id	sid	tid	token	lemma	pos
ejpam-6452	1	1	european	european	PROPN
ejpam-6452	1	2	journal	journal	PROPN
ejpam-6452	1	3	of	of	ADP
ejpam-6452	1	4	pure	pure	ADJ
ejpam-6452	1	5	and	and	CCONJ
ejpam-6452	1	6	applied	applied	ADJ
ejpam-6452	1	7	mathematics	mathematic	NOUN
ejpam-6452	1	8	2025	2025	NUM
ejpam-6452	1	9	,	,	PUNCT
ejpam-6452	1	10	vol	vol	NOUN
ejpam-6452	1	11	.	.	PROPN
ejpam-6452	1	12	18	18	NUM
ejpam-6452	1	13	,	,	PUNCT
ejpam-6452	1	14	issue	issue	NOUN
ejpam-6452	1	15	3	3	NUM
ejpam-6452	1	16	,	,	PUNCT
ejpam-6452	1	17	article	article	NOUN
ejpam-6452	1	18	number	number	NOUN
ejpam-6452	1	19	6452	6452	NUM
ejpam-6452	1	20	issn	issn	VERB
ejpam-6452	1	21	1307	1307	NUM
ejpam-6452	1	22	-	-	SYM
ejpam-6452	1	23	5543	5543	NUM
ejpam-6452	1	24	–	–	PUNCT
ejpam-6452	1	25	ejpam.com	ejpam.com	X
ejpam-6452	1	26	published	publish	VERB
ejpam-6452	1	27	by	by	ADP
ejpam-6452	1	28	new	new	PROPN
ejpam-6452	1	29	york	york	PROPN
ejpam-6452	1	30	business	business	PROPN
ejpam-6452	1	31	global	global	ADJ
ejpam-6452	1	32	integral	integral	ADJ
ejpam-6452	1	33	representations	representation	NOUN
ejpam-6452	1	34	of	of	ADP
ejpam-6452	1	35	generalizations	generalization	NOUN
ejpam-6452	1	36	of	of	ADP
ejpam-6452	1	37	pell	pell	NOUN
ejpam-6452	1	38	numbers	number	NOUN
ejpam-6452	1	39	and	and	CCONJ
ejpam-6452	1	40	their	their	PRON
ejpam-6452	1	41	companion	companion	NOUN
ejpam-6452	1	42	numbers	number	NOUN
ejpam-6452	1	43	weerayuth	weerayuth	NOUN
ejpam-6452	1	44	nilsrakoo1	nilsrakoo1	PROPN
ejpam-6452	1	45	,	,	PUNCT
ejpam-6452	1	46	achariya	achariya	ADV
ejpam-6452	1	47	nilsrakoo2,∗	nilsrakoo2,∗	PROPN
ejpam-6452	1	48	1	1	NUM
ejpam-6452	1	49	department	department	NOUN
ejpam-6452	1	50	of	of	ADP
ejpam-6452	1	51	mathematics	mathematic	NOUN
ejpam-6452	1	52	,	,	PUNCT
ejpam-6452	1	53	statistics	statistic	NOUN
ejpam-6452	1	54	and	and	CCONJ
ejpam-6452	1	55	computer	computer	NOUN
ejpam-6452	1	56	,	,	PUNCT
ejpam-6452	1	57	faculty	faculty	NOUN
ejpam-6452	1	58	of	of	ADP
ejpam-6452	1	59	science	science	NOUN
ejpam-6452	1	60	,	,	PUNCT
ejpam-6452	1	61	ubon	ubon	PROPN
ejpam-6452	1	62	ratchathani	ratchathani	PROPN
ejpam-6452	1	63	university	university	PROPN
ejpam-6452	1	64	,	,	PUNCT
ejpam-6452	1	65	ubon	ubon	PROPN
ejpam-6452	1	66	ratchathani	ratchathani	PROPN
ejpam-6452	1	67	,	,	PUNCT
ejpam-6452	1	68	34190	34190	NUM
ejpam-6452	1	69	,	,	PUNCT
ejpam-6452	1	70	thailand	thailand	PROPN
ejpam-6452	1	71	2	2	NUM
ejpam-6452	1	72	department	department	NOUN
ejpam-6452	1	73	of	of	ADP
ejpam-6452	1	74	mathematics	mathematic	NOUN
ejpam-6452	1	75	,	,	PUNCT
ejpam-6452	1	76	faculty	faculty	NOUN
ejpam-6452	1	77	of	of	ADP
ejpam-6452	1	78	science	science	NOUN
ejpam-6452	1	79	,	,	PUNCT
ejpam-6452	1	80	ubon	ubon	PROPN
ejpam-6452	1	81	ratchathani	ratchathani	PROPN
ejpam-6452	1	82	rajabhat	rajabhat	PROPN
ejpam-6452	1	83	university	university	PROPN
ejpam-6452	1	84	,	,	PUNCT
ejpam-6452	1	85	ubon	ubon	PROPN
ejpam-6452	1	86	ratchathani	ratchathani	PROPN
ejpam-6452	1	87	,	,	PUNCT
ejpam-6452	1	88	34000	34000	NUM
ejpam-6452	1	89	,	,	PUNCT
ejpam-6452	1	90	thailand	thailand	PROPN
ejpam-6452	1	91	abstract	abstract	NOUN
ejpam-6452	1	92	.	.	PUNCT
ejpam-6452	2	1	this	this	DET
ejpam-6452	2	2	paper	paper	NOUN
ejpam-6452	2	3	discusses	discuss	VERB
ejpam-6452	2	4	a	a	DET
ejpam-6452	2	5	one	one	NUM
ejpam-6452	2	6	-	-	PUNCT
ejpam-6452	2	7	parameter	parameter	NOUN
ejpam-6452	2	8	generalization	generalization	NOUN
ejpam-6452	2	9	of	of	ADP
ejpam-6452	2	10	pell	pell	PROPN
ejpam-6452	2	11	numbers	number	NOUN
ejpam-6452	2	12	that	that	PRON
ejpam-6452	2	13	preserves	preserve	VERB
ejpam-6452	2	14	the	the	DET
ejpam-6452	2	15	recurrence	recurrence	NOUN
ejpam-6452	2	16	relation	relation	NOUN
ejpam-6452	2	17	with	with	ADP
ejpam-6452	2	18	arbitrary	arbitrary	ADJ
ejpam-6452	2	19	initial	initial	ADJ
ejpam-6452	2	20	conditions	condition	NOUN
ejpam-6452	2	21	.	.	PUNCT
ejpam-6452	3	1	we	we	PRON
ejpam-6452	3	2	introduce	introduce	VERB
ejpam-6452	3	3	generalized	generalized	ADJ
ejpam-6452	3	4	pell	pell	NOUN
ejpam-6452	3	5	-	-	PUNCT
ejpam-6452	3	6	lucas	lucas	NOUN
ejpam-6452	3	7	-	-	PUNCT
ejpam-6452	3	8	like	like	ADJ
ejpam-6452	3	9	numbers	number	NOUN
ejpam-6452	3	10	,	,	PUNCT
ejpam-6452	3	11	which	which	PRON
ejpam-6452	3	12	are	be	AUX
ejpam-6452	3	13	simple	simple	ADJ
ejpam-6452	3	14	associations	association	NOUN
ejpam-6452	3	15	of	of	ADP
ejpam-6452	3	16	generalized	generalized	ADJ
ejpam-6452	3	17	pell	pell	NOUN
ejpam-6452	3	18	numbers	number	NOUN
ejpam-6452	3	19	.	.	PUNCT
ejpam-6452	4	1	consequently	consequently	ADV
ejpam-6452	4	2	,	,	PUNCT
ejpam-6452	4	3	we	we	PRON
ejpam-6452	4	4	give	give	VERB
ejpam-6452	4	5	some	some	DET
ejpam-6452	4	6	new	new	ADJ
ejpam-6452	4	7	and	and	CCONJ
ejpam-6452	4	8	well	well	ADV
ejpam-6452	4	9	-	-	PUNCT
ejpam-6452	4	10	known	know	VERB
ejpam-6452	4	11	identities	identity	NOUN
ejpam-6452	4	12	.	.	PUNCT
ejpam-6452	5	1	furthermore	furthermore	ADV
ejpam-6452	5	2	,	,	PUNCT
