN
ejpam-6492	422	17	(	(	PUNCT
ejpam-6492	422	18	αm−n	αm−n	NOUN
ejpam-6492	422	19	+	+	SYM
ejpam-6492	422	20	βm−n	βm−n	NOUN
ejpam-6492	422	21	)	)	PUNCT
ejpam-6492	423	1	[	[	PUNCT
ejpam-6492	423	2	αm−n	αm−n	NOUN
ejpam-6492	423	3	(	(	PUNCT
ejpam-6492	423	4	a	a	DET
ejpam-6492	423	5	+	+	NOUN
ejpam-6492	423	6	bq	bq	NOUN
ejpam-6492	423	7	√	√	PROPN
ejpam-6492	423	8	∆	∆	PROPN
ejpam-6492	423	9	)	)	PUNCT
ejpam-6492	424	1	+	+	CCONJ
ejpam-6492	424	2	βm−n	βm−n	PROPN
ejpam-6492	424	3	(	(	PUNCT
ejpam-6492	424	4	a	a	DET
ejpam-6492	424	5	−	−	PROPN
ejpam-6492	424	6	bq	bq	NOUN
ejpam-6492	424	7	√	√	PROPN
ejpam-6492	424	8	∆	∆	PROPN
ejpam-6492	424	9	)	)	PUNCT
ejpam-6492	424	10	]	]	PUNCT
ejpam-6492	425	1	=	=	PUNCT
ejpam-6492	425	2	(	(	PUNCT
ejpam-6492	425	3	αβ)2n	αβ)2n	NOUN
ejpam-6492	425	4	(	(	PUNCT
ejpam-6492	425	5	αm−n	αm−n	NOUN
ejpam-6492	425	6	+	+	CCONJ
ejpam-6492	425	7	βm−n	βm−n	NOUN
ejpam-6492	425	8	)	)	PUNCT
ejpam-6492	425	9	[	[	PUNCT
ejpam-6492	425	10	a	a	DET
ejpam-6492	425	11	(	(	PUNCT
ejpam-6492	425	12	αm−n	αm−n	NOUN
ejpam-6492	425	13	+	+	CCONJ
ejpam-6492	425	14	βm−n	βm−n	NOUN
ejpam-6492	425	15	)	)	PUNCT
ejpam-6492	426	1	+	+	CCONJ
ejpam-6492	426	2	bq	bq	INTJ
ejpam-6492	426	3	√	√	ADJ
ejpam-6492	426	4	∆	∆	PROPN
ejpam-6492	426	5	(	(	PUNCT
ejpam-6492	426	6	αm−n	αm−n	NOUN
ejpam-6492	426	7	−	−	NOUN
ejpam-6492	426	8	βm−n	βm−n	NOUN
ejpam-6492	426	9	)	)	PUNCT
ejpam-6492	426	10	]	]	PUNCT
ejpam-6492	427	1	=	=	PUNCT
ejpam-6492	427	2	(	(	PUNCT
ejpam-6492	427	3	−q)2n	−q)2n	NUM
ejpam-6492	427	4	vm−n	vm−n	PROPN
ejpam-6492	427	5	[	[	X
ejpam-6492	427	6	avm−n	avm−n	ADJ
ejpam-6492	427	7	+	+	X
ejpam-6492	427	8	bq∆um−n	bq∆um−n	PROPN
ejpam-6492	427	9	]	]	PUNCT
ejpam-6492	427	10	.	.	PUNCT
ejpam-6492	428	1	theorem	theorem	VERB
ejpam-6492	428	2	14	14	NUM
ejpam-6492	428	3	.	.	PUNCT
ejpam-6492	429	1	for	for	ADP
ejpam-6492	429	2	all	all	DET
ejpam-6492	429	3	m	m	PROPN
ejpam-6492	429	4	,	,	PUNCT
ejpam-6492	429	5	n	n	PROPN
ejpam-6492	429	6	∈	∈	PROPN
ejpam-6492	429	7	z	z	NOUN
ejpam-6492	429	8	,	,	PUNCT
ejpam-6492	429	9	it	it	PRON
ejpam-6492	429	10	follows	follow	VERB
ejpam-6492	429	11	that	that	SCONJ
ejpam-6492	429	12	k2mk2n	k2mk2n	PROPN
ejpam-6492	429	13	−	−	PROPN
ejpam-6492	429	14	k2	k2	PROPN
ejpam-6492	429	15	m+n	m+n	PROPN
ejpam-6492	430	1	=	=	SYM
ejpam-6492	430	2	(	(	PUNCT
ejpam-6492	430	3	−q)2n	−q)2n	NUM
ejpam-6492	430	4	∆um−n	∆um−n	PROPN
ejpam-6492	430	5	[	[	X
ejpam-6492	430	6	aum−n	aum−n	NOUN
ejpam-6492	430	7	+	+	X
ejpam-6492	430	8	bqvm−n	bqvm−n	PROPN
ejpam-6492	430	9	]	]	X
ejpam-6492	430	10	,	,	PUNCT
ejpam-6492	430	11	where	where	SCONJ
ejpam-6492	430	12	a	a	DET
ejpam-6492	430	13	=	=	X
ejpam-6492	430	14	k0	k0	PROPN
ejpam-6492	430	15	−	−	PROPN
ejpam-6492	430	16	(	(	PUNCT
ejpam-6492	430	17	1	1	NUM
ejpam-6492	430	18	−	−	NUM
ejpam-6492	430	19	q)(1	q)(1	PROPN
ejpam-6492	430	20	+	+	X
ejpam-6492	430	21	q2	q2	NOUN
ejpam-6492	430	22	)	)	PUNCT
ejpam-6492	430	23	and	and	CCONJ
ejpam-6492	430	24	b	b	X
ejpam-6492	430	25	=	=	SYM
ejpam-6492	430	26	(	(	PUNCT
ejpam-6492	430	27	−q)i	−q)i	NOUN
ejpam-6492	430	28	+	+	CCONJ
ejpam-6492	430	29	(	(	PUNCT
ejpam-6492	430	30	−p)j	−p)j	NOUN
ejpam-6492	430	31	+	+	CCONJ
ejpam-6492	430	32	k.	k.	PROPN
ejpam-6492	430	33	proof	proof	NOUN
ejpam-6492	430	34	.	.	PUNCT
ejpam-6492	431	1	k2mk2n	k2mk2n	PROPN
ejpam-6492	432	1	−	−	PROPN
ejpam-6492	432	2	k2	k2	PROPN
ejpam-6492	432	3	m+n	m+n	PROPN
ejpam-6492	432	4	=	=	PUNCT
ejpam-6492	432	5	(	(	PUNCT
ejpam-6492	432	6	α̂α2	α̂α2	PROPN
ejpam-6492	432	7	m	m	PROPN
ejpam-6492	432	8	+	+	NOUN
ejpam-6492	432	9	β̂β2	β̂β2	PROPN
ejpam-6492	432	10	m	m	NOUN
ejpam-6492	432	11	)	)	PUNCT
ejpam-6492	432	12	(	(	PUNCT
ejpam-6492	432	13	α̂α2n	α̂α2n	ADP
ejpam-6492	432	14	+	+	CCONJ
ejpam-6492	432	15	β̂β2n	β̂β2n	X
ejpam-6492	432	16	)	)	PUNCT
ejpam-6492	432	17	−	−	PROPN
ejpam-6492	432	18	(	(	PUNCT
ejpam-6492	432	19	α̂αm+n	α̂αm+n	NOUN
ejpam-6492	432	20	+	+	CCONJ
ejpam-6492	432	21	β̂βm+n	β̂βm+n	ADJ
ejpam-6492	432	22	)	)	PUNCT
ejpam-6492	432	23	2	2	NUM
ejpam-6492	432	24	=	=	PUNCT
ejpam-6492	432	25	[	[	PUNCT
ejpam-6492	432	26	α̂α̂α2m+2n	α̂α̂α2m+2n	NUM
ejpam-6492	432	27	+	+	CCONJ
ejpam-6492	432	28	β̂β̂β2m+2n	β̂β̂β2m+2n	PUNCT
ejpam-6492	432	29	+	+	X
ejpam-6492	432	30	α̂β̂α2mβ2n	α̂β̂α2mβ2n	VERB
ejpam-6492	432	31	+	+	CCONJ
ejpam-6492	432	32	β̂α̂α2nβ2	β̂α̂α2nβ2	PROPN
ejpam-6492	432	33	m	m	VERB
ejpam-6492	432	34	]	]	X
ejpam-6492	432	35	−	−	PROPN
ejpam-6492	433	1	[	[	PUNCT
ejpam-6492	433	2	α̂α̂α2m+2n	α̂α̂α2m+2n	X
ejpam-6492	433	3	+	+	CCONJ
ejpam-6492	433	4	β̂β̂β2m+2n	β̂β̂β2m+2n	NUM
ejpam-6492	433	5	+	+	CCONJ
ejpam-6492	433	6	(	(	PUNCT
ejpam-6492	433	7	αβ)m+n	αβ)m+n	INTJ
ejpam-6492	433	8	(	(	PUNCT
ejpam-6492	433	9	α̂β̂	α̂β̂	X
ejpam-6492	433	10	+	+	CCONJ
ejpam-6492	433	11	β̂α̂	β̂α̂	PUNCT
ejpam-6492	433	12	)	)	PUNCT
ejpam-6492	433	13	]	]	PUNCT
ejpam-6492	434	1	=	=	SYM
ejpam-6492	434	2	α̂β̂	α̂β̂	NOUN
ejpam-6492	434	3	(	(	PUNCT
ejpam-6492	434	4	α2mβ2n	α2mβ2n	NUM
ejpam-6492	434	5	−	−	PROPN
ejpam-6492	434	6	αm+nβm+n	αm+nβm+n	NUM
ejpam-6492	434	7	)	)	PUNCT
ejpam-6492	435	1	+	+	CCONJ
ejpam-6492	435	2	β̂α̂	β̂α̂	PUNCT
ejpam-6492	435	3	(	(	PUNCT
ejpam-6492	435	4	α2nβ2	α2nβ2	PROPN
ejpam-6492	435	5	m	m	VERB
ejpam-6492	435	6	−	−	PROPN
ejpam-6492	435	7	αm+nβm+n	αm+nβm+n	PROPN
ejpam-6492	435	8	)	)	PUNCT
ejpam-6492	435	9	b.	b.	PROPN
ejpam-6492	435	10	demirtürk	demirtürk	PROPN
ejpam-6492	435	11	,	,	PUNCT
ejpam-6492	435	12	n.	n.	NOUN
ejpam-6492	435	13	topal	topal	PROPN
ejpam-6492	435	14	/	/	SYM
ejpam-6492	435	15	eur	eur	PROPN
ejpam-6492	435	16	.	.	PUNCT
ejpam-6492	436	1	j.	j.	PROPN
ejpam-6492	436	2	pure	pure	PROPN
ejpam-6492	436	3	appl	appl	PROPN
ejpam-6492	436	4	.	.	PROPN
ejpam-6492	436	5	math	math	PROPN
ejpam-6492	436	6	,	,	PUNCT
ejpam-6492	436	7	18	18	NUM
ejpam-6492	436	8	(	(	PUNCT
ejpam-6492	436	9	3	3	NUM
ejpam-6492	436	10	)	)	PUNCT
ejpam-6492	436	11	(	(	PUNCT
ejpam-6492	436	12	2025	2025	NUM
ejpam-6492	436	13	)	)	PUNCT
ejpam-6492	436	14	,	,	PUNCT
ejpam-6492	436	15	6492	6492	NUM
ejpam-6492	436	16	18	18	NUM
ejpam-6492	436	17	of	of	ADP
ejpam-6492	436	18	22	22	NUM
ejpam-6492	436	19	=	=	SYM
ejpam-6492	436	20	αm+nβ2nα̂β̂	αm+nβ2nα̂β̂	NOUN
ejpam-6492	436	21	(	(	PUNCT
ejpam-6492	436	22	αm−n	αm−n	NOUN
ejpam-6492	436	23	−	−	NOUN
ejpam-6492	436	24	βm−n	βm−n	NOUN
ejpam-6492	436	25	)	)	PUNCT
ejpam-6492	437	1	+	+	CCONJ
ejpam-6492	437	2	α2nβm+nβ̂α̂	α2nβm+nβ̂α̂	PROPN
ejpam-6492	437	3	(	(	PUNCT
ejpam-6492	437	4	βm−n	βm−n	PROPN
ejpam-6492	437	5	−	−	PROPN
ejpam-6492	437	6	αm−n	αm−n	NOUN
ejpam-6492	437	7	)	)	PUNCT
ejpam-6492	437	8	=	=	PUNCT
ejpam-6492	437	9	(	(	PUNCT
ejpam-6492	437	10	αm−n	αm−n	NOUN
ejpam-6492	437	11	−	−	NOUN
ejpam-6492	437	12	βm−n	βm−n	NOUN
ejpam-6492	437	13	)	)	PUNCT
ejpam-6492	437	14	[	[	PUNCT
ejpam-6492	437	15	αm+nβ2nα̂β̂	αm+nβ2nα̂β̂	SYM
ejpam-6492	437	16	−	−	PROPN
ejpam-6492	437	17	α2nβm+nβ̂α̂	α2nβm+nβ̂α̂	NOUN
ejpam-6492	437	18	]	]	X
ejpam-6492	438	1	=	=	PUNCT
ejpam-6492	438	2	(	(	PUNCT
ejpam-6492	438	3	αβ)2n	αβ)2n	NOUN
ejpam-6492	438	4	(	(	PUNCT
ejpam-6492	438	5	αm−n	αm−n	NOUN
ejpam-6492	438	6	−	−	NOUN
ejpam-6492	438	7	βm−n	βm−n	NOUN
ejpam-6492	438	8	)	)	PUNCT
ejpam-6492	438	9	[	[	PUNCT
ejpam-6492	438	10	αm−nα̂β̂	αm−nα̂β̂	NUM
ejpam-6492	438	11	−	−	NOUN
ejpam-6492	438	12	βm−nβ̂α̂	βm−nβ̂α̂	PROPN
ejpam-6492	438	13	]	]	X
ejpam-6492	438	14	=	=	PUNCT
ejpam-6492	438	15	(	(	PUNCT
ejpam-6492	438	16	αβ)2n	αβ)2n	NOUN
ejpam-6492	438	17	(	(	PUNCT
ejpam-6492	438	18	αm−n	αm−n	NOUN
ejpam-6492	438	19	−	−	NOUN
ejpam-6492	438	20	βm−n	βm−n	NOUN
ejpam-6492	438	21	)	)	PUNCT
ejpam-6492	439	1	[	[	PUNCT
ejpam-6492	439	2	αm−n	αm−n	NOUN
ejpam-6492	439	3	(	(	PUNCT
ejpam-6492	439	4	a	a	DET
ejpam-6492	439	5	+	+	NOUN
ejpam-6492	439	6	bq	bq	NOUN
ejpam-6492	439	7	√	√	PROPN
ejpam-6492	439	8	∆	∆	PROPN
ejpam-6492	439	9	)	)	PUNCT
ejpam-6492	440	1	−	−	ADP
ejpam-6492	440	2	βm−n	βm−n	PROPN
ejpam-6492	440	3	(	(	PUNCT
ejpam-6492	440	4	a	a	DET
ejpam-6492	440	5	−	−	PROPN
ejpam-6492	440	6	bq	bq	NOUN
ejpam-6492	440	7	√	√	PROPN
ejpam-6492	440	8	∆	∆	PROPN
ejpam-6492	440	9	)	)	PUNCT
ejpam-6492	440	10	]	]	PUNCT
ejpam-6492	441	1	=	=	PUNCT
ejpam-6492	441	2	(	(	PUNCT
ejpam-6492	441	3	αβ)2n	αβ)2n	NOUN
ejpam-6492	441	4	(	(	PUNCT
ejpam-6492	441	5	αm−n	αm−n	NOUN
ejpam-6492	441	6	−	−	NOUN
ejpam-6492	441	7	βm−n	βm−n	NOUN
ejpam-6492	441	8	)	)	PUNCT
ejpam-6492	441	9	[	[	PUNCT
ejpam-6492	441	10	a	a	DET
ejpam-6492	441	11	(	(	PUNCT
ejpam-6492	441	12	αm−n	αm−n	NOUN
ejpam-6492	441	13	−	−	NOUN
ejpam-6492	441	14	βm−n	βm−n	NOUN
ejpam-6492	441	15	)	)	PUNCT
ejpam-6492	442	1	+	+	CCONJ
ejpam-6492	442	2	bq	bq	INTJ
ejpam-6492	442	3	√	√	ADJ
ejpam-6492	442	4	∆	∆	PROPN
ejpam-6492	442	5	(	(	PUNCT
ejpam-6492	442	6	αm−n	αm−n	NOUN
ejpam-6492	442	7	+	+	NUM
ejpam-6492	442	8	βm−n	βm−n	NOUN