ejpam-6452	5	3	we	we	PRON
ejpam-6452	5	4	propose	propose	VERB
ejpam-6452	5	5	integral	integral	ADJ
ejpam-6452	5	6	representations	representation	NOUN
ejpam-6452	5	7	of	of	ADP
ejpam-6452	5	8	these	these	DET
ejpam-6452	5	9	numbers	number	NOUN
ejpam-6452	5	10	associated	associate	VERB
ejpam-6452	5	11	with	with	ADP
ejpam-6452	5	12	generalized	generalized	ADJ
ejpam-6452	5	13	pell	pell	NOUN
ejpam-6452	5	14	and	and	CCONJ
ejpam-6452	5	15	pell	pell	NOUN
ejpam-6452	5	16	-	-	PUNCT
ejpam-6452	5	17	lucas	lucas	NOUN
ejpam-6452	5	18	-	-	PUNCT
ejpam-6452	5	19	like	like	ADJ
ejpam-6452	5	20	numbers	number	NOUN
ejpam-6452	5	21	.	.	PUNCT
ejpam-6452	6	1	our	our	PRON
ejpam-6452	6	2	results	result	NOUN
ejpam-6452	6	3	not	not	PART
ejpam-6452	6	4	only	only	ADV
ejpam-6452	6	5	generalize	generalize	VERB
ejpam-6452	6	6	the	the	DET
ejpam-6452	6	7	integral	integral	ADJ
ejpam-6452	6	8	representations	representation	NOUN
ejpam-6452	6	9	of	of	ADP
ejpam-6452	6	10	the	the	DET
ejpam-6452	6	11	pell	pell	NOUN
ejpam-6452	6	12	and	and	CCONJ
ejpam-6452	6	13	pell	pell	NOUN
ejpam-6452	6	14	-	-	PUNCT
ejpam-6452	6	15	lucas	lucas	NOUN
ejpam-6452	6	16	numbers	number	NOUN
ejpam-6452	6	17	but	but	CCONJ
ejpam-6452	6	18	also	also	ADV
ejpam-6452	6	19	apply	apply	VERB
ejpam-6452	6	20	to	to	ADP
ejpam-6452	6	21	all	all	DET
ejpam-6452	6	22	the	the	DET
ejpam-6452	6	23	companion	companion	NOUN
ejpam-6452	6	24	numbers	number	NOUN
ejpam-6452	6	25	of	of	ADP
ejpam-6452	6	26	generalized	generalized	ADJ
ejpam-6452	6	27	pell	pell	NOUN
ejpam-6452	6	28	numbers	number	NOUN
ejpam-6452	6	29	.	.	PUNCT
ejpam-6452	7	1	2020	2020	NUM
ejpam-6452	7	2	mathematics	mathematic	NOUN
ejpam-6452	7	3	subject	subject	NOUN
ejpam-6452	7	4	classifications	classification	NOUN
ejpam-6452	7	5	:	:	PUNCT
ejpam-6452	7	6	11b39	11b39	NUM
ejpam-6452	7	7	,	,	PUNCT
ejpam-6452	7	8	11b37	11b37	NUM
ejpam-6452	7	9	key	key	ADJ
ejpam-6452	7	10	words	word	NOUN
ejpam-6452	7	11	and	and	CCONJ
ejpam-6452	7	12	phrases	phrase	NOUN
ejpam-6452	7	13	:	:	PUNCT
ejpam-6452	7	14	generalized	generalize	VERB
ejpam-6452	7	15	pell	pell	NOUN
ejpam-6452	7	16	number	number	NOUN
ejpam-6452	7	17	,	,	PUNCT
ejpam-6452	7	18	generalized	generalize	VERB
ejpam-6452	7	19	pell	pell	NOUN
ejpam-6452	7	20	-	-	PUNCT
ejpam-6452	7	21	lucas	lucas	NOUN
ejpam-6452	7	22	-	-	PUNCT
ejpam-6452	7	23	like	like	ADJ
ejpam-6452	7	24	number	number	NOUN
ejpam-6452	7	25	,	,	PUNCT
ejpam-6452	7	26	integral	integral	ADJ
ejpam-6452	7	27	representation	representation	NOUN
ejpam-6452	7	28	1	1	NUM
ejpam-6452	7	29	.	.	PUNCT
ejpam-6452	8	1	introduction	introduction	NOUN
ejpam-6452	8	2	recall	recall	NOUN
ejpam-6452	8	3	that	that	PRON
ejpam-6452	8	4	pell	pell	PROPN
ejpam-6452	8	5	numbers	number	NOUN
ejpam-6452	8	6	pn	pn	PROPN
ejpam-6452	8	7	are	be	AUX
ejpam-6452	8	8	defined	define	VERB
ejpam-6452	8	9	by	by	ADP
ejpam-6452	8	10	the	the	DET
ejpam-6452	8	11	recurrence	recurrence	NOUN
ejpam-6452	8	12	relations	relation	NOUN
ejpam-6452	8	13	p0	p0	NOUN
ejpam-6452	8	14	=	=	SYM
ejpam-6452	8	15	0	0	NUM
ejpam-6452	8	16	,	,	PUNCT
ejpam-6452	8	17	p1	p1	NOUN
ejpam-6452	8	18	=	=	SYM
ejpam-6452	8	19	1	1	NUM
ejpam-6452	8	20	,	,	PUNCT
ejpam-6452	8	21	and	and	CCONJ
ejpam-6452	8	22	pn	pn	X
ejpam-6452	8	23	=	=	PROPN
ejpam-6452	8	24	2pn−1	2pn−1	PROPN
ejpam-6452	8	25	+	+	CCONJ
ejpam-6452	8	26	pn−2	pn−2	PROPN
ejpam-6452	8	27	,	,	PUNCT
ejpam-6452	8	28	n	n	PRON
ejpam-6452	8	29	≥	≥	NOUN
ejpam-6452	8	30	2	2	NUM
ejpam-6452	8	31	,	,	PUNCT
ejpam-6452	8	32	and	and	CCONJ
ejpam-6452	8	33	its	its	PRON
ejpam-6452	8	34	associated	associated	ADJ
ejpam-6452	8	35	numbers	number	NOUN
ejpam-6452	8	36	or	or	CCONJ
ejpam-6452	8	37	pell	pell	NOUN
ejpam-6452	8	38	-	-	PUNCT
ejpam-6452	8	39	lucas	lucas	NOUN
ejpam-6452	8	40	numbers	number	NOUN
ejpam-6452	8	41	qn	qn	NOUN
ejpam-6452	8	42	are	be	AUX
ejpam-6452	8	43	defined	define	VERB
ejpam-6452	8	44	by	by	ADP
ejpam-6452	8	45	the	the	DET
ejpam-6452	8	46	recurrence	recurrence	NOUN
ejpam-6452	8	47	relations	relation	NOUN
ejpam-6452	8	48	q0	q0	NOUN
ejpam-6452	8	49	=	=	SYM
ejpam-6452	8	50	2	2	NUM
ejpam-6452	8	51	,	,	PUNCT
ejpam-6452	8	52	q1	q1	NOUN
ejpam-6452	8	53	=	=	SYM
ejpam-6452	8	54	1	1	NUM
ejpam-6452	8	55	,	,	PUNCT
ejpam-6452	8	56	and	and	CCONJ
ejpam-6452	8	57	qn	qn	NOUN
ejpam-6452	8	58	=	=	ADJ
ejpam-6452	8	59	2qn−1	2qn−1	PROPN
ejpam-6452	8	60	+	+	CCONJ
ejpam-6452	8	61	qn−2	qn−2	PROPN
ejpam-6452	8	62	,	,	PUNCT
ejpam-6452	8	63	n	n	PRON
ejpam-6452	8	64	≥	≥	NOUN
ejpam-6452	8	65	2	2	NUM
ejpam-6452	8	66	.	.	PUNCT
ejpam-6452	9	1	the	the	DET
ejpam-6452	9	2	binet	binet	NOUN
ejpam-6452	9	3	’s	’s	PART
ejpam-6452	9	4	formulas	formula	NOUN
ejpam-6452	9	5	for	for	ADP
ejpam-6452	9	6	the	the	DET
ejpam-6452	9	7	pell	pell	NOUN
ejpam-6452	9	8	and	and	CCONJ
ejpam-6452	9	9	pell	pell	NOUN
ejpam-6452	9	10	-	-	PUNCT
ejpam-6452	9	11	lucas	lucas	NOUN
ejpam-6452	9	12	numbers	number	NOUN
ejpam-6452	9	13	are	be	AUX
ejpam-6452	9	14	related	relate	VERB
ejpam-6452	9	15	to	to	ADP