ejpam-6492	442	9	)	)	PUNCT
ejpam-6492	442	10	]	]	PUNCT
ejpam-6492	443	1	=	=	PUNCT
ejpam-6492	443	2	(	(	PUNCT
ejpam-6492	443	3	−q)2n	−q)2n	NUM
ejpam-6492	443	4	√	√	ADJ
ejpam-6492	443	5	∆um−n	∆um−n	PROPN
ejpam-6492	443	6	[	[	PUNCT
ejpam-6492	443	7	a	a	DET
ejpam-6492	443	8	√	√	ADJ
ejpam-6492	443	9	∆um−n	∆um−n	PROPN
ejpam-6492	443	10	+	+	CCONJ
ejpam-6492	443	11	bq	bq	NOUN
ejpam-6492	443	12	√	√	NOUN
ejpam-6492	443	13	∆vm−n	∆vm−n	PROPN
ejpam-6492	443	14	]	]	PUNCT
ejpam-6492	444	1	=	=	PUNCT
ejpam-6492	444	2	(	(	PUNCT
ejpam-6492	444	3	−q)2n	−q)2n	NUM
ejpam-6492	444	4	∆um−n	∆um−n	PROPN
ejpam-6492	444	5	[	[	X
ejpam-6492	444	6	aum−n	aum−n	NOUN
ejpam-6492	444	7	+	+	X
ejpam-6492	444	8	bqvm−n	bqvm−n	PROPN
ejpam-6492	444	9	]	]	PUNCT
ejpam-6492	444	10	.	.	PUNCT
ejpam-6492	445	1	let	let	VERB
ejpam-6492	445	2	us	we	PRON
ejpam-6492	445	3	generalize	generalize	VERB
ejpam-6492	445	4	everman	everman	NOUN
ejpam-6492	445	5	’s	’s	PART
ejpam-6492	445	6	product	product	NOUN
ejpam-6492	445	7	difference	difference	NOUN
ejpam-6492	445	8	of	of	ADP
ejpam-6492	445	9	fibonacci	fibonacci	NOUN
ejpam-6492	445	10	identities	identity	NOUN
ejpam-6492	445	11	to	to	ADP
ejpam-6492	445	12	generalized	generalize	VERB
ejpam-6492	445	13	fibonacci	fibonacci	NOUN
ejpam-6492	445	14	quaternions	quaternion	NOUN
ejpam-6492	445	15	in	in	ADP
ejpam-6492	445	16	theorem	theorem	ADJ
ejpam-6492	445	17	15	15	NUM
ejpam-6492	445	18	.	.	PUNCT
ejpam-6492	445	19	theorem	theorem	VERB
ejpam-6492	445	20	15	15	NUM
ejpam-6492	445	21	.	.	PUNCT
ejpam-6492	446	1	for	for	ADP
ejpam-6492	446	2	all	all	DET
ejpam-6492	446	3	n	n	NOUN
ejpam-6492	446	4	,	,	PUNCT
ejpam-6492	446	5	k	k	PROPN
ejpam-6492	446	6	and	and	CCONJ
ejpam-6492	446	7	h	h	NOUN
ejpam-6492	446	8	∈	∈	PROPN
ejpam-6492	447	1	z	z	X
ejpam-6492	447	2	,	,	PUNCT
ejpam-6492	447	3	it	it	PRON
ejpam-6492	447	4	follows	follow	VERB
ejpam-6492	447	5	that	that	SCONJ
ejpam-6492	448	1	qn+hqn+k	qn+hqn+k	PROPN
ejpam-6492	448	2	−	−	NOUN
ejpam-6492	448	3	qnqn+k+h	qnqn+k+h	PROPN
ejpam-6492	448	4	=	=	SYM
ejpam-6492	448	5	(	(	PUNCT
ejpam-6492	448	6	−q)n	−q)n	NOUN
ejpam-6492	448	7	uh	uh	INTJ
ejpam-6492	448	8	[	[	X
ejpam-6492	448	9	auk	auk	NOUN
ejpam-6492	448	10	−	−	PROPN
ejpam-6492	448	11	qbvk	qbvk	NOUN
ejpam-6492	448	12	]	]	PUNCT
ejpam-6492	448	13	,	,	PUNCT
ejpam-6492	448	14	where	where	SCONJ
ejpam-6492	448	15	a	a	DET
ejpam-6492	448	16	=	=	X
ejpam-6492	448	17	k0	k0	PROPN
ejpam-6492	448	18	−	−	PROPN
ejpam-6492	448	19	(	(	PUNCT
ejpam-6492	448	20	1	1	NUM
ejpam-6492	448	21	−	−	NUM
ejpam-6492	448	22	q)(1	q)(1	PROPN
ejpam-6492	448	23	+	+	X
ejpam-6492	448	24	q2	q2	NOUN
ejpam-6492	448	25	)	)	PUNCT
ejpam-6492	448	26	and	and	CCONJ
ejpam-6492	448	27	b	b	X
ejpam-6492	448	28	=	=	SYM
ejpam-6492	448	29	(	(	PUNCT
ejpam-6492	448	30	−q)i	−q)i	NOUN
ejpam-6492	448	31	+	+	CCONJ
ejpam-6492	448	32	(	(	PUNCT
ejpam-6492	448	33	−p)j	−p)j	NOUN
ejpam-6492	448	34	+	+	CCONJ
ejpam-6492	448	35	k.	k.	PROPN
ejpam-6492	448	36	proof	proof	NOUN
ejpam-6492	448	37	.	.	PUNCT
ejpam-6492	449	1	qn+hqn+k−qnqn+k+h	qn+hqn+k−qnqn+k+h	X
ejpam-6492	449	2	=	=	SYM
ejpam-6492	450	1	αn+hα̂	αn+hα̂	NOUN
ejpam-6492	450	2	−	−	NOUN
ejpam-6492	450	3	βn+hβ̂	βn+hβ̂	NOUN
ejpam-6492	451	1	α	α	NOUN
ejpam-6492	451	2	−	−	NOUN
ejpam-6492	451	3	β	β	X
ejpam-6492	451	4	αn+kα̂	αn+kα̂	PROPN
ejpam-6492	451	5	−	−	PROPN
ejpam-6492	452	1	βn+kβ̂	βn+kβ̂	NOUN
ejpam-6492	452	2	α	α	NOUN
ejpam-6492	452	3	−	−	NOUN
ejpam-6492	452	4	β	β	NOUN
ejpam-6492	452	5	−αnα̂	−αnα̂	NOUN
ejpam-6492	452	6	−	−	PROPN
ejpam-6492	452	7	βnβ̂	βnβ̂	PROPN
ejpam-6492	453	1	α	α	NOUN
ejpam-6492	453	2	−	−	NOUN
ejpam-6492	453	3	β	β	X
ejpam-6492	453	4	αn+h+kα̂	αn+h+kα̂	PUNCT
ejpam-6492	454	1	−	−	PROPN
ejpam-6492	454	2	βn+h+kβ̂	βn+h+kβ̂	PUNCT
ejpam-6492	455	1	α	α	NOUN
ejpam-6492	455	2	−	−	NOUN
ejpam-6492	455	3	β	β	X
ejpam-6492	455	4	=	=	PUNCT
ejpam-6492	456	1	[	[	PUNCT
ejpam-6492	456	2	αn+hα̂	αn+hα̂	NOUN
ejpam-6492	456	3	−	−	NOUN
ejpam-6492	456	4	βn+hβ̂	βn+hβ̂	SYM
ejpam-6492	456	5	]	]	X
ejpam-6492	456	6	[	[	PUNCT
ejpam-6492	456	7	αn+kα̂	αn+kα̂	PROPN
ejpam-6492	456	8	−	−	PROPN
ejpam-6492	456	9	βn+kβ̂	βn+kβ̂	PROPN
ejpam-6492	456	10	]	]	PUNCT
ejpam-6492	456	11	−	−	PROPN
ejpam-6492	457	1	[	[	PUNCT
ejpam-6492	457	2	αnα̂	αnα̂	PROPN
ejpam-6492	457	3	−	−	PROPN
ejpam-6492	457	4	βnβ̂	βnβ̂	PROPN
ejpam-6492	457	5	]	]	X
ejpam-6492	458	1	[	[	PUNCT
ejpam-6492	458	2	αn+h+kα̂	αn+h+kα̂	NOUN
ejpam-6492	458	3	−	−	PROPN
ejpam-6492	458	4	βn+h+kβ̂	βn+h+kβ̂	PUNCT
ejpam-6492	458	5	]	]	PUNCT
ejpam-6492	458	6	(	(	PUNCT
ejpam-6492	458	7	α	α	NOUN
ejpam-6492	458	8	−	−	NOUN
ejpam-6492	458	9	β)2	β)2	NOUN
ejpam-6492	458	10	=	=	SYM
ejpam-6492	458	11	[	[	PUNCT
ejpam-6492	458	12	α2n+h+k	α2n+h+k	NOUN
ejpam-6492	458	13	α̂α̂	α̂α̂	NUM
ejpam-6492	458	14	−	−	NOUN
ejpam-6492	459	1	αn+hβn+k	αn+hβn+k	NOUN
ejpam-6492	459	2	α̂β̂	α̂β̂	NUM
ejpam-6492	459	3	−	−	PROPN
ejpam-6492	459	4	αn+kβn+h	αn+kβn+h	NUM
ejpam-6492	459	5	β̂α̂	β̂α̂	PUNCT
ejpam-6492	459	6	+	+	CCONJ
ejpam-6492	459	7	β2n+h+k	β2n+h+k	X
ejpam-6492	459	8	β̂β̂	β̂β̂	PUNCT
ejpam-6492	459	9	]	]	PUNCT
ejpam-6492	460	1	−	−	X
ejpam-6492	460	2	[	[	PUNCT
ejpam-6492	460	3	α2n+h+k	α2n+h+k	NOUN
ejpam-6492	460	4	α̂α̂	α̂α̂	NUM
ejpam-6492	460	5	−	−	PROPN
ejpam-6492	460	6	αnβn+h+k	αnβn+h+k	PROPN
ejpam-6492	460	7	α̂β̂	α̂β̂	X
ejpam-6492	460	8	−	−	PROPN
ejpam-6492	460	9	αn+h+kβn	αn+h+kβn	NOUN
ejpam-6492	460	10	β̂α̂	β̂α̂	PUNCT
ejpam-6492	460	11	+	+	NUM
ejpam-6492	460	12	β2n+h+k	β2n+h+k	X
ejpam-6492	460	13	β̂β̂	β̂β̂	PUNCT
ejpam-6492	460	14	]	]	PUNCT
ejpam-6492	460	15	(	(	PUNCT
ejpam-6492	460	16	α	α	NOUN
ejpam-6492	460	17	−	−	NOUN
ejpam-6492	460	18	β)2	β)2	NOUN
ejpam-6492	460	19	=	=	X
ejpam-6492	460	20	[	[	PUNCT
ejpam-6492	460	21	−αn+hβn+kα̂β̂	−αn+hβn+kα̂β̂	PROPN
ejpam-6492	460	22	−	−	PROPN
ejpam-6492	460	23	αn+kβn+hβ̂α̂	αn+kβn+hβ̂α̂	PROPN
ejpam-6492	460	24	]	]	PUNCT
ejpam-6492	460	25	−	−	PROPN
ejpam-6492	460	26	[	[	PUNCT
ejpam-6492	460	27	−αnβn+h+kα̂β̂	−αnβn+h+kα̂β̂	PUNCT
ejpam-6492	460	28	−	−	NOUN
ejpam-6492	460	29	αn+h+kβnβ̂α̂	αn+h+kβnβ̂α̂	NOUN
ejpam-6492	460	30	]	]	X
ejpam-6492	461	1	(	(	PUNCT
ejpam-6492	461	2	α	α	NOUN
ejpam-6492	461	3	−	−	NOUN
ejpam-6492	461	4	β)2	β)2	NOUN
ejpam-6492	461	5	=	=	SYM
ejpam-6492	461	6	−αn+hβn+kα̂β̂	−αn+hβn+kα̂β̂	PROPN
ejpam-6492	461	7	−	−	PROPN
ejpam-6492	461	8	αn+kβn+hβ̂α̂	αn+kβn+hβ̂α̂	PROPN
ejpam-6492	461	9	+	+	CCONJ
ejpam-6492	461	10	αnβn+h+kα̂β̂	αnβn+h+kα̂β̂	PROPN
ejpam-6492	461	11	+	+	CCONJ
ejpam-6492	461	12	αn+h+kβnβ̂α̂	αn+h+kβnβ̂α̂	NOUN
ejpam-6492	461	13	(	(	PUNCT
ejpam-6492	461	14	α	α	NOUN
ejpam-6492	461	15	−	−	NOUN
ejpam-6492	461	16	β)2	β)2	NOUN
ejpam-6492	461	17	=	=	NOUN
ejpam-6492	461	18	αnβn	αnβn	NOUN
ejpam-6492	461	19	[	[	PUNCT
ejpam-6492	461	20	α̂β̂	α̂β̂	X
ejpam-6492	461	21	(	(	PUNCT
ejpam-6492	461	22	βh+k	βh+k	X
ejpam-6492	461	23	−	−	PROPN
ejpam-6492	461	24	αhβk	αhβk	NOUN
ejpam-6492	461	25	)	)	PUNCT
ejpam-6492	462	1	+	+	CCONJ
ejpam-6492	462	2	β̂α̂	β̂α̂	PUNCT
ejpam-6492	462	3	(	(	PUNCT
ejpam-6492	462	4	αh+k	αh+k	NUM
ejpam-6492	462	5	−	−	NOUN
ejpam-6492	462	6	αkβh	αkβh	NOUN
ejpam-6492	462	7	)	)	PUNCT
ejpam-6492	462	8	]	]	PUNCT
ejpam-6492	463	1	(	(	PUNCT
ejpam-6492	463	2	α	α	NOUN
ejpam-6492	463	3	−	−	NOUN
ejpam-6492	463	4	β)2	β)2	NOUN
ejpam-6492	463	5	=	=	NOUN
ejpam-6492	463	6	αnβn	αnβn	NOUN
ejpam-6492	463	7	[	[	PUNCT
ejpam-6492	463	8	−βkα̂β̂	−βkα̂β̂	X
ejpam-6492	463	9	(	(	PUNCT
ejpam-6492	463	10	αh	αh	NOUN
ejpam-6492	463	11	−	−	PROPN
ejpam-6492	463	12	βh	βh	ADP
ejpam-6492	463	13	)	)	PUNCT
ejpam-6492	464	1	+	+	CCONJ
ejpam-6492	465	1	αkβ̂α̂	αkβ̂α̂	NUM
ejpam-6492	465	2	(	(	PUNCT
ejpam-6492	465	3	αh	αh	NOUN
ejpam-6492	465	4	−	−	PROPN
ejpam-6492	465	5	βh	βh	ADP
ejpam-6492	465	6	)	)	PUNCT
ejpam-6492	465	7	]	]	PUNCT
ejpam-6492	465	8	(	(	PUNCT
ejpam-6492	465	9	α	α	NOUN
ejpam-6492	465	10	−	−	NOUN
ejpam-6492	465	11	β)2	β)2	NOUN
ejpam-6492	465	12	=	=	NOUN
ejpam-6492	465	13	αnβn	αnβn	NOUN
ejpam-6492	465	14	(	(	PUNCT
ejpam-6492	465	15	αh	αh	NOUN
ejpam-6492	465	16	−	−	PROPN
ejpam-6492	465	17	βh	βh	ADP
ejpam-6492	465	18	)	)	PUNCT
ejpam-6492	466	1	[	[	PUNCT
ejpam-6492	466	2	αkβ̂α̂	αkβ̂α̂	NOUN
ejpam-6492	466	3	−	−	NOUN
ejpam-6492	466	4	βkα̂β̂	βkα̂β̂	PUNCT
ejpam-6492	466	5	]	]	PUNCT
ejpam-6492	466	6	(	(	PUNCT
ejpam-6492	466	7	α	α	NOUN
ejpam-6492	466	8	−	−	NOUN
ejpam-6492	466	9	β)2	β)2	NOUN
ejpam-6492	466	10	=	=	NOUN
ejpam-6492	466	11	αnβn	αnβn	NOUN
ejpam-6492	466	12	(	(	PUNCT
ejpam-6492	466	13	αh	αh	NOUN
ejpam-6492	466	14	−	−	PROPN
ejpam-6492	466	15	βh	βh	ADP
ejpam-6492	466	16	)	)	PUNCT
ejpam-6492	466	17	[	[	PUNCT
ejpam-6492	466	18	αk	αk	X
ejpam-6492	466	19	(	(	PUNCT
ejpam-6492	466	20	a	a	DET
ejpam-6492	466	21	−	−	PROPN
ejpam-6492	466	22	qb	qb	PROPN
ejpam-6492	466	23	√	√	PROPN
ejpam-6492	466	24	∆	∆	PROPN
ejpam-6492	466	25	)	)	PUNCT
ejpam-6492	467	1	−	−	PROPN
ejpam-6492	467	2	βk	βk	INTJ
ejpam-6492	467	3	(	(	PUNCT
ejpam-6492	467	4	a	a	DET
ejpam-6492	467	5	+	+	X
ejpam-6492	467	6	qb	qb	NOUN
ejpam-6492	467	7	√	√	NOUN
ejpam-6492	467	8	∆	∆	PROPN
ejpam-6492	467	9	)	)	PUNCT
ejpam-6492	467	10	]	]	PUNCT
ejpam-6492	467	11	(	(	PUNCT
ejpam-6492	467	12	α	α	NOUN
ejpam-6492	467	13	−	−	PROPN
ejpam-6492	467	14	β)2	β)2	NOUN
ejpam-6492	467	15	b.	b.	PROPN
ejpam-6492	467	16	demirtürk	demirtürk	PROPN
ejpam-6492	467	17	,	,	PUNCT
ejpam-6492	467	18	n.	n.	NOUN
ejpam-6492	467	19	topal	topal	PROPN
ejpam-6492	467	20	/	/	SYM
ejpam-6492	467	21	eur	eur	PROPN
ejpam-6492	467	22	.	.	PUNCT
ejpam-6492	468	1	j.	j.	PROPN
ejpam-6492	468	2	pure	pure	PROPN
ejpam-6492	468	3	appl	appl	PROPN
ejpam-6492	468	4	.	.	PROPN
ejpam-6492	468	5	math	math	PROPN
ejpam-6492	468	6	,	,	PUNCT
ejpam-6492	468	7	18	18	NUM
ejpam-6492	468	8	(	(	PUNCT
ejpam-6492	468	9	3	3	NUM
ejpam-6492	468	10	)	)	PUNCT
ejpam-6492	468	11	(	(	PUNCT
ejpam-6492	468	12	2025	2025	NUM
ejpam-6492	468	13	)	)	PUNCT
ejpam-6492	468	14	,	,	PUNCT
ejpam-6492	468	15	6492	6492	NUM
ejpam-6492	468	16	19	19	NUM
ejpam-6492	468	17	of	of	ADP
ejpam-6492	468	18	22	22	NUM
ejpam-6492	468	19	=	=	SYM
ejpam-6492	468	20	(	(	PUNCT
ejpam-6492	468	21	αβ)n	αβ)n	PROPN
ejpam-6492	469	1	αh	αh	NOUN
ejpam-6492	469	2	−	−	PROPN
ejpam-6492	469	3	βh	βh	ADP
ejpam-6492	469	4	α	α	PRON
ejpam-6492	469	5	−	−	NOUN
ejpam-6492	469	6	β	β	X
ejpam-6492	469	7	a	a	VERB
ejpam-6492	469	8	(	(	PUNCT
ejpam-6492	469	9	αk	αk	ADP
ejpam-6492	469	10	−	−	PROPN
ejpam-6492	469	11	βk	βk	NOUN
ejpam-6492	469	12	)	)	PUNCT
ejpam-6492	469	13	α	α	NOUN
ejpam-6492	469	14	−	−	NOUN
ejpam-6492	469	15	β	β	NOUN
ejpam-6492	469	16	−	−	NOUN
ejpam-6492	469	17	q	q	PROPN
ejpam-6492	469	18	√	√	PROPN
ejpam-6492	469	19	∆b	∆b	PROPN
ejpam-6492	469	20	αk	αk	NOUN
ejpam-6492	469	21	+	+	CCONJ
ejpam-6492	469	22	βk	βk	ADP
ejpam-6492	469	23	α	α	PRON
ejpam-6492	469	24	−	−	NOUN
ejpam-6492	469	25	β	β	X
ejpam-6492	469	26			NOUN
ejpam-6492	469	27	=	=	SYM
ejpam-6492	469	28	(	(	PUNCT
ejpam-6492	469	29	−q)n	−q)n	NOUN
ejpam-6492	469	30	uh	uh	INTJ
ejpam-6492	469	31	[	[	X
ejpam-6492	469	32	auk	auk	NOUN
ejpam-6492	469	33	−	−	PROPN
ejpam-6492	469	34	qbvk	qbvk	NOUN
ejpam-6492	469	35	]	]	PUNCT
ejpam-6492	469	36	.	.	PUNCT
ejpam-6492	470	1	now	now	ADV
ejpam-6492	470	2	we	we	PRON
ejpam-6492	470	3	will	will	AUX
ejpam-6492	470	4	generalize	generalize	VERB
ejpam-6492	470	5	everman	everman	NOUN
ejpam-6492	470	6	’s	’s	PART
ejpam-6492	470	7	product	product	NOUN
ejpam-6492	470	8	difference	difference	NOUN
ejpam-6492	470	9	of	of	ADP
ejpam-6492	470	10	fibonacci	fibonacci	NOUN
ejpam-6492	470	11	identity	identity	NOUN
ejpam-6492	470	12	to	to	ADP
ejpam-6492	470	13	generalized	generalized	ADJ
ejpam-6492	470	14	lucas	lucas	NOUN
ejpam-6492	470	15	quaternions	quaternion	NOUN
ejpam-6492	470	16	in	in	ADP
ejpam-6492	470	17	theorem	theorem	ADJ
ejpam-6492	470	18	16	16	NUM
ejpam-6492	470	19	.	.	PUNCT
ejpam-6492	471	1	theorem	theorem	VERB
ejpam-6492	471	2	16	16	NUM
ejpam-6492	471	3	.	.	PUNCT
ejpam-6492	472	1	for	for	ADP
ejpam-6492	472	2	all	all	DET
ejpam-6492	472	3	n	n	NOUN
ejpam-6492	472	4	,	,	PUNCT
ejpam-6492	472	5	k	k	PROPN
ejpam-6492	472	6	and	and	CCONJ
ejpam-6492	472	7	h	h	NOUN
ejpam-6492	472	8	∈	∈	PROPN
ejpam-6492	472	9	z	z	X
ejpam-6492	472	10	,	,	PUNCT
ejpam-6492	472	11	it	it	PRON
ejpam-6492	472	12	follows	follow	VERB
ejpam-6492	472	13	that	that	SCONJ
ejpam-6492	472	14	kn+hkn+k	kn+hkn+k	PROPN
ejpam-6492	472	15	−	−	PROPN
ejpam-6492	472	16	knkn+k+h	knkn+k+h	X
ejpam-6492	472	17	=	=	SYM
ejpam-6492	472	18	∆	∆	PROPN
ejpam-6492	472	19	(	(	PUNCT
ejpam-6492	472	20	−q)n	−q)n	NOUN
ejpam-6492	472	21	uh	uh	INTJ
ejpam-6492	472	22	[	[	X
ejpam-6492	472	23	qbvk	qbvk	ADJ
ejpam-6492	472	24	−	−	PROPN
ejpam-6492	472	25	auk	auk	PROPN
ejpam-6492	472	26	]	]	PUNCT
ejpam-6492	472	27	,	,	PUNCT
ejpam-6492	472	28	where	where	SCONJ
ejpam-6492	472	29	a	a	DET
ejpam-6492	472	30	=	=	X
ejpam-6492	472	31	k0	k0	PROPN
ejpam-6492	472	32	−	−	PROPN
ejpam-6492	472	33	(	(	PUNCT
ejpam-6492	472	34	1	1	NUM
ejpam-6492	472	35	−	−	NUM
ejpam-6492	472	36	q)(1	q)(1	PROPN
ejpam-6492	472	37	+	+	X
ejpam-6492	472	38	q2	q2	NOUN
ejpam-6492	472	39	)	)	PUNCT
ejpam-6492	472	40	and	and	CCONJ
ejpam-6492	472	41	b	b	X
ejpam-6492	472	42	=	=	SYM
ejpam-6492	472	43	(	(	PUNCT
ejpam-6492	472	44	−q)i	−q)i	NOUN
ejpam-6492	472	45	+	+	CCONJ
ejpam-6492	472	46	(	(	PUNCT
ejpam-6492	472	47	−p)j	−p)j	NOUN
ejpam-6492	472	48	+	+	CCONJ
ejpam-6492	472	49	k.	k.	PROPN
ejpam-6492	472	50	proof	proof	NOUN
ejpam-6492	472	51	.	.	PUNCT
ejpam-6492	473	1	kn+hkn+k−knkn+k+h	kn+hkn+k−knkn+k+h	X
ejpam-6492	474	1	=	=	PUNCT
ejpam-6492	474	2	[	[	PUNCT
ejpam-6492	474	3	αn+hα̂	αn+hα̂	NOUN
ejpam-6492	474	4	+	+	X
ejpam-6492	474	5	βn+hβ̂	βn+hβ̂	X
ejpam-6492	474	6	]	]	X
ejpam-6492	474	7	[	[	PUNCT
ejpam-6492	474	8	αn+kα̂	αn+kα̂	PROPN
ejpam-6492	474	9	+	+	X
ejpam-6492	474	10	βn+kβ̂	βn+kβ̂	PROPN
ejpam-6492	474	11	]	]	PUNCT
ejpam-6492	474	12	−	−	PROPN
ejpam-6492	475	1	[	[	PUNCT
ejpam-6492	475	2	αnα̂	αnα̂	PROPN
ejpam-6492	475	3	+	+	PROPN
ejpam-6492	475	4	βnβ̂	βnβ̂	PROPN
ejpam-6492	475	5	]	]	X
ejpam-6492	475	6	[	[	PUNCT
ejpam-6492	475	7	αn+h+kα̂	αn+h+kα̂	PUNCT
ejpam-6492	476	1	+	+	X
ejpam-6492	476	2	βn+h+kβ̂	βn+h+kβ̂	PUNCT
ejpam-6492	476	3	]	]	PUNCT
ejpam-6492	477	1	=	=	PUNCT
ejpam-6492	477	2	[	[	PUNCT
ejpam-6492	477	3	α2n+h+kα̂α̂	α2n+h+kα̂α̂	PROPN
ejpam-6492	477	4	+	+	CCONJ
ejpam-6492	477	5	αn+hβn+kα̂β̂	αn+hβn+kα̂β̂	PROPN
ejpam-6492	477	6	+	+	CCONJ
ejpam-6492	477	7	αn+kβn+hβ̂α̂	αn+kβn+hβ̂α̂	PROPN
ejpam-6492	477	8	+	+	PUNCT
ejpam-6492	477	9	β2n+h+kβ̂β̂	β2n+h+kβ̂β̂	PUNCT
ejpam-6492	477	10	]	]	PUNCT
ejpam-6492	478	1	−	−	X
ejpam-6492	478	2	[	[	PUNCT
ejpam-6492	478	3	α2n+h+kα̂α̂	α2n+h+kα̂α̂	PROPN
ejpam-6492	479	1	+	+	PROPN
ejpam-6492	479	2	αnβn+h+kα̂β̂	αnβn+h+kα̂β̂	PROPN
ejpam-6492	479	3	+	+	CCONJ
ejpam-6492	479	4	αn+h+kβnβ̂α̂	αn+h+kβnβ̂α̂	NOUN
ejpam-6492	479	5	+	+	X
ejpam-6492	479	6	β2n+h+kβ̂β̂	β2n+h+kβ̂β̂	SYM
ejpam-6492	479	7	]	]	PUNCT
ejpam-6492	480	1	=	=	PUNCT
ejpam-6492	480	2	[	[	PUNCT
ejpam-6492	480	3	αn+hβn+kα̂β̂	αn+hβn+kα̂β̂	X
ejpam-6492	480	4	+	+	X
ejpam-6492	480	5	αn+kβn+hβ̂α̂	αn+kβn+hβ̂α̂	PROPN
ejpam-6492	480	6	]	]	PUNCT
ejpam-6492	480	7	−	−	PROPN
ejpam-6492	481	1	[	[	PUNCT
ejpam-6492	481	2	αnβn+h+kα̂β̂	αnβn+h+kα̂β̂	PROPN
ejpam-6492	481	3	+	+	PRON
ejpam-6492	481	4	αn+h+kβnβ̂α̂	αn+h+kβnβ̂α̂	NOUN
ejpam-6492	481	5	]	]	PUNCT
ejpam-6492	481	6	=	=	PUNCT
ejpam-6492	481	7	αn+hβn+kα̂β̂	αn+hβn+kα̂β̂	X
ejpam-6492	481	8	+	+	NOUN
ejpam-6492	481	9	αn+kβn+hβ̂α̂	αn+kβn+hβ̂α̂	PROPN
ejpam-6492	481	10	−	−	PROPN
ejpam-6492	481	11	αnβn+h+kα̂β̂	αnβn+h+kα̂β̂	PROPN
ejpam-6492	481	12	−	−	PROPN
ejpam-6492	481	13	αn+h+kβnβ̂α̂	αn+h+kβnβ̂α̂	NOUN
ejpam-6492	481	14	=	=	NOUN
ejpam-6492	481	15	αnβn	αnβn	NOUN
ejpam-6492	481	16	[	[	PUNCT
ejpam-6492	481	17	α̂β̂	α̂β̂	X
ejpam-6492	481	18	(	(	PUNCT
ejpam-6492	481	19	αhβk	αhβk	NOUN
ejpam-6492	481	20	−	−	PROPN
ejpam-6492	481	21	βh+k	βh+k	NUM
ejpam-6492	481	22	)	)	PUNCT
ejpam-6492	482	1	+	+	CCONJ
ejpam-6492	482	2	β̂α̂	β̂α̂	PUNCT
ejpam-6492	482	3	(	(	PUNCT
ejpam-6492	482	4	αkβh	αkβh	NOUN
ejpam-6492	482	5	−	−	NOUN
ejpam-6492	482	6	αh+k	αh+k	NUM
ejpam-6492	482	7	)	)	PUNCT
ejpam-6492	482	8	]	]	PUNCT
ejpam-6492	483	1	=	=	PUNCT
ejpam-6492	483	2	αnβn	αnβn	NOUN
ejpam-6492	483	3	[	[	PUNCT
ejpam-6492	483	4	α̂β̂βk	α̂β̂βk	NOUN
ejpam-6492	483	5	(	(	PUNCT
ejpam-6492	483	6	αh	αh	NOUN
ejpam-6492	483	7	−	−	PROPN
ejpam-6492	483	8	βh	βh	ADP
ejpam-6492	483	9	)	)	PUNCT
ejpam-6492	484	1	+	+	CCONJ
ejpam-6492	484	2	β̂α̂αk	β̂α̂αk	NOUN
ejpam-6492	484	3	(	(	PUNCT
ejpam-6492	484	4	βh	βh	ADV
ejpam-6492	484	5	−	−	NOUN
ejpam-6492	484	6	αh	αh	NOUN
ejpam-6492	484	7	)	)	PUNCT
ejpam-6492	484	8	]	]	PUNCT
ejpam-6492	485	1	=	=	PUNCT
ejpam-6492	485	2	(	(	PUNCT
ejpam-6492	485	3	αβ)n	αβ)n	PROPN
ejpam-6492	485	4	(	(	PUNCT
ejpam-6492	485	5	αh	αh	NOUN
ejpam-6492	485	6	−	−	PROPN
ejpam-6492	485	7	βh	βh	ADP
ejpam-6492	485	8	)	)	PUNCT
ejpam-6492	485	9	[	[	PUNCT
ejpam-6492	485	10	α̂β̂βk	α̂β̂βk	NOUN
ejpam-6492	485	11	−	−	PROPN
ejpam-6492	485	12	β̂α̂αk	β̂α̂αk	NOUN
ejpam-6492	485	13	]	]	PUNCT
ejpam-6492	485	14	=	=	PUNCT
ejpam-6492	485	15	(	(	PUNCT
ejpam-6492	485	16	αβ)n	αβ)n	PROPN
ejpam-6492	485	17	(	(	PUNCT
ejpam-6492	485	18	αh	αh	NOUN
ejpam-6492	485	19	−	−	PROPN
ejpam-6492	485	20	βh	βh	ADP
ejpam-6492	485	21	)	)	PUNCT
ejpam-6492	485	22	[	[	PUNCT
ejpam-6492	485	23	βk	βk	X
ejpam-6492	485	24	(	(	PUNCT
ejpam-6492	485	25	a	a	DET
ejpam-6492	485	26	+	+	X
ejpam-6492	485	27	qb	qb	NOUN
ejpam-6492	485	28	√	√	PROPN
ejpam-6492	485	29	∆	∆	PROPN
ejpam-6492	485	30	)	)	PUNCT
ejpam-6492	486	1	−	−	ADP
ejpam-6492	487	1	αk	αk	INTJ
ejpam-6492	487	2	(	(	PUNCT
ejpam-6492	487	3	a	a	DET