ejpam-6452	9	16	the	the	DET
ejpam-6452	9	17	silver	silver	NOUN
ejpam-6452	9	18	ratio	ratio	NOUN
ejpam-6452	9	19	φ	φ	NOUN
ejpam-6452	9	20	=	=	SYM
ejpam-6452	9	21	1	1	NUM
ejpam-6452	9	22	+	+	CCONJ
ejpam-6452	9	23	√	√	NUM
ejpam-6452	9	24	2	2	NUM
ejpam-6452	9	25	.	.	PUNCT
ejpam-6452	10	1	there	there	PRON
ejpam-6452	10	2	are	be	VERB
ejpam-6452	10	3	some	some	DET
ejpam-6452	10	4	generalizations	generalization	NOUN
ejpam-6452	10	5	of	of	ADP
ejpam-6452	10	6	pell	pell	NOUN
ejpam-6452	10	7	and	and	CCONJ
ejpam-6452	10	8	pell	pell	NOUN
ejpam-6452	10	9	-	-	PUNCT
ejpam-6452	10	10	lucas	lucas	NOUN
ejpam-6452	10	11	numbers	number	NOUN
ejpam-6452	10	12	defined	define	VERB
ejpam-6452	10	13	in	in	ADP
ejpam-6452	10	14	different	different	ADJ
ejpam-6452	10	15	ways	way	NOUN
ejpam-6452	10	16	.	.	PUNCT
ejpam-6452	11	1	∗corresponding	∗corresponde	VERB
ejpam-6452	11	2	author	author	NOUN
ejpam-6452	11	3	.	.	PUNCT
ejpam-6452	12	1	doi	doi	NOUN
ejpam-6452	12	2	:	:	PUNCT
ejpam-6452	12	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6452	https://doi.org/10.29020/nybg.ejpam.v18i3.6452	NOUN
ejpam-6452	12	4	email	email	NOUN
ejpam-6452	12	5	addresses	address	NOUN
ejpam-6452	12	6	:	:	PUNCT
ejpam-6452	12	7	weerayuth.ni@ubu.ac.th	weerayuth.ni@ubu.ac.th	NUM
ejpam-6452	12	8	(	(	PUNCT
ejpam-6452	12	9	w.	w.	NOUN
ejpam-6452	12	10	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	12	11	)	)	PUNCT
ejpam-6452	12	12	,	,	PUNCT
ejpam-6452	12	13	achariya.n@ubru.ac.th	achariya.n@ubru.ac.th	PROPN
ejpam-6452	12	14	(	(	PUNCT
ejpam-6452	12	15	a.	a.	NOUN
ejpam-6452	12	16	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	12	17	)	)	PUNCT
ejpam-6452	12	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6452	12	19	1	1	NUM
ejpam-6452	12	20	copyright	copyright	NOUN
ejpam-6452	12	21	:	:	PUNCT
ejpam-6452	13	1	©	©	PROPN
ejpam-6452	13	2	2025	2025	NUM
ejpam-6452	13	3	the	the	DET
ejpam-6452	13	4	author(s	author(s	NOUN
ejpam-6452	13	5	)	)	PUNCT
ejpam-6452	13	6	.	.	PUNCT
ejpam-6452	14	1	(	(	PUNCT
ejpam-6452	14	2	cc	cc	NOUN
ejpam-6452	14	3	by	by	ADP
ejpam-6452	14	4	-	-	PUNCT
ejpam-6452	14	5	nc	nc	PROPN
ejpam-6452	14	6	4.0	4.0	NUM
ejpam-6452	14	7	)	)	PUNCT
ejpam-6452	14	8	w.	w.	NOUN
ejpam-6452	14	9	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	14	10	,	,	PUNCT
ejpam-6452	14	11	a.	a.	NOUN
ejpam-6452	14	12	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	14	13	/	/	SYM
ejpam-6452	14	14	eur	eur	PROPN
ejpam-6452	14	15	.	.	PUNCT
ejpam-6452	15	1	j.	j.	PROPN
ejpam-6452	15	2	pure	pure	PROPN
ejpam-6452	15	3	appl	appl	PROPN
ejpam-6452	15	4	.	.	PROPN
ejpam-6452	15	5	math	math	PROPN
ejpam-6452	15	6	,	,	PUNCT
ejpam-6452	15	7	18	18	NUM
ejpam-6452	15	8	(	(	PUNCT
ejpam-6452	15	9	3	3	NUM
ejpam-6452	15	10	)	)	PUNCT
ejpam-6452	15	11	(	(	PUNCT
ejpam-6452	15	12	2025	2025	NUM
ejpam-6452	15	13	)	)	PUNCT
ejpam-6452	15	14	,	,	PUNCT
ejpam-6452	15	15	6452	6452	NUM
ejpam-6452	15	16	2	2	NUM
ejpam-6452	15	17	of	of	ADP
ejpam-6452	15	18	18	18	NUM
ejpam-6452	15	19	in	in	ADP
ejpam-6452	15	20	2007	2007	NUM
ejpam-6452	15	21	,	,	PUNCT
ejpam-6452	15	22	falcón	falcón	ADJ
ejpam-6452	15	23	and	and	CCONJ
ejpam-6452	15	24	plaza	plaza	VERB
ejpam-6452	15	25	[	[	X
ejpam-6452	15	26	1	1	X
ejpam-6452	15	27	]	]	PUNCT
ejpam-6452	15	28	introduced	introduce	VERB
ejpam-6452	15	29	the	the	DET
ejpam-6452	15	30	first	first	ADJ
ejpam-6452	15	31	kind	kind	NOUN
ejpam-6452	15	32	of	of	ADP
ejpam-6452	15	33	one	one	NUM
ejpam-6452	15	34	-	-	PUNCT
ejpam-6452	15	35	parameter	parameter	NOUN
ejpam-6452	15	36	generalization	generalization	NOUN
ejpam-6452	15	37	of	of	ADP
ejpam-6452	15	38	fibonacci	fibonacci	PROPN
ejpam-6452	15	39	and	and	CCONJ
ejpam-6452	15	40	pell	pell	VERB
ejpam-6452	15	41	numbers	number	NOUN
ejpam-6452	15	42	as	as	SCONJ
ejpam-6452	15	43	follows	follow	VERB
ejpam-6452	15	44	:	:	PUNCT
ejpam-6452	15	45	the	the	DET
ejpam-6452	15	46	k	k	PROPN
ejpam-6452	15	47	-	-	PUNCT
ejpam-6452	15	48	fibonacci	fibonacci	NOUN
ejpam-6452	15	49	numbers	number	NOUN
ejpam-6452	15	50	fk	fk	INTJ
ejpam-6452	15	51	,	,	PUNCT
ejpam-6452	15	52	n	n	PRON
ejpam-6452	15	53	are	be	AUX
ejpam-6452	15	54	defined	define	VERB
ejpam-6452	15	55	by	by	ADP
ejpam-6452	15	56	the	the	DET
ejpam-6452	15	57	recurrence	recurrence	NOUN
ejpam-6452	15	58	relations	relation	NOUN
ejpam-6452	15	59	fk,0	fk,0	PROPN
ejpam-6452	15	60	=	=	SYM
ejpam-6452	15	61	0	0	NUM
ejpam-6452	15	62	,	,	PUNCT
ejpam-6452	15	63	fk,1	fk,1	NOUN
ejpam-6452	15	64	=	=	SYM
ejpam-6452	15	65	1	1	NUM
ejpam-6452	15	66	,	,	PUNCT
ejpam-6452	15	67	and	and	CCONJ
ejpam-6452	15	68	fk	fk	INTJ
ejpam-6452	15	69	,	,	PUNCT
ejpam-6452	15	70	n	n	PROPN
ejpam-6452	15	71	=	=	SYM
ejpam-6452	15	72	kfk	kfk	NOUN
ejpam-6452	15	73	,	,	PUNCT
ejpam-6452	15	74	n−1	n−1	PROPN
ejpam-6452	15	75	+	+	NUM
ejpam-6452	15	76	fk	fk	PROPN
ejpam-6452	15	77	,	,	PUNCT
ejpam-6452	15	78	n−2	n−2	PROPN
ejpam-6452	15	79	,	,	PUNCT
ejpam-6452	15	80	n	n	PRON
ejpam-6452	15	81	≥	≥	NOUN
ejpam-6452	15	82	2	2	NUM