ejpam-6492	487	4	−	−	PROPN
ejpam-6492	487	5	qb	qb	PROPN
ejpam-6492	487	6	√	√	PROPN
ejpam-6492	487	7	∆	∆	PROPN
ejpam-6492	487	8	)	)	PUNCT
ejpam-6492	487	9	]	]	PUNCT
ejpam-6492	488	1	=	=	PUNCT
ejpam-6492	488	2	(	(	PUNCT
ejpam-6492	488	3	αβ)n	αβ)n	PROPN
ejpam-6492	488	4	(	(	PUNCT
ejpam-6492	488	5	αh	αh	NOUN
ejpam-6492	488	6	−	−	PROPN
ejpam-6492	488	7	βh	βh	ADP
ejpam-6492	488	8	)	)	PUNCT
ejpam-6492	488	9	[	[	PUNCT
ejpam-6492	488	10	qb	qb	PROPN
ejpam-6492	488	11	√	√	PROPN
ejpam-6492	488	12	∆	∆	PROPN
ejpam-6492	488	13	(	(	PUNCT
ejpam-6492	488	14	αk	αk	X
ejpam-6492	488	15	+	+	CCONJ
ejpam-6492	488	16	βk	βk	NOUN
ejpam-6492	488	17	)	)	PUNCT
ejpam-6492	489	1	−	−	PROPN
ejpam-6492	489	2	a	a	DET
ejpam-6492	489	3	(	(	PUNCT
ejpam-6492	489	4	αk	αk	NOUN
ejpam-6492	489	5	−	−	PROPN
ejpam-6492	489	6	βk	βk	NOUN
ejpam-6492	489	7	)	)	PUNCT
ejpam-6492	489	8	]	]	PUNCT
ejpam-6492	490	1	=	=	PUNCT
ejpam-6492	490	2	∆	∆	X
ejpam-6492	490	3	(	(	PUNCT
ejpam-6492	490	4	αβ)n	αβ)n	PROPN
ejpam-6492	490	5	(	(	PUNCT
ejpam-6492	490	6	αh	αh	NOUN
ejpam-6492	490	7	−	−	PROPN
ejpam-6492	490	8	βh	βh	ADP
ejpam-6492	490	9	α	α	NOUN
ejpam-6492	490	10	−	−	NOUN
ejpam-6492	490	11	β	β	X
ejpam-6492	490	12	)	)	PUNCT
ejpam-6492	490	13	qb	qb	NOUN
ejpam-6492	490	14	√	√	NUM
ejpam-6492	490	15	∆	∆	PROPN
ejpam-6492	490	16	(	(	PUNCT
ejpam-6492	490	17	αk	αk	X
ejpam-6492	490	18	+	+	CCONJ
ejpam-6492	490	19	βk	βk	NOUN
ejpam-6492	490	20	)	)	PUNCT
ejpam-6492	491	1	−	−	PROPN
ejpam-6492	491	2	a	a	DET
ejpam-6492	491	3	(	(	PUNCT
ejpam-6492	491	4	αk	αk	NOUN
ejpam-6492	491	5	−	−	PROPN
ejpam-6492	491	6	βk	βk	NOUN
ejpam-6492	491	7	)	)	PUNCT
ejpam-6492	491	8	α	α	NOUN
ejpam-6492	491	9	−	−	NOUN
ejpam-6492	491	10	β	β	NOUN
ejpam-6492	491	11			NOUN
ejpam-6492	491	12	=	=	SYM
ejpam-6492	491	13	∆	∆	X
ejpam-6492	491	14	(	(	PUNCT
ejpam-6492	491	15	−q)n	−q)n	NOUN
ejpam-6492	491	16	uh	uh	INTJ
ejpam-6492	491	17	[	[	X
ejpam-6492	491	18	qbvk	qbvk	ADJ
ejpam-6492	491	19	−	−	PROPN
ejpam-6492	491	20	auk	auk	NOUN
ejpam-6492	491	21	]	]	X
ejpam-6492	491	22	.	.	PUNCT
ejpam-6492	492	1	as	as	ADP
ejpam-6492	492	2	a	a	DET
ejpam-6492	492	3	consequence	consequence	NOUN
ejpam-6492	492	4	of	of	ADP
ejpam-6492	492	5	theorem	theorem	ADJ
ejpam-6492	492	6	15	15	NUM
ejpam-6492	492	7	and	and	CCONJ
ejpam-6492	492	8	theorem	theorem	VERB
ejpam-6492	492	9	16	16	NUM
ejpam-6492	492	10	,	,	PUNCT
ejpam-6492	492	11	we	we	PRON
ejpam-6492	492	12	can	can	AUX
ejpam-6492	492	13	give	give	VERB
ejpam-6492	492	14	corollary	corollary	ADJ
ejpam-6492	492	15	5	5	NUM
ejpam-6492	492	16	.	.	PUNCT
ejpam-6492	492	17	corollary	corollary	ADJ
ejpam-6492	492	18	5	5	NUM
ejpam-6492	492	19	.	.	PUNCT
ejpam-6492	493	1	for	for	ADP
ejpam-6492	493	2	all	all	DET
ejpam-6492	493	3	n	n	NOUN
ejpam-6492	493	4	,	,	PUNCT
ejpam-6492	493	5	k	k	PROPN
ejpam-6492	493	6	and	and	CCONJ
ejpam-6492	493	7	h	h	NOUN
ejpam-6492	493	8	∈	∈	PROPN
ejpam-6492	493	9	z	z	X
ejpam-6492	493	10	,	,	PUNCT
ejpam-6492	493	11	it	it	PRON
ejpam-6492	493	12	follows	follow	VERB
ejpam-6492	493	13	that	that	SCONJ
ejpam-6492	493	14	kn+hkn+k	kn+hkn+k	PROPN
ejpam-6492	493	15	−	−	PROPN
ejpam-6492	493	16	knkn+k+h	knkn+k+h	X
ejpam-6492	493	17	=	=	SYM
ejpam-6492	493	18	−∆	−∆	NOUN
ejpam-6492	494	1	[	[	X
ejpam-6492	494	2	qn+hqn+k	qn+hqn+k	ADP
ejpam-6492	494	3	−	−	NOUN
ejpam-6492	494	4	qnqn+k+h	qnqn+k+h	AUX
ejpam-6492	494	5	]	]	PUNCT
ejpam-6492	494	6	.	.	PUNCT
ejpam-6492	495	1	b.	b.	PROPN
ejpam-6492	495	2	demirtürk	demirtürk	PROPN
ejpam-6492	495	3	,	,	PUNCT
ejpam-6492	495	4	n.	n.	NOUN
ejpam-6492	495	5	topal	topal	PROPN
ejpam-6492	495	6	/	/	SYM
ejpam-6492	495	7	eur	eur	PROPN
ejpam-6492	495	8	.	.	PUNCT
ejpam-6492	496	1	j.	j.	PROPN
ejpam-6492	496	2	pure	pure	PROPN
ejpam-6492	496	3	appl	appl	PROPN
ejpam-6492	496	4	.	.	PROPN
ejpam-6492	496	5	math	math	PROPN
ejpam-6492	496	6	,	,	PUNCT
ejpam-6492	496	7	18	18	NUM
ejpam-6492	496	8	(	(	PUNCT
ejpam-6492	496	9	3	3	NUM
ejpam-6492	496	10	)	)	PUNCT
ejpam-6492	496	11	(	(	PUNCT
ejpam-6492	496	12	2025	2025	NUM
ejpam-6492	496	13	)	)	PUNCT
ejpam-6492	496	14	,	,	PUNCT
ejpam-6492	496	15	6492	6492	NUM
ejpam-6492	496	16	20	20	NUM
ejpam-6492	496	17	of	of	ADP
ejpam-6492	496	18	22	22	NUM
ejpam-6492	496	19	4	4	NUM
ejpam-6492	496	20	.	.	PUNCT
ejpam-6492	496	21	conclusion	conclusion	NOUN
ejpam-6492	496	22	this	this	DET
ejpam-6492	496	23	paper	paper	NOUN
ejpam-6492	496	24	presents	present	VERB
ejpam-6492	496	25	new	new	ADJ
ejpam-6492	496	26	identities	identity	NOUN
ejpam-6492	496	27	for	for	ADP
ejpam-6492	496	28	both	both	DET
ejpam-6492	496	29	theoretical	theoretical	ADJ
ejpam-6492	496	30	mathematical	mathematical	ADJ
ejpam-6492	496	31	theories	theory	NOUN
ejpam-6492	496	32	and	and	CCONJ
ejpam-6492	496	33	various	various	ADJ
ejpam-6492	496	34	applied	apply	VERB
ejpam-6492	496	35	fields	field	NOUN
ejpam-6492	496	36	,	,	PUNCT
ejpam-6492	496	37	by	by	ADP
ejpam-6492	496	38	integrating	integrate	VERB
ejpam-6492	496	39	generalized	generalized	ADJ
ejpam-6492	496	40	fibonacci	fibonacci	NOUN
ejpam-6492	496	41	and	and	CCONJ
ejpam-6492	496	42	lucas	lucas	PROPN
ejpam-6492	496	43	sequences	sequence	NOUN
ejpam-6492	496	44	with	with	ADP
ejpam-6492	496	45	quaternion	quaternion	NOUN
ejpam-6492	496	46	structures	structure	NOUN
ejpam-6492	496	47	.	.	PUNCT
ejpam-6492	497	1	known	know	VERB
ejpam-6492	497	2	identities	identity	NOUN
ejpam-6492	497	3	of	of	ADP
ejpam-6492	497	4	fibonacci	fibonacci	NOUN
ejpam-6492	497	5	and	and	CCONJ
ejpam-6492	497	6	lucas	lucas	PROPN
ejpam-6492	497	7	sequences	sequence	NOUN
ejpam-6492	497	8	,	,	PUNCT
ejpam-6492	497	9	such	such	ADJ
ejpam-6492	497	10	as	as	ADP
ejpam-6492	497	11	the	the	DET
ejpam-6492	497	12	classical	classical	ADJ
ejpam-6492	497	13	binet	binet	NOUN
ejpam-6492	497	14	formulas	formula	NOUN
ejpam-6492	497	15	,	,	PUNCT
ejpam-6492	497	16	cassini	cassini	NOUN
ejpam-6492	497	17	’s	’s	PART
ejpam-6492	497	18	,	,	PUNCT
ejpam-6492	497	19	catalan	catalan	NOUN
ejpam-6492	497	20	’s	’s	PART
ejpam-6492	497	21	and	and	CCONJ
ejpam-6492	497	22	d’ocagne	d’ocagne	PROPN
ejpam-6492	497	23	’s	’s	PART
ejpam-6492	497	24	identities	identity	NOUN
ejpam-6492	497	25	are	be	AUX
ejpam-6492	497	26	generalized	generalize	VERB
ejpam-6492	497	27	,	,	PUNCT
ejpam-6492	497	28	as	as	ADV
ejpam-6492	497	29	well	well	ADV
ejpam-6492	497	30	as	as	ADP
ejpam-6492	497	31	new	new	ADJ
ejpam-6492	497	32	identities	identity	NOUN
ejpam-6492	497	33	,	,	PUNCT
ejpam-6492	497	34	are	be	AUX
ejpam-6492	497	35	rederived	rederive	VERB
ejpam-6492	497	36	through	through	ADP
ejpam-6492	497	37	quaternion	quaternion	NOUN
ejpam-6492	497	38	extensions	extension	NOUN
ejpam-6492	497	39	.	.	PUNCT
ejpam-6492	498	1	therefore	therefore	ADV
ejpam-6492	498	2	,	,	PUNCT
ejpam-6492	498	3	this	this	DET
ejpam-6492	498	4	paper	paper	NOUN
ejpam-6492	498	5	demonstrates	demonstrate	VERB
ejpam-6492	498	6	the	the	DET
ejpam-6492	498	7	interaction	interaction	NOUN
ejpam-6492	498	8	between	between	ADP
ejpam-6492	498	9	number	number	NOUN
ejpam-6492	498	10	sequences	sequence	NOUN
ejpam-6492	498	11	and	and	CCONJ
ejpam-6492	498	12	quaternions	quaternion	NOUN
ejpam-6492	498	13	by	by	ADP
ejpam-6492	498	14	extending	extend	VERB
ejpam-6492	498	15	some	some	DET
ejpam-6492	498	16	fundamental	fundamental	ADJ
ejpam-6492	498	17	equations	equation	NOUN
ejpam-6492	498	18	in	in	ADP
ejpam-6492	498	19	the	the	DET
ejpam-6492	498	20	existing	exist	VERB
ejpam-6492	498	21	literature	literature	NOUN
ejpam-6492	498	22	to	to	ADP
ejpam-6492	498	23	high	high	ADJ
ejpam-6492	498	24	-	-	PUNCT
ejpam-6492	498	25	dimensional	dimensional	ADJ
ejpam-6492	498	26	hypercomplex	hypercomplex	NOUN
ejpam-6492	498	27	structures	structure	NOUN
ejpam-6492	498	28	.	.	PUNCT
ejpam-6492	499	1	in	in	ADP
ejpam-6492	499	2	particular	particular	ADJ
ejpam-6492	499	3	,	,	PUNCT
ejpam-6492	499	4	the	the	DET
ejpam-6492	499	5	original	original	ADJ
ejpam-6492	499	6	part	part	NOUN
ejpam-6492	499	7	of	of	ADP
ejpam-6492	499	8	the	the	DET
ejpam-6492	499	9	paper	paper	NOUN
ejpam-6492	499	10	is	be	AUX
ejpam-6492	499	11	the	the	DET
ejpam-6492	499	12	reconstruction	reconstruction	NOUN
ejpam-6492	499	13	of	of	ADP
ejpam-6492	499	14	the	the	DET
ejpam-6492	499	15	classical	classical	ADJ
ejpam-6492	499	16	results	result	NOUN
ejpam-6492	499	17	of	of	ADP
ejpam-6492	499	18	researchers	researcher	NOUN
ejpam-6492	499	19	such	such	ADJ
ejpam-6492	499	20	as	as	ADP
ejpam-6492	499	21	koshy	koshy	ADJ
ejpam-6492	499	22	and	and	CCONJ
ejpam-6492	499	23	everman	everman	NOUN
ejpam-6492	499	24	by	by	ADP
ejpam-6492	499	25	means	mean	NOUN
ejpam-6492	499	26	of	of	ADP
ejpam-6492	499	27	generalized	generalized	ADJ
ejpam-6492	499	28	fibonacci	fibonacci	NOUN
ejpam-6492	499	29	and	and	CCONJ
ejpam-6492	499	30	lucas	lucas	PROPN