ejpam-6452	15	83	,	,	PUNCT
ejpam-6452	15	84	where	where	SCONJ
ejpam-6452	15	85	k	k	PROPN
ejpam-6452	15	86	and	and	CCONJ
ejpam-6452	15	87	n	n	PROPN
ejpam-6452	15	88	are	be	AUX
ejpam-6452	15	89	non	non	ADJ
ejpam-6452	15	90	-	-	ADJ
ejpam-6452	15	91	negative	negative	ADJ
ejpam-6452	15	92	integers	integer	NOUN
ejpam-6452	15	93	with	with	ADP
ejpam-6452	15	94	k	k	PROPN
ejpam-6452	15	95	̸=	̸=	PROPN
ejpam-6452	15	96	0	0	NUM
ejpam-6452	15	97	.	.	PUNCT
ejpam-6452	16	1	the	the	DET
ejpam-6452	16	2	associated	associated	ADJ
ejpam-6452	16	3	numbers	number	NOUN
ejpam-6452	16	4	of	of	ADP
ejpam-6452	16	5	kfibonacci	kfibonacci	ADJ
ejpam-6452	16	6	numbers	number	NOUN
ejpam-6452	16	7	introduced	introduce	VERB
ejpam-6452	16	8	in	in	ADP
ejpam-6452	16	9	2011	2011	NUM
ejpam-6452	16	10	by	by	ADP
ejpam-6452	16	11	falcón	falcón	NOUN
ejpam-6452	16	12	[	[	X
ejpam-6452	16	13	2	2	NUM
ejpam-6452	16	14	]	]	PUNCT
ejpam-6452	16	15	as	as	SCONJ
ejpam-6452	16	16	follows	follow	VERB
ejpam-6452	16	17	:	:	PUNCT
ejpam-6452	16	18	the	the	DET
ejpam-6452	16	19	k	k	PROPN
ejpam-6452	16	20	-	-	PUNCT
ejpam-6452	16	21	lucas	lucas	PROPN
ejpam-6452	16	22	numbers	number	NOUN
ejpam-6452	16	23	lk	lk	PROPN
ejpam-6452	16	24	,	,	PUNCT
ejpam-6452	16	25	n	n	PRON
ejpam-6452	16	26	are	be	AUX
ejpam-6452	16	27	defined	define	VERB
ejpam-6452	16	28	by	by	ADP
ejpam-6452	16	29	the	the	DET
ejpam-6452	16	30	recurrence	recurrence	NOUN
ejpam-6452	16	31	relations	relation	NOUN
ejpam-6452	16	32	lk,0	lk,0	NOUN
ejpam-6452	16	33	=	=	PUNCT
ejpam-6452	16	34	2	2	NUM
ejpam-6452	16	35	,	,	PUNCT
ejpam-6452	16	36	lk,1	lk,1	PROPN
ejpam-6452	16	37	=	=	SYM
ejpam-6452	16	38	k	k	PROPN
ejpam-6452	16	39	,	,	PUNCT
ejpam-6452	16	40	and	and	CCONJ
ejpam-6452	16	41	lk	lk	PROPN
ejpam-6452	16	42	,	,	PUNCT
ejpam-6452	16	43	n	n	PROPN
ejpam-6452	16	44	=	=	SYM
ejpam-6452	16	45	klk	klk	PROPN
ejpam-6452	16	46	,	,	PUNCT
ejpam-6452	16	47	n−1	n−1	PROPN
ejpam-6452	16	48	+	+	CCONJ
ejpam-6452	16	49	lk	lk	PROPN
ejpam-6452	16	50	,	,	PUNCT
ejpam-6452	16	51	n−2	n−2	PROPN
ejpam-6452	16	52	,	,	PUNCT
ejpam-6452	16	53	n	n	PRON
ejpam-6452	16	54	≥	≥	NOUN
ejpam-6452	16	55	2	2	NUM
ejpam-6452	16	56	.	.	PUNCT
ejpam-6452	16	57	in	in	ADP
ejpam-6452	16	58	2013	2013	NUM
ejpam-6452	16	59	,	,	PUNCT
ejpam-6452	16	60	catarino	catarino	NOUN
ejpam-6452	17	1	[	[	X
ejpam-6452	17	2	3	3	NUM
ejpam-6452	17	3	]	]	PUNCT
ejpam-6452	17	4	introduced	introduce	VERB
ejpam-6452	17	5	the	the	DET
ejpam-6452	17	6	second	second	ADJ
ejpam-6452	17	7	kind	kind	NOUN
ejpam-6452	17	8	of	of	ADP
ejpam-6452	17	9	one	one	NUM
ejpam-6452	17	10	-	-	PUNCT
ejpam-6452	17	11	parameter	parameter	NOUN
ejpam-6452	17	12	generalization	generalization	NOUN
ejpam-6452	17	13	of	of	ADP
ejpam-6452	17	14	pell	pell	PROPN
ejpam-6452	17	15	numbers	number	NOUN
ejpam-6452	17	16	as	as	SCONJ
ejpam-6452	17	17	follows	follow	VERB
ejpam-6452	17	18	:	:	PUNCT
ejpam-6452	17	19	the	the	DET
ejpam-6452	17	20	k	k	ADJ
ejpam-6452	17	21	-	-	PUNCT
ejpam-6452	17	22	pell	pell	ADJ
ejpam-6452	17	23	numbers	number	NOUN
ejpam-6452	17	24	pk	pk	NOUN
ejpam-6452	17	25	,	,	PUNCT
ejpam-6452	17	26	n	n	PRON
ejpam-6452	17	27	are	be	AUX
ejpam-6452	17	28	defined	define	VERB
ejpam-6452	17	29	by	by	ADP
ejpam-6452	17	30	the	the	DET
ejpam-6452	17	31	recurrence	recurrence	NOUN
ejpam-6452	17	32	relations	relation	NOUN
ejpam-6452	17	33	pk,0	pk,0	PROPN
ejpam-6452	17	34	=	=	SYM
ejpam-6452	17	35	0	0	NUM
ejpam-6452	17	36	,	,	PUNCT
ejpam-6452	17	37	pk,1	pk,1	NOUN
ejpam-6452	17	38	=	=	NOUN
ejpam-6452	17	39	1	1	NUM
ejpam-6452	17	40	,	,	PUNCT
ejpam-6452	17	41	and	and	CCONJ
ejpam-6452	17	42	pk	pk	NOUN
ejpam-6452	17	43	,	,	PUNCT
ejpam-6452	17	44	n	n	NOUN
ejpam-6452	17	45	=	=	SYM
ejpam-6452	17	46	2pk	2pk	NOUN
ejpam-6452	17	47	,	,	PUNCT
ejpam-6452	17	48	n−1	n−1	PROPN
ejpam-6452	17	49	+	+	CCONJ
ejpam-6452	17	50	kpk	kpk	PROPN
ejpam-6452	17	51	,	,	PUNCT
ejpam-6452	17	52	n−2	n−2	PROPN
ejpam-6452	17	53	,	,	PUNCT
ejpam-6452	17	54	n	n	PRON
ejpam-6452	17	55	≥	≥	NOUN
ejpam-6452	17	56	2	2	NUM
ejpam-6452	17	57	,	,	PUNCT
ejpam-6452	17	58	subsequently	subsequently	ADV
ejpam-6452	17	59	,	,	PUNCT
ejpam-6452	17	60	catarino	catarino	NOUN
ejpam-6452	17	61	and	and	CCONJ
ejpam-6452	17	62	vasco	vasco	PROPN
ejpam-6452	17	63	[	[	X
ejpam-6452	17	64	4	4	NUM
ejpam-6452	17	65	]	]	PUNCT
ejpam-6452	17	66	introduced	introduce	VERB
ejpam-6452	17	67	the	the	DET
ejpam-6452	17	68	association	association	NOUN
ejpam-6452	17	69	of	of	ADP
ejpam-6452	17	70	pell	pell	PROPN
ejpam-6452	17	71	numbers	number	NOUN
ejpam-6452	17	72	as	as	SCONJ
ejpam-6452	17	73	follows	follow	VERB
ejpam-6452	17	74	:	:	PUNCT
ejpam-6452	17	75	the	the	DET
ejpam-6452	17	76	k	k	NOUN
ejpam-6452	17	77	-	-	PUNCT
ejpam-6452	17	78	pell	pell	ADJ
ejpam-6452	17	79	-	-	PUNCT
ejpam-6452	17	80	lucas	lucas	NOUN
ejpam-6452	17	81	numbers	number	NOUN
ejpam-6452	17	82	qk	qk	PROPN
ejpam-6452	17	83	,	,	PUNCT
ejpam-6452	17	84	n	n	PRON
ejpam-6452	17	85	are	be	AUX
ejpam-6452	17	86	defined	define	VERB
ejpam-6452	17	87	by	by	ADP
ejpam-6452	17	88	the	the	DET
ejpam-6452	17	89	recurrence	recurrence	NOUN
ejpam-6452	17	90	relations	relation	NOUN
ejpam-6452	17	91	qk,0	qk,0	PROPN
ejpam-6452	17	92	=	=	SYM
ejpam-6452	17	93	2	2	NUM
ejpam-6452	17	94	,	,	PUNCT
ejpam-6452	17	95	qk,1	qk,1	NOUN
ejpam-6452	17	96	=	=	SYM
ejpam-6452	17	97	2	2	NUM
ejpam-6452	17	98	,	,	PUNCT
ejpam-6452	17	99	and	and	CCONJ
ejpam-6452	17	100	qk	qk	NOUN
ejpam-6452	17	101	,	,	PUNCT
ejpam-6452	17	102	n	n	NOUN
ejpam-6452	17	103	=	=	SYM
ejpam-6452	17	104	2qk	2qk	NOUN
ejpam-6452	17	105	,	,	PUNCT
ejpam-6452	17	106	n−1	n−1	PROPN
ejpam-6452	17	107	+	+	CCONJ
ejpam-6452	17	108	kqk	kqk	PROPN
ejpam-6452	17	109	,	,	PUNCT
ejpam-6452	17	110	n−2	n−2	PROPN
ejpam-6452	17	111	,	,	PUNCT
ejpam-6452	17	112	n	n	PRON
ejpam-6452	17	113	≥	≥	NOUN
ejpam-6452	17	114	2	2	NUM
ejpam-6452	17	115	,	,	PUNCT
ejpam-6452	17	116	in	in	ADP
ejpam-6452	17	117	2019	2019	NUM
ejpam-6452	17	118	,	,	PUNCT
ejpam-6452	17	119	trojnar	trojnar	NOUN
ejpam-6452	17	120	-	-	PUNCT
ejpam-6452	17	121	spenlina	spenlina	NOUN
ejpam-6452	17	122	and	and	CCONJ
ejpam-6452	17	123	w	w	PROPN
ejpam-6452	17	124	loch	loch	PROPN
ejpam-6452	18	1	[	[	X
ejpam-6452	18	2	5	5	NUM
ejpam-6452	18	3	]	]	PUNCT
ejpam-6452	18	4	introduced	introduce	VERB
ejpam-6452	18	5	the	the	DET
ejpam-6452	18	6	third	third	ADJ
ejpam-6452	18	7	kind	kind	NOUN
ejpam-6452	18	8	of	of	ADP
ejpam-6452	18	9	one	one	NUM
ejpam-6452	18	10	-	-	PUNCT
ejpam-6452	18	11	parameter	parameter	NOUN
ejpam-6452	18	12	generalization	generalization	NOUN
ejpam-6452	18	13	of	of	ADP
ejpam-6452	18	14	pell	pell	PROPN
ejpam-6452	18	15	numbers	number	NOUN
ejpam-6452	18	16	as	as	SCONJ
ejpam-6452	18	17	follows	follow	VERB
ejpam-6452	18	18	:	:	PUNCT
ejpam-6452	18	19	the	the	DET
ejpam-6452	18	20	generalized	generalized	ADJ
ejpam-6452	18	21	pell	pell	NOUN
ejpam-6452	18	22	numbers	number	NOUN
ejpam-6452	18	23	pk	pk	NOUN
ejpam-6452	18	24	,	,	PUNCT
ejpam-6452	18	25	n	n	PRON
ejpam-6452	18	26	are	be	AUX
ejpam-6452	18	27	defined	define	VERB
ejpam-6452	18	28	by	by	ADP
ejpam-6452	18	29	the	the	DET
ejpam-6452	18	30	recurrence	recurrence	NOUN
ejpam-6452	18	31	relations	relation	NOUN
ejpam-6452	18	32	pk,0	pk,0	PROPN
ejpam-6452	18	33	=	=	PUNCT
ejpam-6452	18	34	0,pk,1	0,pk,1	PUNCT
ejpam-6452	19	1	=	=	SYM
ejpam-6452	19	2	1	1	NUM
ejpam-6452	19	3	,	,	PUNCT
ejpam-6452	19	4	and	and	CCONJ
ejpam-6452	19	5	pk	pk	NOUN
ejpam-6452	19	6	,	,	PUNCT
ejpam-6452	19	7	n	n	PROPN
ejpam-6452	19	8	=	=	SYM
ejpam-6452	19	9	kpk	kpk	PROPN
ejpam-6452	19	10	,	,	PUNCT
ejpam-6452	19	11	n−1	n−1	PROPN
ejpam-6452	19	12	+	+	CCONJ
ejpam-6452	19	13	(	(	PUNCT
ejpam-6452	19	14	k	k	PROPN
ejpam-6452	20	1	−	−	PROPN
ejpam-6452	20	2	1)pk	1)pk	PROPN
ejpam-6452	20	3	,	,	PUNCT
ejpam-6452	20	4	n−2	n−2	PROPN
ejpam-6452	20	5	,	,	PUNCT
ejpam-6452	20	6	n	n	PRON
ejpam-6452	20	7	≥	≥	NOUN
ejpam-6452	20	8	2	2	NUM
ejpam-6452	20	9	,	,	PUNCT
ejpam-6452	20	10	(	(	PUNCT
ejpam-6452	20	11	1	1	X
ejpam-6452	20	12	)	)	PUNCT
ejpam-6452	20	13	subsequently	subsequently	ADV
ejpam-6452	20	14	,	,	PUNCT
ejpam-6452	20	15	the	the	DET
ejpam-6452	20	16	association	association	NOUN
ejpam-6452	20	17	of	of	ADP
ejpam-6452	20	18	generalized	generalized	ADJ
ejpam-6452	20	19	pell	pell	NOUN
ejpam-6452	20	20	numbers	number	NOUN
ejpam-6452	20	21	qk	qk	ADP
ejpam-6452	20	22	,	,	PUNCT
ejpam-6452	20	23	n	n	CCONJ
ejpam-6452	20	24	,	,	PUNCT
ejpam-6452	20	25	so	so	ADV
ejpam-6452	20	26	-	-	PUNCT
ejpam-6452	20	27	called	call	VERB
ejpam-6452	20	28	generalized	generalized	ADJ
ejpam-6452	20	29	pelllucas	pellluca	NOUN
ejpam-6452	20	30	numbers	number	NOUN
ejpam-6452	20	31	,	,	PUNCT
ejpam-6452	20	32	are	be	AUX
ejpam-6452	20	33	defined	define	VERB
ejpam-6452	20	34	by	by	ADP
ejpam-6452	20	35	the	the	DET
ejpam-6452	20	36	recurrence	recurrence	NOUN
ejpam-6452	20	37	relations	relation	NOUN
ejpam-6452	20	38	qk,0	qk,0	PROPN
ejpam-6452	20	39	=	=	SYM
ejpam-6452	21	1	2,qk,1	2,qk,1	NUM
ejpam-6452	21	2	=	=	SYM
ejpam-6452	21	3	2	2	NUM
ejpam-6452	21	4	,	,	PUNCT
ejpam-6452	21	5	and	and	CCONJ
ejpam-6452	21	6	qk	qk	NOUN
ejpam-6452	21	7	,	,	PUNCT
ejpam-6452	21	8	n	n	NOUN
ejpam-6452	21	9	=	=	SYM
ejpam-6452	21	10	kqk	kqk	X
ejpam-6452	21	11	,	,	PUNCT
ejpam-6452	21	12	n−1	n−1	PROPN
ejpam-6452	21	13	+	+	CCONJ
ejpam-6452	21	14	(	(	PUNCT
ejpam-6452	21	15	k	k	PROPN
ejpam-6452	21	16	−	−	PROPN
ejpam-6452	22	1	1)qk	1)qk	PROPN
ejpam-6452	22	2	,	,	PUNCT
ejpam-6452	22	3	n−2	n−2	PROPN
ejpam-6452	22	4	,	,	PUNCT
ejpam-6452	22	5	n	n	PRON
ejpam-6452	22	6	≥	≥	NOUN
ejpam-6452	22	7	2	2	NUM
ejpam-6452	22	8	,	,	PUNCT
ejpam-6452	22	9	in	in	ADP
ejpam-6452	22	10	the	the	DET
ejpam-6452	22	11	paper	paper	NOUN
ejpam-6452	22	12	[	[	X
ejpam-6452	22	13	6	6	NUM
ejpam-6452	22	14	]	]	PUNCT
ejpam-6452	22	15	,	,	PUNCT
ejpam-6452	22	16	the	the	DET
ejpam-6452	22	17	other	other	ADJ
ejpam-6452	22	18	associated	associated	ADJ
ejpam-6452	22	19	numbers	number	NOUN
ejpam-6452	22	20	of	of	ADP
ejpam-6452	22	21	generalized	generalized	ADJ