ejpam-6492	499	31	sequences	sequence	NOUN
ejpam-6492	499	32	and	and	CCONJ
ejpam-6492	499	33	the	the	DET
ejpam-6492	499	34	systematic	systematic	ADJ
ejpam-6492	499	35	presentation	presentation	NOUN
ejpam-6492	499	36	of	of	ADP
ejpam-6492	499	37	the	the	DET
ejpam-6492	499	38	quaternion	quaternion	NOUN
ejpam-6492	499	39	forms	form	NOUN
ejpam-6492	499	40	of	of	ADP
ejpam-6492	499	41	these	these	DET
ejpam-6492	499	42	equations	equation	NOUN
ejpam-6492	499	43	and	and	CCONJ
ejpam-6492	499	44	product	product	NOUN
ejpam-6492	499	45	differences	difference	NOUN
ejpam-6492	499	46	generalizations	generalization	NOUN
ejpam-6492	499	47	.	.	PUNCT
ejpam-6492	500	1	generalized	generalize	VERB
ejpam-6492	500	2	quaternions	quaternion	NOUN
ejpam-6492	500	3	provide	provide	VERB
ejpam-6492	500	4	algebraic	algebraic	ADJ
ejpam-6492	500	5	models	model	NOUN
ejpam-6492	500	6	that	that	PRON
ejpam-6492	500	7	can	can	AUX
ejpam-6492	500	8	be	be	AUX
ejpam-6492	500	9	used	use	VERB
ejpam-6492	500	10	in	in	ADP
ejpam-6492	500	11	many	many	ADJ
ejpam-6492	500	12	fields	field	NOUN
ejpam-6492	500	13	such	such	ADJ
ejpam-6492	500	14	as	as	ADP
ejpam-6492	500	15	computer	computer	NOUN
ejpam-6492	500	16	graphics	graphic	NOUN
ejpam-6492	500	17	,	,	PUNCT
ejpam-6492	500	18	cryptography	cryptography	NOUN
ejpam-6492	500	19	,	,	PUNCT
ejpam-6492	500	20	signal	signal	NOUN
ejpam-6492	500	21	processing	processing	NOUN
ejpam-6492	500	22	,	,	PUNCT
ejpam-6492	500	23	modeling	modeling	NOUN
ejpam-6492	500	24	of	of	ADP
ejpam-6492	500	25	physical	physical	ADJ
ejpam-6492	500	26	systems	system	NOUN
ejpam-6492	500	27	and	and	CCONJ
ejpam-6492	500	28	bioinformatics	bioinformatic	NOUN
ejpam-6492	500	29	.	.	PUNCT
ejpam-6492	501	1	in	in	ADP
ejpam-6492	501	2	particular	particular	ADJ
ejpam-6492	501	3	,	,	PUNCT
ejpam-6492	501	4	quaternionbased	quaternionbase	VERB
ejpam-6492	501	5	structures	structure	NOUN
ejpam-6492	501	6	provide	provide	VERB
ejpam-6492	501	7	natural	natural	ADJ
ejpam-6492	501	8	solutions	solution	NOUN
ejpam-6492	501	9	to	to	ADP
ejpam-6492	501	10	transformation	transformation	NOUN
ejpam-6492	501	11	and	and	CCONJ
ejpam-6492	501	12	rotation	rotation	NOUN
ejpam-6492	501	13	problems	problem	NOUN
ejpam-6492	501	14	in	in	ADP
ejpam-6492	501	15	a	a	DET
ejpam-6492	501	16	three	three	NUM
ejpam-6492	501	17	-	-	PUNCT
ejpam-6492	501	18	dimensional	dimensional	ADJ
ejpam-6492	501	19	space	space	NOUN
ejpam-6492	501	20	,	,	PUNCT
ejpam-6492	501	21	and	and	CCONJ
ejpam-6492	501	22	the	the	DET
ejpam-6492	501	23	use	use	NOUN
ejpam-6492	501	24	of	of	ADP
ejpam-6492	501	25	generalized	generalized	ADJ
ejpam-6492	501	26	number	number	NOUN
ejpam-6492	501	27	sequences	sequence	NOUN
ejpam-6492	501	28	in	in	ADP
ejpam-6492	501	29	these	these	DET
ejpam-6492	501	30	structures	structure	NOUN
ejpam-6492	501	31	is	be	AUX
ejpam-6492	501	32	important	important	ADJ
ejpam-6492	501	33	both	both	CCONJ
ejpam-6492	501	34	theoretically	theoretically	ADV
ejpam-6492	501	35	and	and	CCONJ
ejpam-6492	501	36	practically	practically	ADV
ejpam-6492	501	37	.	.	PUNCT
ejpam-6492	502	1	in	in	ADP
ejpam-6492	502	2	this	this	DET
ejpam-6492	502	3	respect	respect	NOUN
ejpam-6492	502	4	,	,	PUNCT
ejpam-6492	502	5	this	this	DET
ejpam-6492	502	6	paper	paper	NOUN
ejpam-6492	502	7	provides	provide	VERB
ejpam-6492	502	8	an	an	DET
ejpam-6492	502	9	extension	extension	NOUN
ejpam-6492	502	10	of	of	ADP
ejpam-6492	502	11	algebraic	algebraic	ADJ
ejpam-6492	502	12	properties	property	NOUN
ejpam-6492	502	13	in	in	ADP
ejpam-6492	502	14	the	the	DET
ejpam-6492	502	15	intersection	intersection	NOUN
ejpam-6492	502	16	set	set	NOUN
ejpam-6492	502	17	of	of	ADP
ejpam-6492	502	18	quaternions	quaternion	NOUN
ejpam-6492	502	19	and	and	CCONJ
ejpam-6492	502	20	generalized	generalized	ADJ
ejpam-6492	502	21	number	number	NOUN
ejpam-6492	502	22	sequences	sequence	NOUN
ejpam-6492	502	23	.	.	PUNCT
ejpam-6492	503	1	[	[	X
ejpam-6492	503	2	1	1	X
ejpam-6492	503	3	]	]	PUNCT
ejpam-6492	503	4	s.	s.	PROPN
ejpam-6492	503	5	sharma	sharma	PROPN
ejpam-6492	503	6	,	,	PUNCT
ejpam-6492	503	7	a.	a.	NOUN
ejpam-6492	503	8	tomar	tomar	PROPN
ejpam-6492	503	9	,	,	PUNCT
ejpam-6492	503	10	and	and	CCONJ
ejpam-6492	503	11	s.	s.	PROPN
ejpam-6492	503	12	k.	k.	PROPN
ejpam-6492	503	13	padaliya	padaliya	PROPN
ejpam-6492	503	14	.	.	PUNCT
ejpam-6492	504	1	on	on	ADP
ejpam-6492	504	2	the	the	DET
ejpam-6492	504	3	evolution	evolution	NOUN
ejpam-6492	504	4	and	and	CCONJ
ejpam-6492	504	5	importance	importance	NOUN
ejpam-6492	504	6	of	of	ADP
ejpam-6492	504	7	the	the	DET
ejpam-6492	504	8	fibonacci	fibonacci	NOUN
ejpam-6492	504	9	sequence	sequence	NOUN
ejpam-6492	504	10	in	in	ADP
ejpam-6492	504	11	visualization	visualization	NOUN
ejpam-6492	504	12	of	of	ADP
ejpam-6492	504	13	fractals	fractal	NOUN
ejpam-6492	504	14	.	.	PUNCT
ejpam-6492	505	1	chaos	chaos	NOUN
ejpam-6492	505	2	,	,	PUNCT
ejpam-6492	505	3	solitons	soliton	NOUN
ejpam-6492	505	4	&	&	CCONJ
ejpam-6492	505	5	fractals	fractal	NOUN
ejpam-6492	505	6	,	,	PUNCT
ejpam-6492	505	7	191	191	NUM
ejpam-6492	505	8	,	,	PUNCT
ejpam-6492	505	9	2025	2025	NUM
ejpam-6492	505	10	.	.	PUNCT
ejpam-6492	506	1	[	[	X
ejpam-6492	506	2	2	2	X
ejpam-6492	506	3	]	]	PUNCT
ejpam-6492	506	4	s.	s.	PROPN
ejpam-6492	506	5	h.	h.	PROPN
ejpam-6492	506	6	larsen	larsen	PROPN
ejpam-6492	506	7	.	.	PROPN
ejpam-6492	507	1	dna	dna	PROPN
ejpam-6492	507	2	structure	structure	NOUN
ejpam-6492	507	3	and	and	CCONJ
ejpam-6492	507	4	the	the	DET
ejpam-6492	507	5	golden	golden	ADJ
ejpam-6492	507	6	ratio	ratio	NOUN
ejpam-6492	507	7	revisited	revisit	VERB
ejpam-6492	507	8	.	.	PUNCT
ejpam-6492	508	1	symmetry	symmetry	NOUN
ejpam-6492	508	2	,	,	PUNCT
ejpam-6492	508	3	13(10):1949	13(10):1949	NUM
ejpam-6492	508	4	,	,	PUNCT
ejpam-6492	508	5	2021	2021	NUM
ejpam-6492	508	6	.	.	PUNCT
ejpam-6492	509	1	[	[	X
ejpam-6492	509	2	3	3	X
ejpam-6492	509	3	]	]	X
ejpam-6492	509	4	v.	v.	ADP
ejpam-6492	509	5	pletser	pletser	NOUN
ejpam-6492	509	6	.	.	PUNCT
ejpam-6492	510	1	fibonacci	fibonacci	NOUN
ejpam-6492	510	2	numbers	number	NOUN
ejpam-6492	510	3	and	and	CCONJ
ejpam-6492	510	4	the	the	DET
ejpam-6492	510	5	golden	golden	ADJ
ejpam-6492	510	6	ratio	ratio	NOUN
ejpam-6492	510	7	in	in	ADP
ejpam-6492	510	8	biology	biology	NOUN
ejpam-6492	510	9	,	,	PUNCT
ejpam-6492	510	10	physics	physics	NOUN
ejpam-6492	510	11	,	,	PUNCT
ejpam-6492	510	12	astrophysics	astrophysic	NOUN
ejpam-6492	510	13	,	,	PUNCT
ejpam-6492	510	14	chemistry	chemistry	NOUN
ejpam-6492	510	15	and	and	CCONJ
ejpam-6492	510	16	technology	technology	NOUN
ejpam-6492	510	17	:	:	PUNCT
ejpam-6492	510	18	a	a	DET
ejpam-6492	510	19	non	non	ADJ
ejpam-6492	510	20	-	-	ADJ
ejpam-6492	510	21	exhaustive	exhaustive	ADJ
ejpam-6492	510	22	review	review	NOUN
ejpam-6492	510	23	.	.	PUNCT
ejpam-6492	511	1	arxiv	arxiv	PROPN
ejpam-6492	511	2	preprint	preprint	PROPN
ejpam-6492	511	3	arxiv:1801.01369	arxiv:1801.01369	PROPN
ejpam-6492	511	4	,	,	PUNCT
ejpam-6492	511	5	2017	2017	NUM
ejpam-6492	511	6	.	.	PUNCT
ejpam-6492	512	1	[	[	X
ejpam-6492	512	2	4	4	X
ejpam-6492	512	3	]	]	X
ejpam-6492	512	4	s.	s.	PROPN
ejpam-6492	512	5	sinha	sinha	PROPN
ejpam-6492	512	6	.	.	PUNCT
ejpam-6492	513	1	the	the	DET
ejpam-6492	513	2	fibonacci	fibonacci	NOUN
ejpam-6492	513	3	numbers	number	NOUN
ejpam-6492	513	4	and	and	CCONJ
ejpam-6492	513	5	its	its	PRON
ejpam-6492	513	6	amazing	amazing	ADJ
ejpam-6492	513	7	applications	application	NOUN
ejpam-6492	513	8	.	.	PUNCT
ejpam-6492	514	1	international	international	ADJ
ejpam-6492	514	2	journal	journal	PROPN
ejpam-6492	514	3	of	of	ADP
ejpam-6492	514	4	engineering	engineering	NOUN
ejpam-6492	514	5	science	science	NOUN
ejpam-6492	514	6	invention	invention	NOUN
ejpam-6492	514	7	,	,	PUNCT
ejpam-6492	514	8	6:7–14	6:7–14	NUM
ejpam-6492	514	9	,	,	PUNCT
ejpam-6492	514	10	2017	2017	NUM
ejpam-6492	514	11	.	.	PUNCT
ejpam-6492	515	1	[	[	X
ejpam-6492	515	2	5	5	X
ejpam-6492	515	3	]	]	PUNCT
ejpam-6492	515	4	h.	h.	PROPN
ejpam-6492	515	5	tran	tran	PROPN
ejpam-6492	515	6	-	-	PUNCT
ejpam-6492	515	7	ngoc	ngoc	PROPN
ejpam-6492	515	8	,	,	PUNCT
ejpam-6492	515	9	t.	t.	PROPN
ejpam-6492	515	10	le	le	PROPN
ejpam-6492	515	11	-	-	PROPN
ejpam-6492	515	12	xuan	xuan	PROPN
ejpam-6492	515	13	,	,	PUNCT
ejpam-6492	515	14	s.	s.	PROPN
ejpam-6492	515	15	khatir	khatir	PROPN
ejpam-6492	515	16	,	,	PUNCT
ejpam-6492	515	17	g.	g.	PROPN
ejpam-6492	515	18	de	de	PROPN
ejpam-6492	515	19	roeck	roeck	PROPN
ejpam-6492	515	20	,	,	PUNCT
ejpam-6492	515	21	t.	t.	NOUN
ejpam-6492	515	22	bui	bui	PROPN
ejpam-6492	515	23	-	-	PUNCT
ejpam-6492	515	24	tien	tien	NOUN
ejpam-6492	515	25	,	,	PUNCT
ejpam-6492	515	26	and	and	CCONJ
ejpam-6492	515	27	m.	m.	PROPN
ejpam-6492	515	28	abdel	abdel	PROPN
ejpam-6492	515	29	wahab	wahab	PROPN
ejpam-6492	515	30	.	.	PUNCT
ejpam-6492	516	1	a	a	DET
ejpam-6492	516	2	promising	promising	ADJ
ejpam-6492	516	3	approach	approach	NOUN
ejpam-6492	516	4	using	use	VERB
ejpam-6492	516	5	fibonacci	fibonacci	NOUN
ejpam-6492	516	6	sequence	sequence	NOUN
ejpam-6492	516	7	-	-	PUNCT
ejpam-6492	516	8	based	base	VERB
ejpam-6492	516	9	optimization	optimization	NOUN
ejpam-6492	516	10	algorithms	algorithm	NOUN
ejpam-6492	516	11	and	and	CCONJ
ejpam-6492	516	12	advanced	advanced	ADJ
ejpam-6492	516	13	computing	computing	NOUN
ejpam-6492	516	14	.	.	PUNCT
ejpam-6492	517	1	scientific	scientific	ADJ
ejpam-6492	517	2	reports	report	NOUN
ejpam-6492	517	3	,	,	PUNCT
ejpam-6492	517	4	13:3405	13:3405	NUM
ejpam-6492	517	5	,	,	PUNCT
ejpam-6492	517	6	2023	2023	NUM
ejpam-6492	517	7	.	.	PUNCT
ejpam-6492	518	1	[	[	X
ejpam-6492	518	2	6	6	NUM
ejpam-6492	518	3	]	]	PUNCT
ejpam-6492	518	4	r.	r.	PROPN
ejpam-6492	518	5	retnaningsih	retnaningsih	PROPN
ejpam-6492	518	6	.	.	PUNCT
ejpam-6492	519	1	fractal	fractal	PROPN
ejpam-6492	519	2	geometry	geometry	NOUN
ejpam-6492	519	3	,	,	PUNCT
ejpam-6492	519	4	fibonacci	fibonacci	NOUN
ejpam-6492	519	5	numbers	number	NOUN
ejpam-6492	519	6	,	,	PUNCT
ejpam-6492	519	7	golden	golden	ADJ
ejpam-6492	519	8	ratios	ratio	NOUN
ejpam-6492	519	9	,	,	PUNCT
ejpam-6492	519	10	and	and	CCONJ
ejpam-6492	519	11	pascal	pascal	ADJ
ejpam-6492	519	12	triangles	triangle	NOUN
ejpam-6492	519	13	as	as	ADP
ejpam-6492	519	14	designs	design	NOUN
ejpam-6492	519	15	.	.	PUNCT
ejpam-6492	520	1	the	the	DET
ejpam-6492	520	2	journal	journal	NOUN
ejpam-6492	520	3	of	of	ADP
ejpam-6492	520	4	academic	academic	ADJ
ejpam-6492	520	5	science	science	NOUN
ejpam-6492	520	6	,	,	PUNCT
ejpam-6492	520	7	1(1):51–66	1(1):51–66	NUM
ejpam-6492	520	8	,	,	PUNCT
ejpam-6492	520	9	2024	2024	NUM
ejpam-6492	520	10	.	.	PUNCT
ejpam-6492	521	1	[	[	X
ejpam-6492	521	2	7	7	X
ejpam-6492	521	3	]	]	PUNCT
ejpam-6492	521	4	w.	w.	PROPN
ejpam-6492	521	5	n.	n.	PROPN
ejpam-6492	521	6	goetzmann	goetzmann	PROPN
ejpam-6492	521	7	.	.	PUNCT
ejpam-6492	522	1	fibonacci	fibonacci	PROPN
ejpam-6492	522	2	and	and	CCONJ
ejpam-6492	522	3	the	the	DET
ejpam-6492	522	4	financial	financial	ADJ
ejpam-6492	522	5	revolution	revolution	NOUN
ejpam-6492	522	6	.	.	PUNCT
ejpam-6492	523	1	technical	technical	ADJ
ejpam-6492	523	2	report	report	NOUN
ejpam-6492	523	3	10352	10352	NUM
ejpam-6492	523	4	,	,	PUNCT
ejpam-6492	523	5	national	national	ADJ
ejpam-6492	523	6	bureau	bureau	NOUN
ejpam-6492	523	7	of	of	ADP
ejpam-6492	523	8	economic	economic	ADJ
ejpam-6492	523	9	research	research	NOUN
ejpam-6492	523	10	,	,	PUNCT
ejpam-6492	523	11	2004	2004	NUM
ejpam-6492	523	12	.	.	PUNCT
ejpam-6492	524	1	[	[	X
ejpam-6492	524	2	8	8	NUM
ejpam-6492	524	3	]	]	PUNCT
ejpam-6492	524	4	s.	s.	PROPN
ejpam-6492	524	5	halici	halici	PROPN
ejpam-6492	524	6	.	.	PUNCT
ejpam-6492	525	1	on	on	ADP
ejpam-6492	525	2	fibonacci	fibonacci	NOUN
ejpam-6492	525	3	quaternions	quaternion	NOUN
ejpam-6492	525	4	.	.	PUNCT
ejpam-6492	526	1	advances	advance	NOUN
ejpam-6492	526	2	in	in	ADP
ejpam-6492	526	3	applied	apply	VERB
ejpam-6492	526	4	clifford	clifford	PROPN
ejpam-6492	526	5	algebras	algebras	PROPN
ejpam-6492	526	6	,	,	PUNCT
ejpam-6492	526	7	22:321	22:321	NUM
ejpam-6492	526	8	–	–	PUNCT
ejpam-6492	526	9	327	327	NUM
ejpam-6492	526	10	,	,	PUNCT
ejpam-6492	526	11	2012	2012	NUM
ejpam-6492	526	12	.	.	PUNCT
ejpam-6492	527	1	[	[	X
ejpam-6492	527	2	9	9	NUM
ejpam-6492	527	3	]	]	PUNCT
ejpam-6492	527	4	s.	s.	PROPN
ejpam-6492	527	5	halıcı	halıcı	PROPN
ejpam-6492	527	6	.	.	PUNCT
ejpam-6492	528	1	on	on	ADP
ejpam-6492	528	2	complex	complex	ADJ
ejpam-6492	528	3	fibonacci	fibonacci	NOUN
ejpam-6492	528	4	quaternions	quaternion	NOUN
ejpam-6492	528	5	.	.	PUNCT
ejpam-6492	529	1	advances	advance	NOUN
ejpam-6492	529	2	in	in	ADP
ejpam-6492	529	3	applied	apply	VERB
ejpam-6492	529	4	clifford	clifford	PROPN
ejpam-6492	529	5	algebras	algebras	PROPN
ejpam-6492	529	6	,	,	PUNCT
ejpam-6492	529	7	23:105–112	23:105–112	PROPN
ejpam-6492	529	8	,	,	PUNCT
ejpam-6492	529	9	2013	2013	NUM
ejpam-6492	529	10	.	.	PUNCT
ejpam-6492	530	1	b.	b.	PROPN
ejpam-6492	530	2	demirtürk	demirtürk	PROPN
ejpam-6492	530	3	,	,	PUNCT
ejpam-6492	530	4	n.	n.	NOUN
ejpam-6492	530	5	topal	topal	PROPN
ejpam-6492	530	6	/	/	SYM
ejpam-6492	530	7	eur	eur	PROPN
ejpam-6492	530	8	.	.	PUNCT
ejpam-6492	531	1	j.	j.	PROPN
ejpam-6492	531	2	pure	pure	PROPN
ejpam-6492	531	3	appl	appl	PROPN
ejpam-6492	531	4	.	.	PROPN
ejpam-6492	531	5	math	math	PROPN
ejpam-6492	531	6	,	,	PUNCT
ejpam-6492	531	7	18	18	NUM
ejpam-6492	531	8	(	(	PUNCT
ejpam-6492	531	9	3	3	NUM
ejpam-6492	531	10	)	)	PUNCT
ejpam-6492	531	11	(	(	PUNCT
ejpam-6492	531	12	2025	2025	NUM
ejpam-6492	531	13	)	)	PUNCT
ejpam-6492	531	14	,	,	PUNCT
ejpam-6492	531	15	6492	6492	NUM
ejpam-6492	531	16	21	21	NUM
ejpam-6492	531	17	of	of	ADP
ejpam-6492	531	18	22	22	NUM
ejpam-6492	532	1	[	[	X
ejpam-6492	532	2	10	10	NUM
ejpam-6492	532	3	]	]	PUNCT
ejpam-6492	532	4	s.	s.	PROPN
ejpam-6492	532	5	halıcı	halıcı	PROPN
ejpam-6492	532	6	and	and	CCONJ
ejpam-6492	532	7	a.	a.	NOUN
ejpam-6492	532	8	karataş.	karataş.	PROPN
ejpam-6492	533	1	[	[	X
ejpam-6492	533	2	11	11	NUM
ejpam-6492	533	3	]	]	PUNCT
ejpam-6492	533	4	s.	s.	PROPN
ejpam-6492	533	5	kesim	kesim	PROPN
ejpam-6492	533	6	.	.	PUNCT
ejpam-6492	534	1	exponential	exponential	ADJ
ejpam-6492	534	2	generating	generating	NOUN
ejpam-6492	534	3	functions	function	NOUN
ejpam-6492	534	4	for	for	ADP
ejpam-6492	534	5	generalized	generalized	ADJ
ejpam-6492	534	6	fibonacci	fibonacci	NOUN
ejpam-6492	534	7	and	and	CCONJ
ejpam-6492	534	8	lucas	lucas	PROPN
ejpam-6492	534	9	quaternions	quaternion	NOUN
ejpam-6492	534	10	.	.	PUNCT
ejpam-6492	535	1	advances	advance	NOUN
ejpam-6492	535	2	in	in	ADP
ejpam-6492	535	3	difference	difference	NOUN
ejpam-6492	535	4	equations	equation	NOUN
ejpam-6492	535	5	,	,	PUNCT
ejpam-6492	535	6	2015(1):169	2015(1):169	NUM
ejpam-6492	535	7	,	,	PUNCT
ejpam-6492	535	8	2015	2015	NUM
ejpam-6492	535	9	.	.	PUNCT
ejpam-6492	536	1	[	[	X
ejpam-6492	536	2	12	12	NUM
ejpam-6492	536	3	]	]	PUNCT
ejpam-6492	536	4	s.	s.	PROPN
ejpam-6492	536	5	köme	köme	PROPN
ejpam-6492	536	6	,	,	PUNCT
ejpam-6492	536	7	c.	c.	PROPN
ejpam-6492	536	8	köme	köme	PROPN
ejpam-6492	536	9	,	,	PUNCT
ejpam-6492	536	10	and	and	CCONJ
ejpam-6492	536	11	y.	y.	PROPN
ejpam-6492	536	12	yazlık	yazlık	PROPN
ejpam-6492	536	13	.	.	PUNCT
ejpam-6492	537	1	modified	modify	VERB
ejpam-6492	537	2	generalized	generalized	ADJ
ejpam-6492	537	3	fibonacci	fibonacci	NOUN
ejpam-6492	537	4	and	and	CCONJ
ejpam-6492	537	5	lucas	lucas	PROPN
ejpam-6492	537	6	quaternions	quaternion	NOUN
ejpam-6492	537	7	.	.	PUNCT
ejpam-6492	538	1	journal	journal	NOUN
ejpam-6492	538	2	of	of	ADP
ejpam-6492	538	3	science	science	NOUN
ejpam-6492	538	4	and	and	CCONJ
ejpam-6492	538	5	arts	art	NOUN
ejpam-6492	538	6	,	,	PUNCT
ejpam-6492	538	7	19(1):49–60	19(1):49–60	NUM
ejpam-6492	538	8	,	,	PUNCT
ejpam-6492	538	9	2019	2019	NUM
ejpam-6492	538	10	.	.	PUNCT
ejpam-6492	539	1	[	[	X
ejpam-6492	539	2	13	13	NUM
ejpam-6492	539	3	]	]	PUNCT
ejpam-6492	539	4	s.	s.	PROPN
ejpam-6492	539	5	aydinyüz	aydinyüz	PROPN
ejpam-6492	539	6	and	and	CCONJ
ejpam-6492	539	7	m.	m.	PROPN
ejpam-6492	539	8	asci	asci	PROPN
ejpam-6492	539	9	.	.	PUNCT
ejpam-6492	540	1	generalized	generalize	VERB
ejpam-6492	540	2	k	k	ADJ
ejpam-6492	540	3	-	-	PUNCT
ejpam-6492	540	4	order	order	NOUN
ejpam-6492	540	5	fibonacci	fibonacci	NOUN
ejpam-6492	540	6	and	and	CCONJ
ejpam-6492	540	7	lucas	lucas	PROPN
ejpam-6492	540	8	quaternions	quaternion	NOUN
ejpam-6492	540	9	.	.	PUNCT
ejpam-6492	541	1	turkish	turkish	ADJ
ejpam-6492	541	2	journal	journal	NOUN
ejpam-6492	541	3	of	of	ADP
ejpam-6492	541	4	analysis	analysis	NOUN
ejpam-6492	541	5	and	and	CCONJ
ejpam-6492	541	6	number	number	NOUN
ejpam-6492	541	7	theory	theory	NOUN
ejpam-6492	541	8	,	,	PUNCT
ejpam-6492	541	9	11(1):1–6	11(1):1–6	NUM
ejpam-6492	541	10	,	,	PUNCT
ejpam-6492	541	11	2023	2023	NUM
ejpam-6492	541	12	.	.	PUNCT
ejpam-6492	542	1	[	[	X
ejpam-6492	542	2	14	14	NUM
ejpam-6492	542	3	]	]	X
ejpam-6492	542	4	c.	c.	PROPN
ejpam-6492	542	5	kızılateş	kızılateş	PROPN
ejpam-6492	542	6	,	,	PUNCT
ejpam-6492	542	7	w.	w.	PROPN
ejpam-6492	542	8	s.	s.	PROPN
ejpam-6492	542	9	du	du	PROPN
ejpam-6492	542	10	,	,	PUNCT
ejpam-6492	542	11	n.	n.	PROPN
ejpam-6492	542	12	terzioğlu	terzioğlu	PROPN
ejpam-6492	542	13	,	,	PUNCT
ejpam-6492	542	14	and	and	CCONJ
ejpam-6492	542	15	r.	r.	PROPN
ejpam-6492	542	16	c.	c.	PROPN
ejpam-6492	542	17	chen	chen	PROPN
ejpam-6492	542	18	.	.	PUNCT
ejpam-6492	543	1	new	new	ADJ
ejpam-6492	543	2	properties	property	NOUN
ejpam-6492	543	3	and	and	CCONJ
ejpam-6492	543	4	matrix	matrix	VERB
ejpam-6492	543	5	representations	representation	NOUN