ejpam-6452	22	22	pell	pell	NOUN
ejpam-6452	22	23	numbers	number	NOUN
ejpam-6452	22	24	,	,	PUNCT
ejpam-6452	22	25	which	which	PRON
ejpam-6452	22	26	are	be	AUX
ejpam-6452	22	27	the	the	DET
ejpam-6452	22	28	so	so	ADV
ejpam-6452	22	29	-	-	PUNCT
ejpam-6452	22	30	called	call	VERB
ejpam-6452	22	31	generalized	generalized	ADJ
ejpam-6452	22	32	modified	modify	VERB
ejpam-6452	22	33	pell	pell	NOUN
ejpam-6452	22	34	numbers	number	NOUN
ejpam-6452	22	35	qk	qk	ADP
ejpam-6452	22	36	,	,	PUNCT
ejpam-6452	22	37	n	n	CCONJ
ejpam-6452	22	38	,	,	PUNCT
ejpam-6452	22	39	are	be	AUX
ejpam-6452	22	40	defined	define	VERB
ejpam-6452	22	41	by	by	ADP
ejpam-6452	22	42	the	the	DET
ejpam-6452	22	43	recurrence	recurrence	NOUN
ejpam-6452	22	44	relations	relation	NOUN
ejpam-6452	22	45	qk,0	qk,0	PROPN
ejpam-6452	22	46	=	=	SYM
ejpam-6452	22	47	1	1	NUM
ejpam-6452	22	48	,	,	PUNCT
ejpam-6452	22	49	qk,1	qk,1	NOUN
ejpam-6452	22	50	=	=	SYM
ejpam-6452	22	51	1	1	NUM
ejpam-6452	22	52	,	,	PUNCT
ejpam-6452	22	53	and	and	CCONJ
ejpam-6452	22	54	qk	qk	INTJ
ejpam-6452	22	55	,	,	PUNCT
ejpam-6452	22	56	n	n	NOUN
ejpam-6452	22	57	=	=	SYM
ejpam-6452	22	58	kqk	kqk	X
ejpam-6452	22	59	,	,	PUNCT
ejpam-6452	22	60	n−1	n−1	PROPN
ejpam-6452	22	61	+	+	CCONJ
ejpam-6452	22	62	(	(	PUNCT
ejpam-6452	22	63	k	k	PROPN
ejpam-6452	22	64	−	−	PROPN
ejpam-6452	22	65	1)qk	1)qk	PROPN
ejpam-6452	22	66	,	,	PUNCT
ejpam-6452	22	67	n−2	n−2	PROPN
ejpam-6452	22	68	,	,	PUNCT
ejpam-6452	22	69	n	n	PRON
ejpam-6452	22	70	≥	≥	NOUN
ejpam-6452	22	71	2	2	NUM
ejpam-6452	22	72	,	,	PUNCT
ejpam-6452	22	73	it	it	PRON
ejpam-6452	22	74	is	be	AUX
ejpam-6452	22	75	known	know	VERB
ejpam-6452	22	76	that	that	SCONJ
ejpam-6452	22	77	qk	qk	NOUN
ejpam-6452	22	78	,	,	PUNCT
ejpam-6452	22	79	n	n	NOUN
ejpam-6452	22	80	=	=	SYM
ejpam-6452	22	81	2qk	2qk	NOUN
ejpam-6452	22	82	,	,	PUNCT
ejpam-6452	22	83	n.	n.	NOUN
ejpam-6452	22	84	recall	recall	VERB
ejpam-6452	22	85	that	that	SCONJ
ejpam-6452	22	86	a	a	DET
ejpam-6452	22	87	pair	pair	NOUN
ejpam-6452	22	88	of	of	ADP
ejpam-6452	22	89	lucas	lucas	PROPN
ejpam-6452	22	90	sequences	sequence	NOUN
ejpam-6452	22	91	(	(	PUNCT
ejpam-6452	22	92	{	{	PUNCT
ejpam-6452	22	93	un	un	PROPN
ejpam-6452	22	94	}	}	PUNCT
ejpam-6452	22	95	,	,	PUNCT
ejpam-6452	22	96	{	{	PUNCT
ejpam-6452	22	97	vn	vn	NOUN
ejpam-6452	22	98	}	}	PUNCT
ejpam-6452	22	99	)	)	PUNCT
ejpam-6452	23	1	[	[	X
ejpam-6452	23	2	7	7	X
ejpam-6452	23	3	]	]	PUNCT
ejpam-6452	23	4	is	be	AUX
ejpam-6452	23	5	defined	define	VERB
ejpam-6452	23	6	by	by	ADP
ejpam-6452	23	7	the	the	DET
ejpam-6452	23	8	formulas	formula	NOUN
ejpam-6452	23	9	u0	u0	NOUN
ejpam-6452	23	10	=	=	SYM
ejpam-6452	23	11	0	0	NUM
ejpam-6452	23	12	,	,	PUNCT
ejpam-6452	23	13	u1	u1	NOUN
ejpam-6452	23	14	=	=	SYM
ejpam-6452	23	15	1	1	NUM
ejpam-6452	23	16	,	,	PUNCT
ejpam-6452	23	17	and	and	CCONJ
ejpam-6452	23	18	un	un	PROPN
ejpam-6452	23	19	=	=	PROPN
ejpam-6452	23	20	αun−1	αun−1	PROPN
ejpam-6452	23	21	−	−	PROPN
ejpam-6452	23	22	βun−2	βun−2	PROPN
ejpam-6452	23	23	,	,	PUNCT
ejpam-6452	23	24	n	n	PRON
ejpam-6452	23	25	≥	≥	NOUN
ejpam-6452	23	26	2	2	NUM
ejpam-6452	23	27	,	,	PUNCT
ejpam-6452	23	28	v0	v0	NOUN
ejpam-6452	23	29	=	=	SYM
ejpam-6452	23	30	2	2	NUM
ejpam-6452	23	31	,	,	PUNCT
ejpam-6452	23	32	v1	v1	NOUN
ejpam-6452	23	33	=	=	SYM
ejpam-6452	23	34	α	α	NOUN
ejpam-6452	23	35	,	,	PUNCT
ejpam-6452	23	36	and	and	CCONJ
ejpam-6452	23	37	vn	vn	X
ejpam-6452	23	38	=	=	SYM
ejpam-6452	23	39	αvn−1	αvn−1	PROPN
ejpam-6452	23	40	−	−	PROPN
ejpam-6452	23	41	βvn−2	βvn−2	PROPN
ejpam-6452	23	42	,	,	PUNCT
ejpam-6452	23	43	n	n	PRON
ejpam-6452	23	44	≥	≥	NOUN
ejpam-6452	23	45	2	2	NUM
ejpam-6452	23	46	.	.	PUNCT
ejpam-6452	23	47	where	where	SCONJ
ejpam-6452	23	48	α	α	NOUN
ejpam-6452	23	49	and	and	CCONJ
ejpam-6452	23	50	β	β	X
ejpam-6452	23	51	are	be	AUX
ejpam-6452	23	52	integers	integer	NOUN
ejpam-6452	23	53	such	such	ADJ
ejpam-6452	23	54	that	that	SCONJ
ejpam-6452	23	55	the	the	DET
ejpam-6452	23	56	discriminant	discriminant	NOUN
ejpam-6452	23	57	∆	∆	X
ejpam-6452	23	58	=	=	SYM
ejpam-6452	23	59	α2	α2	PROPN
ejpam-6452	23	60	+4β	+4β	PROPN
ejpam-6452	23	61	̸=	̸=	PROPN
ejpam-6452	23	62	0	0	NUM
ejpam-6452	23	63	.	.	PUNCT
ejpam-6452	24	1	in	in	ADP
ejpam-6452	24	2	this	this	DET
ejpam-6452	24	3	case	case	NOUN
ejpam-6452	24	4	,	,	PUNCT
ejpam-6452	24	5	{	{	PUNCT
ejpam-6452	24	6	un	un	PROPN
ejpam-6452	24	7	}	}	PUNCT
ejpam-6452	24	8	and	and	CCONJ
ejpam-6452	24	9	{	{	PUNCT
ejpam-6452	24	10	vn	vn	NOUN
ejpam-6452	24	11	}	}	PUNCT
ejpam-6452	24	12	are	be	AUX
ejpam-6452	24	13	called	call	VERB
ejpam-6452	24	14	the	the	DET
ejpam-6452	24	15	lucas	lucas	PROPN
ejpam-6452	24	16	sequences	sequence	NOUN
ejpam-6452	24	17	of	of	ADP
ejpam-6452	24	18	the	the	DET
ejpam-6452	24	19	first	first	ADJ
ejpam-6452	24	20	and	and	CCONJ
ejpam-6452	24	21	second	second	ADJ
ejpam-6452	24	22	kinds	kind	NOUN
ejpam-6452	24	23	,	,	PUNCT
ejpam-6452	24	24	respectively	respectively	ADV
ejpam-6452	24	25	.	.	PUNCT
ejpam-6452	25	1	note	note	VERB
ejpam-6452	25	2	that	that	SCONJ
ejpam-6452	25	3	(	(	PUNCT
ejpam-6452	25	4	{	{	PUNCT
ejpam-6452	25	5	fk	fk	INTJ
ejpam-6452	25	6	,	,	PUNCT
ejpam-6452	25	7	n	n	CCONJ
ejpam-6452	25	8	}	}	PUNCT
ejpam-6452	25	9	,	,	PUNCT
ejpam-6452	25	10	{	{	PUNCT
ejpam-6452	25	11	lk	lk	NOUN
ejpam-6452	25	12	,	,	PUNCT
ejpam-6452	25	13	n	n	CCONJ
ejpam-6452	25	14	}	}	PUNCT
ejpam-6452	25	15	)	)	PUNCT
ejpam-6452	25	16	and	and	CCONJ
ejpam-6452	25	17	(	(	PUNCT
ejpam-6452	25	18	{	{	PUNCT
ejpam-6452	25	19	pk	pk	NOUN
ejpam-6452	25	20	,	,	PUNCT
ejpam-6452	25	21	n	n	CCONJ
ejpam-6452	25	22	}	}	PUNCT
ejpam-6452	25	23	,	,	PUNCT
ejpam-6452	25	24	{	{	PUNCT
ejpam-6452	25	25	qk	qk	NOUN
ejpam-6452	25	26	,	,	PUNCT
ejpam-6452	25	27	n	n	CCONJ
ejpam-6452	25	28	}	}	PUNCT
ejpam-6452	25	29	)	)	PUNCT
ejpam-6452	25	30	are	be	AUX
ejpam-6452	25	31	included	include	VERB
ejpam-6452	25	32	in	in	ADP
ejpam-6452	25	33	the	the	DET
ejpam-6452	25	34	more	more	ADV
ejpam-6452	25	35	general	general	ADJ
ejpam-6452	25	36	definition	definition	NOUN
ejpam-6452	25	37	by	by	ADP
ejpam-6452	25	38	w.	w.	PROPN
ejpam-6452	25	39	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	25	40	,	,	PUNCT
ejpam-6452	25	41	a.	a.	NOUN
ejpam-6452	25	42	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	25	43	/	/	SYM
ejpam-6452	25	44	eur	eur	PROPN
ejpam-6452	25	45	.	.	PUNCT
ejpam-6452	26	1	j.	j.	PROPN
ejpam-6452	26	2	pure	pure	PROPN
ejpam-6452	26	3	appl	appl	PROPN
ejpam-6452	26	4	.	.	PROPN
ejpam-6452	26	5	math	math	PROPN
ejpam-6452	26	6	,	,	PUNCT
ejpam-6452	26	7	18	18	NUM
ejpam-6452	26	8	(	(	PUNCT
ejpam-6452	26	9	3	3	NUM
ejpam-6452	26	10	)	)	PUNCT
ejpam-6452	26	11	(	(	PUNCT
ejpam-6452	26	12	2025	2025	NUM
ejpam-6452	26	13	)	)	PUNCT
ejpam-6452	26	14	,	,	PUNCT
ejpam-6452	26	15	6452	6452	NUM
ejpam-6452	26	16	3	3	NUM
ejpam-6452	26	17	of	of	ADP
ejpam-6452	26	18	18	18	NUM
ejpam-6452	26	19	assuming	assume	VERB
ejpam-6452	26	20	(	(	PUNCT
ejpam-6452	26	21	α	α	X
ejpam-6452	26	22	,	,	PUNCT
ejpam-6452	26	23	β	β	NOUN
ejpam-6452	26	24	)	)	PUNCT
ejpam-6452	26	25	=	=	SYM
ejpam-6452	26	26	(	(	PUNCT
ejpam-6452	26	27	k,−1	k,−1	PROPN
ejpam-6452	26	28	)	)	PUNCT
ejpam-6452	26	29	and	and	CCONJ
ejpam-6452	26	30	(	(	PUNCT
ejpam-6452	26	31	α	α	NOUN
ejpam-6452	26	32	,	,	PUNCT
ejpam-6452	26	33	β	β	NOUN
ejpam-6452	26	34	)	)	PUNCT
ejpam-6452	26	35	=	=	SYM
ejpam-6452	26	36	(	(	PUNCT
ejpam-6452	26	37	2,−k	2,−k	NUM
ejpam-6452	26	38	)	)	PUNCT
ejpam-6452	26	39	,	,	PUNCT
ejpam-6452	26	40	respectively	respectively	ADV
ejpam-6452	26	41	.	.	PUNCT
ejpam-6452	27	1	however	however	ADV
ejpam-6452	27	2	,	,	PUNCT
ejpam-6452	27	3	the	the	DET
ejpam-6452	27	4	third	third	ADJ
ejpam-6452	27	5	kind	kind	NOUN
ejpam-6452	27	6	of	of	ADP
ejpam-6452	27	7	one	one	NUM
ejpam-6452	27	8	-	-	PUNCT
ejpam-6452	27	9	parameter	parameter	NOUN
ejpam-6452	27	10	generalization	generalization	NOUN
ejpam-6452	27	11	of	of	ADP
ejpam-6452	27	12	pell	pell	PROPN
ejpam-6452	27	13	numbers	number	NOUN
ejpam-6452	27	14	(	(	PUNCT
ejpam-6452	27	15	{	{	PUNCT
ejpam-6452	27	16	pk	pk	NOUN
ejpam-6452	27	17	,	,	PUNCT
ejpam-6452	27	18	n	n	CCONJ
ejpam-6452	27	19	}	}	PUNCT
ejpam-6452	27	20	,	,	PUNCT
ejpam-6452	27	21	{	{	PUNCT
ejpam-6452	27	22	qk	qk	NOUN
ejpam-6452	27	23	,	,	PUNCT
ejpam-6452	27	24	n	n	CCONJ
ejpam-6452	27	25	}	}	PUNCT
ejpam-6452	27	26	)	)	PUNCT
ejpam-6452	27	27	and	and	CCONJ
ejpam-6452	27	28	(	(	PUNCT
ejpam-6452	27	29	{	{	PUNCT
ejpam-6452	27	30	pk	pk	NOUN
ejpam-6452	27	31	,	,	PUNCT
ejpam-6452	27	32	n	n	CCONJ
ejpam-6452	27	33	}	}	PUNCT
ejpam-6452	27	34	,	,	PUNCT
ejpam-6452	27	35	{	{	PUNCT
ejpam-6452	27	36	qk	qk	NOUN
ejpam-6452	27	37	,	,	PUNCT
ejpam-6452	27	38	n	n	CCONJ
ejpam-6452	27	39	}	}	PUNCT
ejpam-6452	27	40	)	)	PUNCT
ejpam-6452	27	41	are	be	AUX
ejpam-6452	27	42	not	not	PART
ejpam-6452	27	43	a	a	DET
ejpam-6452	27	44	pair	pair	NOUN
ejpam-6452	27	45	of	of	ADP
ejpam-6452	27	46	lucas	lucas	PROPN
ejpam-6452	27	47	sequences	sequence	NOUN
ejpam-6452	27	48	.	.	PUNCT
ejpam-6452	28	1	if	if	SCONJ
ejpam-6452	28	2	we	we	PRON
ejpam-6452	28	3	define	define	VERB
ejpam-6452	28	4	lk	lk	PROPN
ejpam-6452	28	5	,	,	PUNCT
ejpam-6452	28	6	n	n	CCONJ
ejpam-6452	28	7	by	by	ADP
ejpam-6452	28	8	lk,0	lk,0	PROPN
ejpam-6452	28	9	=	=	PUNCT
ejpam-6452	28	10	2,lk,1	2,lk,1	PROPN
ejpam-6452	28	11	=	=	SYM
ejpam-6452	28	12	k	k	PROPN
ejpam-6452	28	13	,	,	PUNCT
ejpam-6452	28	14	and	and	CCONJ
ejpam-6452	28	15	lk	lk	PROPN
ejpam-6452	28	16	,	,	PUNCT
ejpam-6452	28	17	n	n	PROPN
ejpam-6452	28	18	=	=	SYM
ejpam-6452	28	19	klk	klk	PROPN
ejpam-6452	28	20	,	,	PUNCT
ejpam-6452	28	21	n−1	n−1	PROPN
ejpam-6452	28	22	+	+	CCONJ
ejpam-6452	28	23	(	(	PUNCT
ejpam-6452	28	24	k	k	PROPN
ejpam-6452	28	25	−	−	PROPN
ejpam-6452	29	1	1)lk	1)lk	PROPN
ejpam-6452	29	2	,	,	PUNCT
ejpam-6452	29	3	n−2	n−2	PROPN
ejpam-6452	29	4	,	,	PUNCT
ejpam-6452	29	5	n	n	PRON
ejpam-6452	29	6	≥	≥	NOUN
ejpam-6452	29	7	2	2	NUM
ejpam-6452	29	8	,	,	PUNCT
ejpam-6452	29	9	then	then	ADV
ejpam-6452	29	10	it	it	PRON