ejpam-6492	543	6	on	on	ADP
ejpam-6492	543	7	higher	high	ADJ
ejpam-6492	543	8	-	-	PUNCT
ejpam-6492	543	9	order	order	NOUN
ejpam-6492	543	10	generalized	generalize	VERB
ejpam-6492	543	11	fibonacci	fibonacci	NOUN
ejpam-6492	543	12	quaternions	quaternion	NOUN
ejpam-6492	543	13	with	with	ADP
ejpam-6492	543	14	q	q	ADJ
ejpam-6492	543	15	-	-	PUNCT
ejpam-6492	543	16	integer	integer	NOUN
ejpam-6492	543	17	components	component	NOUN
ejpam-6492	543	18	.	.	PUNCT
ejpam-6492	544	1	axioms	axiom	NOUN
ejpam-6492	544	2	,	,	PUNCT
ejpam-6492	544	3	13(10):677	13(10):677	NUM
ejpam-6492	544	4	,	,	PUNCT
ejpam-6492	544	5	2024	2024	NUM
ejpam-6492	544	6	.	.	PUNCT
ejpam-6492	545	1	[	[	X
ejpam-6492	545	2	15	15	NUM
ejpam-6492	545	3	]	]	X
ejpam-6492	545	4	c.	c.	PROPN
ejpam-6492	545	5	flaut	flaut	PROPN
ejpam-6492	545	6	and	and	CCONJ
ejpam-6492	545	7	v.	v.	ADP
ejpam-6492	545	8	shpakivskyi	shpakivskyi	NOUN
ejpam-6492	545	9	.	.	PUNCT
ejpam-6492	546	1	on	on	ADP
ejpam-6492	546	2	generalized	generalized	ADJ
ejpam-6492	546	3	fibonacci	fibonacci	NOUN
ejpam-6492	546	4	quaternions	quaternion	NOUN
ejpam-6492	546	5	and	and	CCONJ
ejpam-6492	546	6	fibonacci	fibonacci	NOUN
ejpam-6492	546	7	–	–	PUNCT
ejpam-6492	546	8	narayana	narayana	PROPN
ejpam-6492	546	9	quaternions	quaternion	NOUN
ejpam-6492	546	10	.	.	PUNCT
ejpam-6492	547	1	advances	advance	NOUN
ejpam-6492	547	2	in	in	ADP
ejpam-6492	547	3	applied	apply	VERB
ejpam-6492	547	4	clifford	clifford	PROPN
ejpam-6492	547	5	algebras	algebras	PROPN
ejpam-6492	547	6	,	,	PUNCT
ejpam-6492	547	7	23(3):673–688	23(3):673–688	NUM
ejpam-6492	547	8	,	,	PUNCT
ejpam-6492	547	9	2013	2013	NUM
ejpam-6492	547	10	.	.	PUNCT
ejpam-6492	548	1	[	[	X
ejpam-6492	548	2	16	16	NUM
ejpam-6492	548	3	]	]	PUNCT
ejpam-6492	548	4	a.	a.	PROPN
ejpam-6492	548	5	f.	f.	PROPN
ejpam-6492	548	6	horadam	horadam	PROPN
ejpam-6492	548	7	.	.	PUNCT
ejpam-6492	549	1	a	a	DET
ejpam-6492	549	2	generalized	generalized	ADJ
ejpam-6492	549	3	fibonacci	fibonacci	NOUN
ejpam-6492	549	4	sequence	sequence	NOUN
ejpam-6492	549	5	.	.	PUNCT
ejpam-6492	550	1	the	the	DET
ejpam-6492	550	2	american	american	PROPN
ejpam-6492	550	3	mathematical	mathematical	PROPN
ejpam-6492	550	4	monthly	monthly	ADV
ejpam-6492	550	5	,	,	PUNCT
ejpam-6492	550	6	68(5):455–459	68(5):455–459	NOUN
ejpam-6492	550	7	,	,	PUNCT
ejpam-6492	550	8	1961	1961	NUM
ejpam-6492	550	9	.	.	PUNCT
ejpam-6492	551	1	[	[	X
ejpam-6492	551	2	17	17	NUM
ejpam-6492	551	3	]	]	PUNCT
ejpam-6492	551	4	a.	a.	PROPN
ejpam-6492	551	5	f.	f.	PROPN
ejpam-6492	551	6	horadam	horadam	PROPN
ejpam-6492	551	7	.	.	PUNCT
ejpam-6492	552	1	basic	basic	ADJ
ejpam-6492	552	2	properties	property	NOUN
ejpam-6492	552	3	of	of	ADP
ejpam-6492	552	4	certain	certain	ADJ
ejpam-6492	552	5	generalized	generalized	ADJ
ejpam-6492	552	6	sequences	sequence	NOUN
ejpam-6492	552	7	of	of	ADP
ejpam-6492	552	8	numbers	number	NOUN
ejpam-6492	552	9	.	.	PUNCT
ejpam-6492	553	1	the	the	DET
ejpam-6492	553	2	fibonacci	fibonacci	PROPN
ejpam-6492	553	3	quarterly	quarterly	PROPN
ejpam-6492	553	4	,	,	PUNCT
ejpam-6492	553	5	3(3):161–176	3(3):161–176	PROPN
ejpam-6492	553	6	,	,	PUNCT
ejpam-6492	553	7	1965	1965	NUM
ejpam-6492	553	8	.	.	PUNCT
ejpam-6492	554	1	[	[	X
ejpam-6492	554	2	18	18	NUM
ejpam-6492	554	3	]	]	X
ejpam-6492	554	4	v.	v.	PROPN
ejpam-6492	554	5	e.	e.	PROPN
ejpam-6492	554	6	hoggatt	hoggatt	PROPN
ejpam-6492	554	7	.	.	PUNCT
ejpam-6492	555	1	fibonacci	fibonacci	PROPN
ejpam-6492	555	2	and	and	CCONJ
ejpam-6492	555	3	lucas	lucas	PROPN
ejpam-6492	555	4	numbers	number	NOUN
ejpam-6492	555	5	.	.	PUNCT
ejpam-6492	556	1	houghton	houghton	PROPN
ejpam-6492	556	2	-	-	PUNCT
ejpam-6492	556	3	mifflin	mifflin	PROPN
ejpam-6492	556	4	,	,	PUNCT
ejpam-6492	556	5	boston	boston	PROPN
ejpam-6492	556	6	,	,	PUNCT
ejpam-6492	556	7	1969	1969	NUM
ejpam-6492	556	8	.	.	PUNCT
ejpam-6492	557	1	[	[	X
ejpam-6492	557	2	19	19	NUM
ejpam-6492	557	3	]	]	X
ejpam-6492	557	4	t.	t.	PROPN
ejpam-6492	557	5	koshy	koshy	PROPN
ejpam-6492	557	6	.	.	PUNCT
ejpam-6492	558	1	fibonacci	fibonacci	PROPN
ejpam-6492	558	2	and	and	CCONJ
ejpam-6492	558	3	lucas	lucas	PROPN
ejpam-6492	558	4	numbers	number	NOUN
ejpam-6492	558	5	with	with	ADP
ejpam-6492	558	6	applications	application	NOUN
ejpam-6492	558	7	.	.	PUNCT
ejpam-6492	559	1	wiley	wiley	PROPN
ejpam-6492	559	2	,	,	PUNCT
ejpam-6492	559	3	new	new	PROPN
ejpam-6492	559	4	york	york	PROPN
ejpam-6492	559	5	,	,	PUNCT
ejpam-6492	559	6	2001	2001	NUM
ejpam-6492	559	7	.	.	PUNCT
ejpam-6492	560	1	[	[	X
ejpam-6492	560	2	20	20	NUM
ejpam-6492	560	3	]	]	X
ejpam-6492	560	4	d.	d.	PROPN
ejpam-6492	560	5	kalman	kalman	PROPN
ejpam-6492	560	6	and	and	CCONJ
ejpam-6492	560	7	r.	r.	PROPN
ejpam-6492	560	8	mena	mena	PROPN
ejpam-6492	560	9	.	.	PUNCT
ejpam-6492	561	1	the	the	DET
ejpam-6492	561	2	fibonacci	fibonacci	NOUN
ejpam-6492	561	3	numbers	number	NOUN
ejpam-6492	561	4	exposed	expose	VERB
ejpam-6492	561	5	.	.	PUNCT
ejpam-6492	562	1	mathematics	mathematic	NOUN
ejpam-6492	562	2	magazine	magazine	NOUN
ejpam-6492	562	3	,	,	PUNCT
ejpam-6492	562	4	76(3):167–181	76(3):167–181	PROPN
ejpam-6492	562	5	,	,	PUNCT
ejpam-6492	562	6	2003	2003	NUM
ejpam-6492	562	7	.	.	PUNCT
ejpam-6492	563	1	[	[	X
ejpam-6492	563	2	21	21	NUM
ejpam-6492	563	3	]	]	X
ejpam-6492	563	4	s.	s.	PROPN
ejpam-6492	563	5	vajda	vajda	PROPN
ejpam-6492	563	6	.	.	PUNCT
ejpam-6492	564	1	fibonacci	fibonacci	PROPN
ejpam-6492	564	2	and	and	CCONJ
ejpam-6492	564	3	lucas	lucas	PROPN
ejpam-6492	564	4	numbers	number	NOUN
ejpam-6492	564	5	and	and	CCONJ
ejpam-6492	564	6	the	the	DET
ejpam-6492	564	7	golden	golden	ADJ
ejpam-6492	564	8	section	section	NOUN
ejpam-6492	564	9	.	.	PUNCT
ejpam-6492	565	1	ellis	ellis	PROPN
ejpam-6492	565	2	horwood	horwood	PROPN
ejpam-6492	565	3	limited	limited	PROPN
ejpam-6492	565	4	,	,	PUNCT
ejpam-6492	565	5	chichester	chichester	PROPN
ejpam-6492	565	6	,	,	PUNCT
ejpam-6492	565	7	1989	1989	NUM
ejpam-6492	565	8	.	.	PUNCT
ejpam-6492	566	1	[	[	X
ejpam-6492	566	2	22	22	NUM
ejpam-6492	566	3	]	]	PUNCT
ejpam-6492	566	4	s.	s.	PROPN
ejpam-6492	566	5	rabinowitz	rabinowitz	PROPN
ejpam-6492	566	6	.	.	PUNCT
ejpam-6492	567	1	algorithmic	algorithmic	ADJ
ejpam-6492	567	2	manipulation	manipulation	NOUN
ejpam-6492	567	3	of	of	ADP
ejpam-6492	567	4	fibonacci	fibonacci	NOUN
ejpam-6492	567	5	identities	identity	NOUN
ejpam-6492	567	6	.	.	PUNCT
ejpam-6492	568	1	in	in	ADP
ejpam-6492	568	2	applications	application	NOUN
ejpam-6492	568	3	of	of	ADP
ejpam-6492	568	4	fibonacci	fibonacci	NOUN
ejpam-6492	568	5	numbers	number	NOUN
ejpam-6492	568	6	,	,	PUNCT
ejpam-6492	568	7	volume	volume	NOUN
ejpam-6492	568	8	6	6	NUM
ejpam-6492	568	9	,	,	PUNCT
ejpam-6492	568	10	pages	page	NOUN
ejpam-6492	568	11	389–408	389–408	NUM
ejpam-6492	568	12	.	.	PUNCT
ejpam-6492	569	1	kluwer	kluwer	NOUN
ejpam-6492	569	2	academic	academic	ADJ
ejpam-6492	569	3	publishers	publisher	NOUN
ejpam-6492	569	4	,	,	PUNCT
ejpam-6492	569	5	dordrecht	dordrecht	PROPN
ejpam-6492	569	6	,	,	PUNCT
ejpam-6492	569	7	1996	1996	NUM
ejpam-6492	569	8	.	.	PUNCT
ejpam-6492	570	1	[	[	X
ejpam-6492	570	2	23	23	NUM
ejpam-6492	570	3	]	]	X
ejpam-6492	570	4	s.	s.	PROPN
ejpam-6492	570	5	fairgrieve	fairgrieve	PROPN
ejpam-6492	570	6	and	and	CCONJ
ejpam-6492	570	7	h.	h.	PROPN
ejpam-6492	570	8	w.	w.	PROPN
ejpam-6492	570	9	gould	gould	PROPN
ejpam-6492	570	10	.	.	PUNCT
ejpam-6492	571	1	product	product	NOUN
ejpam-6492	571	2	difference	difference	NOUN
ejpam-6492	571	3	fibonacci	fibonacci	NOUN
ejpam-6492	571	4	identities	identity	NOUN
ejpam-6492	571	5	of	of	ADP
ejpam-6492	571	6	simson	simson	PROPN
ejpam-6492	571	7	,	,	PUNCT
ejpam-6492	571	8	gelin	gelin	PROPN
ejpam-6492	571	9	-	-	PUNCT
ejpam-6492	571	10	cesàro	cesàro	PROPN
ejpam-6492	571	11	,	,	PUNCT
ejpam-6492	571	12	tagiuri	tagiuri	NOUN
ejpam-6492	571	13	and	and	CCONJ
ejpam-6492	571	14	generalizations	generalization	NOUN
ejpam-6492	571	15	.	.	PUNCT
ejpam-6492	572	1	the	the	DET
ejpam-6492	572	2	fibonacci	fibonacci	NOUN
ejpam-6492	572	3	quarterly	quarterly	ADV
ejpam-6492	572	4	,	,	PUNCT
ejpam-6492	572	5	43(2):137–141	43(2):137–141	PROPN
ejpam-6492	572	6	,	,	PUNCT
ejpam-6492	572	7	2005	2005	NUM
ejpam-6492	572	8	.	.	PUNCT
ejpam-6492	573	1	[	[	X
ejpam-6492	573	2	24	24	NUM
ejpam-6492	573	3	]	]	PUNCT
ejpam-6492	573	4	j.	j.	PROPN
ejpam-6492	573	5	morgado	morgado	PROPN
ejpam-6492	573	6	.	.	PUNCT
ejpam-6492	574	1	some	some	DET
ejpam-6492	574	2	remarks	remark	NOUN
ejpam-6492	574	3	on	on	ADP
ejpam-6492	574	4	an	an	DET
ejpam-6492	574	5	identity	identity	NOUN
ejpam-6492	574	6	of	of	ADP
ejpam-6492	574	7	catalan	catalan	NOUN
ejpam-6492	574	8	concerning	concern	VERB
ejpam-6492	574	9	the	the	DET
ejpam-6492	574	10	fibonacci	fibonacci	NOUN
ejpam-6492	574	11	numbers	number	NOUN
ejpam-6492	574	12	.	.	PUNCT
ejpam-6492	575	1	portugaliae	portugaliae	PROPN
ejpam-6492	575	2	mathematica	mathematica	PROPN
ejpam-6492	575	3	,	,	PUNCT
ejpam-6492	575	4	39:341–348	39:341–348	PROPN
ejpam-6492	575	5	,	,	PUNCT
ejpam-6492	575	6	1980	1980	NUM
ejpam-6492	575	7	.	.	PUNCT
ejpam-6492	576	1	[	[	X
ejpam-6492	576	2	25	25	NUM
ejpam-6492	576	3	]	]	PUNCT
ejpam-6492	576	4	r.	r.	PROPN
ejpam-6492	576	5	s.	s.	PROPN
ejpam-6492	576	6	melham	melham	PROPN
ejpam-6492	576	7	.	.	PUNCT
ejpam-6492	577	1	on	on	ADP
ejpam-6492	577	2	product	product	NOUN
ejpam-6492	577	3	difference	difference	NOUN
ejpam-6492	577	4	fibonacci	fibonacci	NOUN
ejpam-6492	577	5	identities	identity	NOUN
ejpam-6492	577	6	.	.	PUNCT
ejpam-6492	578	1	integers	integer	NOUN
ejpam-6492	578	2	,	,	PUNCT
ejpam-6492	578	3	11(2):119–126	11(2):119–126	NUM
ejpam-6492	578	4	,	,	PUNCT
ejpam-6492	578	5	2011	2011	NUM
ejpam-6492	578	6	.	.	PUNCT
ejpam-6492	579	1	[	[	X
ejpam-6492	579	2	26	26	NUM
ejpam-6492	579	3	]	]	X
ejpam-6492	579	4	d.	d.	NOUN
ejpam-6492	579	5	everman	everman	NOUN
ejpam-6492	579	6	,	,	PUNCT
ejpam-6492	579	7	a.	a.	PROPN
ejpam-6492	579	8	e.	e.	PROPN
ejpam-6492	579	9	danese	danese	PROPN
ejpam-6492	579	10	,	,	PUNCT
ejpam-6492	579	11	and	and	CCONJ
ejpam-6492	579	12	k.	k.	PROPN
ejpam-6492	579	13	venkannayah	venkannayah	PROPN
ejpam-6492	579	14	.	.	PUNCT
ejpam-6492	580	1	problem	problem	NOUN
ejpam-6492	580	2	e1396	e1396	ADV
ejpam-6492	580	3	.	.	PUNCT
ejpam-6492	581	1	the	the	DET
ejpam-6492	581	2	american	american	PROPN
ejpam-6492	581	3	mathematical	mathematical	PROPN
ejpam-6492	581	4	monthly	monthly	ADV
ejpam-6492	581	5	,	,	PUNCT
ejpam-6492	581	6	67:81–82	67:81–82	NUM
ejpam-6492	581	7	,	,	PUNCT
ejpam-6492	581	8	1960	1960	NUM
ejpam-6492	581	9	.	.	PUNCT
ejpam-6492	582	1	[	[	X
ejpam-6492	582	2	27	27	NUM
ejpam-6492	582	3	]	]	PUNCT
ejpam-6492	582	4	z.	z.	PROPN
ejpam-6492	582	5	şiar	şiar	PROPN
ejpam-6492	582	6	and	and	CCONJ
ejpam-6492	582	7	r.	r.	PROPN
ejpam-6492	582	8	keskin	keskin	PROPN
ejpam-6492	582	9	.	.	PUNCT
ejpam-6492	583	1	some	some	DET
ejpam-6492	583	2	new	new	ADJ
ejpam-6492	583	3	identities	identity	NOUN
ejpam-6492	583	4	concerning	concern	VERB
ejpam-6492	583	5	generalized	generalized	ADJ
ejpam-6492	583	6	fibonacci	fibonacci	NOUN
ejpam-6492	583	7	and	and	CCONJ
ejpam-6492	583	8	lucas	lucas	PROPN
ejpam-6492	583	9	numbers	number	NOUN
ejpam-6492	583	10	.	.	PUNCT
ejpam-6492	584	1	hacettepe	hacettepe	PROPN
ejpam-6492	584	2	journal	journal	PROPN
ejpam-6492	584	3	of	of	ADP
ejpam-6492	584	4	mathematics	mathematic	NOUN
ejpam-6492	584	5	and	and	CCONJ
ejpam-6492	584	6	statistics	statistic	NOUN
ejpam-6492	584	7	,	,	PUNCT
ejpam-6492	584	8	42:211–222	42:211–222	PROPN
ejpam-6492	584	9	,	,	PUNCT
ejpam-6492	584	10	2013	2013	NUM
ejpam-6492	584	11	.	.	PUNCT
ejpam-6492	585	1	[	[	X
ejpam-6492	585	2	28	28	NUM
ejpam-6492	585	3	]	]	X
ejpam-6492	585	4	w.	w.	PROPN
ejpam-6492	585	5	r.	r.	PROPN
ejpam-6492	585	6	hamilton	hamilton	PROPN
ejpam-6492	585	7	.	.	PUNCT
ejpam-6492	586	1	elements	element	NOUN
ejpam-6492	586	2	of	of	ADP
ejpam-6492	586	3	quaternions	quaternion	NOUN
ejpam-6492	586	4	.	.	PUNCT
ejpam-6492	587	1	longmans	longman	NOUN
ejpam-6492	587	2	,	,	PUNCT
ejpam-6492	587	3	green	green	ADJ
ejpam-6492	587	4	,	,	PUNCT
ejpam-6492	587	5	london	london	PROPN
ejpam-6492	587	6	,	,	PUNCT
ejpam-6492	587	7	1866	1866	NUM
ejpam-6492	587	8	.	.	PUNCT
ejpam-6492	588	1	[	[	X
ejpam-6492	588	2	29	29	NUM
ejpam-6492	588	3	]	]	PUNCT
ejpam-6492	588	4	a.	a.	NOUN
ejpam-6492	588	5	s.	s.	PROPN
ejpam-6492	588	6	hardy	hardy	PROPN
ejpam-6492	588	7	.	.	PUNCT
ejpam-6492	589	1	elements	element	NOUN
ejpam-6492	589	2	of	of	ADP
ejpam-6492	589	3	quaternions	quaternion	NOUN
ejpam-6492	589	4	.	.	PUNCT
ejpam-6492	590	1	ginn	ginn	PROPN
ejpam-6492	590	2	,	,	PUNCT
ejpam-6492	590	3	heath	heath	PROPN
ejpam-6492	590	4	&	&	CCONJ
ejpam-6492	590	5	co.	co.	PROPN
ejpam-6492	590	6	,	,	PUNCT
ejpam-6492	590	7	boston	boston	PROPN
ejpam-6492	590	8	,	,	PUNCT
ejpam-6492	590	9	1881	1881	NUM
ejpam-6492	590	10	.	.	PUNCT
ejpam-6492	591	1	[	[	X
ejpam-6492	591	2	30	30	NUM
ejpam-6492	591	3	]	]	X
ejpam-6492	591	4	g.	g.	PROPN
ejpam-6492	591	5	h.	h.	PROPN
ejpam-6492	591	6	hardy	hardy	PROPN
ejpam-6492	591	7	and	and	CCONJ
ejpam-6492	591	8	e.	e.	PROPN
ejpam-6492	591	9	m.	m.	PROPN
ejpam-6492	591	10	wright	wright	PROPN
ejpam-6492	591	11	.	.	PUNCT
ejpam-6492	592	1	an	an	DET
ejpam-6492	592	2	introduction	introduction	NOUN
ejpam-6492	592	3	to	to	ADP
ejpam-6492	592	4	the	the	DET
ejpam-6492	592	5	theory	theory	NOUN
ejpam-6492	592	6	of	of	ADP
ejpam-6492	592	7	numbers	number	NOUN
ejpam-6492	592	8	.	.	PUNCT
ejpam-6492	593	1	oxford	oxford	PROPN
ejpam-6492	593	2	university	university	PROPN
ejpam-6492	593	3	press	press	PROPN
ejpam-6492	593	4	,	,	PUNCT
ejpam-6492	593	5	london	london	PROPN
ejpam-6492	593	6	,	,	PUNCT
ejpam-6492	593	7	1960	1960	NUM
ejpam-6492	593	8	.	.	PUNCT
ejpam-6492	594	1	[	[	X
ejpam-6492	594	2	31	31	NUM
ejpam-6492	594	3	]	]	PUNCT
ejpam-6492	594	4	a.	a.	NOUN
ejpam-6492	594	5	l.	l.	PROPN
ejpam-6492	594	6	iakin	iakin	PROPN
ejpam-6492	594	7	.	.	PUNCT
ejpam-6492	595	1	generalized	generalize	VERB
ejpam-6492	595	2	quaternions	quaternion	NOUN
ejpam-6492	595	3	with	with	ADP
ejpam-6492	595	4	quaternions	quaternions	ADJ
ejpam-6492	595	5	components	component	NOUN
ejpam-6492	595	6	.	.	PUNCT
ejpam-6492	596	1	the	the	DET
ejpam-6492	596	2	fibonacci	fibonacci	NOUN
ejpam-6492	596	3	quarterly	quarterly	PROPN
ejpam-6492	596	4	,	,	PUNCT
ejpam-6492	596	5	15(4):350–352	15(4):350–352	PROPN
ejpam-6492	596	6	,	,	PUNCT
ejpam-6492	596	7	1977	1977	NUM
ejpam-6492	596	8	.	.	PUNCT
ejpam-6492	597	1	[	[	X
ejpam-6492	597	2	32	32	NUM
ejpam-6492	597	3	]	]	PUNCT
ejpam-6492	597	4	a.	a.	NOUN
ejpam-6492	597	5	f.	f.	PROPN
ejpam-6492	597	6	horadam	horadam	PROPN
ejpam-6492	597	7	.	.	PUNCT
ejpam-6492	598	1	complex	complex	ADJ
ejpam-6492	598	2	fibonacci	fibonacci	NOUN
ejpam-6492	598	3	numbers	number	NOUN
ejpam-6492	598	4	and	and	CCONJ
ejpam-6492	598	5	fibonacci	fibonacci	NOUN
ejpam-6492	598	6	quaternions	quaternion	NOUN
ejpam-6492	598	7	.	.	PUNCT
ejpam-6492	599	1	the	the	DET
ejpam-6492	599	2	amerb	amerb	NOUN
ejpam-6492	599	3	.	.	PUNCT
ejpam-6492	600	1	demirtürk	demirtürk	NOUN
ejpam-6492	600	2	,	,	PUNCT
ejpam-6492	600	3	n.	n.	NOUN
ejpam-6492	600	4	topal	topal	PROPN
ejpam-6492	600	5	/	/	SYM
ejpam-6492	600	6	eur	eur	PROPN
ejpam-6492	600	7	.	.	PUNCT
ejpam-6492	601	1	j.	j.	PROPN
ejpam-6492	601	2	pure	pure	PROPN
ejpam-6492	601	3	appl	appl	PROPN
ejpam-6492	601	4	.	.	PROPN
ejpam-6492	601	5	math	math	PROPN
ejpam-6492	601	6	,	,	PUNCT
ejpam-6492	601	7	18	18	NUM
ejpam-6492	601	8	(	(	PUNCT
ejpam-6492	601	9	3	3	NUM
ejpam-6492	601	10	)	)	PUNCT
ejpam-6492	601	11	(	(	PUNCT
ejpam-6492	601	12	2025	2025	NUM
ejpam-6492	601	13	)	)	PUNCT
ejpam-6492	601	14	,	,	PUNCT
ejpam-6492	601	15	6492	6492	NUM
ejpam-6492	601	16	22	22	NUM
ejpam-6492	601	17	of	of	ADP
ejpam-6492	601	18	22	22	NUM
ejpam-6492	601	19	ican	ican	PROPN
ejpam-6492	601	20	mathematical	mathematical	ADJ
ejpam-6492	601	21	monthly	monthly	ADV
ejpam-6492	601	22	,	,	PUNCT
ejpam-6492	601	23	70:289–291	70:289–291	PROPN
ejpam-6492	601	24	,	,	PUNCT
ejpam-6492	601	25	1963	1963	NUM
ejpam-6492	601	26	.	.	PUNCT
ejpam-6492	602	1	[	[	X
ejpam-6492	602	2	33	33	NUM
ejpam-6492	602	3	]	]	PUNCT
ejpam-6492	602	4	a.	a.	NOUN
ejpam-6492	602	5	l.	l.	PROPN
ejpam-6492	602	6	iakin	iakin	PROPN
ejpam-6492	602	7	.	.	PUNCT
ejpam-6492	603	1	extended	extend	VERB
ejpam-6492	603	2	binet	binet	NOUN
ejpam-6492	603	3	forms	form	NOUN
ejpam-6492	603	4	for	for	ADP
ejpam-6492	603	5	generalized	generalized	ADJ
ejpam-6492	603	6	quaternions	quaternion	NOUN
ejpam-6492	603	7	of	of	ADP
ejpam-6492	603	8	higher	high	ADJ
ejpam-6492	603	9	order	order	NOUN
ejpam-6492	603	10	.	.	PUNCT
ejpam-6492	604	1	the	the	DET
ejpam-6492	604	2	fibonacci	fibonacci	NOUN
ejpam-6492	604	3	quarterly	quarterly	ADV
ejpam-6492	604	4	,	,	PUNCT
ejpam-6492	604	5	19(5):410–413	19(5):410–413	PROPN
ejpam-6492	604	6	,	,	PUNCT
ejpam-6492	604	7	1981	1981	NUM
ejpam-6492	604	8	.	.	PUNCT
ejpam-6492	605	1	[	[	X
ejpam-6492	605	2	34	34	NUM
ejpam-6492	605	3	]	]	X
ejpam-6492	605	4	m.	m.	PROPN
ejpam-6492	605	5	r.	r.	PROPN
ejpam-6492	605	6	iyer	iyer	PROPN
ejpam-6492	605	7	.	.	PUNCT
ejpam-6492	606	1	some	some	DET
ejpam-6492	606	2	results	result	NOUN
ejpam-6492	606	3	on	on	ADP
ejpam-6492	606	4	fibonacci	fibonacci	NOUN
ejpam-6492	606	5	quaternions	quaternion	NOUN
ejpam-6492	606	6	.	.	PUNCT
ejpam-6492	607	1	the	the	DET
ejpam-6492	607	2	fibonacci	fibonacci	NOUN
ejpam-6492	607	3	quarterly	quarterly	ADV
ejpam-6492	607	4	,	,	PUNCT
ejpam-6492	607	5	7(2):201–210	7(2):201–210	NOUN
ejpam-6492	607	6	,	,	PUNCT
ejpam-6492	607	7	1969	1969	NUM
ejpam-6492	607	8	.	.	PUNCT