ejpam-6452	29	11	is	be	AUX
ejpam-6452	29	12	a	a	DET
ejpam-6452	29	13	convenient	convenient	ADJ
ejpam-6452	29	14	lucas	lucas	NOUN
ejpam-6452	29	15	sequence	sequence	NOUN
ejpam-6452	29	16	of	of	ADP
ejpam-6452	29	17	the	the	DET
ejpam-6452	29	18	second	second	ADJ
ejpam-6452	29	19	kind	kind	NOUN
ejpam-6452	29	20	such	such	ADJ
ejpam-6452	29	21	that	that	SCONJ
ejpam-6452	29	22	(	(	PUNCT
ejpam-6452	29	23	{	{	PUNCT
ejpam-6452	29	24	pk	pk	NOUN
ejpam-6452	29	25	,	,	PUNCT
ejpam-6452	29	26	n	n	CCONJ
ejpam-6452	29	27	}	}	PUNCT
ejpam-6452	29	28	,	,	PUNCT
ejpam-6452	29	29	{	{	PUNCT
ejpam-6452	29	30	lk	lk	NOUN
ejpam-6452	29	31	,	,	PUNCT
ejpam-6452	29	32	n	n	CCONJ
ejpam-6452	29	33	}	}	PUNCT
ejpam-6452	29	34	)	)	PUNCT
ejpam-6452	29	35	is	be	AUX
ejpam-6452	29	36	a	a	DET
ejpam-6452	29	37	pair	pair	NOUN
ejpam-6452	29	38	of	of	ADP
ejpam-6452	29	39	lucas	lucas	PROPN
ejpam-6452	29	40	sequences	sequence	NOUN
ejpam-6452	29	41	with	with	ADP
ejpam-6452	29	42	(	(	PUNCT
ejpam-6452	29	43	α	α	X
ejpam-6452	29	44	,	,	PUNCT
ejpam-6452	29	45	β	β	NOUN
ejpam-6452	29	46	)	)	PUNCT
ejpam-6452	30	1	=	=	SYM
ejpam-6452	30	2	(	(	PUNCT
ejpam-6452	30	3	k	k	X
ejpam-6452	30	4	,	,	PUNCT
ejpam-6452	30	5	1−	1−	NUM
ejpam-6452	30	6	k	k	NOUN
ejpam-6452	30	7	)	)	PUNCT
ejpam-6452	30	8	to	to	PART
ejpam-6452	30	9	consider	consider	VERB
ejpam-6452	30	10	it	it	PRON
ejpam-6452	30	11	to	to	PART
ejpam-6452	30	12	be	be	AUX
ejpam-6452	30	13	an	an	DET
ejpam-6452	30	14	associated	associated	ADJ
ejpam-6452	30	15	number	number	NOUN
ejpam-6452	30	16	of	of	ADP
ejpam-6452	30	17	pk	pk	NOUN
ejpam-6452	30	18	,	,	PUNCT
ejpam-6452	30	19	n.	n.	NOUN
ejpam-6452	30	20	here	here	ADV
ejpam-6452	30	21	lk	lk	PROPN
ejpam-6452	30	22	,	,	PUNCT
ejpam-6452	30	23	n	n	PRON
ejpam-6452	30	24	is	be	AUX
ejpam-6452	30	25	called	call	VERB
ejpam-6452	30	26	generalized	generalized	ADJ
ejpam-6452	30	27	pell	pell	NOUN
ejpam-6452	30	28	-	-	PUNCT
ejpam-6452	30	29	lucas	lucas	NOUN
ejpam-6452	30	30	-	-	PUNCT
ejpam-6452	30	31	like	like	ADJ
ejpam-6452	30	32	.	.	PUNCT
ejpam-6452	31	1	the	the	DET
ejpam-6452	31	2	tables	table	NOUN
ejpam-6452	31	3	presented	present	VERB
ejpam-6452	31	4	below	below	ADP
ejpam-6452	31	5	contain	contain	VERB
ejpam-6452	31	6	initial	initial	ADJ
ejpam-6452	31	7	terms	term	NOUN
ejpam-6452	31	8	of	of	ADP
ejpam-6452	31	9	the	the	DET
ejpam-6452	31	10	sequences	sequence	NOUN
ejpam-6452	31	11	{	{	PUNCT
ejpam-6452	31	12	lk	lk	PROPN
ejpam-6452	31	13	,	,	PUNCT
ejpam-6452	31	14	n	n	CCONJ
ejpam-6452	31	15	}	}	PUNCT
ejpam-6452	31	16	for	for	ADP
ejpam-6452	31	17	selected	select	VERB
ejpam-6452	31	18	values	value	NOUN
ejpam-6452	31	19	k	k	NOUN
ejpam-6452	31	20	(	(	PUNCT
ejpam-6452	31	21	table	table	NOUN
ejpam-6452	31	22	1	1	NUM
ejpam-6452	31	23	)	)	PUNCT
ejpam-6452	31	24	.	.	PUNCT
ejpam-6452	32	1	table	table	NOUN
ejpam-6452	32	2	1	1	NUM
ejpam-6452	32	3	:	:	PUNCT
ejpam-6452	32	4	initial	initial	ADJ
ejpam-6452	32	5	terms	term	NOUN
ejpam-6452	32	6	of	of	ADP
ejpam-6452	32	7	the	the	DET
ejpam-6452	32	8	generalized	generalize	VERB
ejpam-6452	32	9	pell	pell	NOUN
ejpam-6452	32	10	-	-	PUNCT
ejpam-6452	32	11	lucas	lucas	NOUN
ejpam-6452	32	12	-	-	PUNCT
ejpam-6452	32	13	like	like	ADJ
ejpam-6452	32	14	numbers	number	NOUN
ejpam-6452	32	15	{	{	PUNCT
ejpam-6452	32	16	lk	lk	NOUN
ejpam-6452	32	17	,	,	PUNCT
ejpam-6452	32	18	n	n	CCONJ
ejpam-6452	32	19	}	}	PUNCT
ejpam-6452	32	20	.	.	PUNCT
ejpam-6452	33	1	n	n	CCONJ
ejpam-6452	33	2	0	0	NUM
ejpam-6452	33	3	1	1	NUM
ejpam-6452	33	4	2	2	NUM
ejpam-6452	33	5	3	3	NUM
ejpam-6452	33	6	4	4	NUM
ejpam-6452	33	7	5	5	NUM
ejpam-6452	33	8	6	6	NUM
ejpam-6452	33	9	7	7	NUM
ejpam-6452	33	10	8	8	NUM
ejpam-6452	33	11	9	9	NUM
ejpam-6452	33	12	l2,n	l2,n	PROPN
ejpam-6452	33	13	2	2	NUM
ejpam-6452	33	14	2	2	NUM
ejpam-6452	33	15	6	6	NUM
ejpam-6452	33	16	14	14	NUM
ejpam-6452	33	17	34	34	NUM
ejpam-6452	33	18	82	82	NUM
ejpam-6452	33	19	198	198	NUM
ejpam-6452	33	20	478	478	NUM
ejpam-6452	33	21	1154	1154	NUM
ejpam-6452	33	22	2786	2786	NUM
ejpam-6452	33	23	l3,n	l3,n	PROPN
ejpam-6452	33	24	2	2	NUM
ejpam-6452	33	25	3	3	NUM
ejpam-6452	33	26	13	13	NUM
ejpam-6452	33	27	45	45	NUM
ejpam-6452	33	28	161	161	NUM
ejpam-6452	33	29	573	573	NUM
ejpam-6452	33	30	2041	2041	NUM
ejpam-6452	33	31	7269	7269	NUM
ejpam-6452	33	32	25889	25889	NUM
ejpam-6452	33	33	92205	92205	NUM
ejpam-6452	34	1	l4,n	l4,n	PROPN
ejpam-6452	34	2	2	2	NUM
ejpam-6452	34	3	4	4	NUM
ejpam-6452	34	4	22	22	NUM
ejpam-6452	34	5	100	100	NUM
ejpam-6452	34	6	466	466	NUM
ejpam-6452	34	7	2164	2164	NUM
ejpam-6452	34	8	10054	10054	NUM
ejpam-6452	34	9	46708	46708	NUM
ejpam-6452	34	10	216994	216994	NUM
ejpam-6452	34	11	1008100	1008100	NUM
ejpam-6452	34	12	l5,n	l5,n	NOUN
ejpam-6452	34	13	2	2	NUM
ejpam-6452	34	14	5	5	NUM
ejpam-6452	34	15	33	33	NUM
ejpam-6452	34	16	185	185	NUM
ejpam-6452	34	17	1057	1057	NUM
ejpam-6452	34	18	6025	6025	NUM
ejpam-6452	34	19	34353	34353	NUM
ejpam-6452	34	20	195865	195865	NUM
ejpam-6452	34	21	1116737	1116737	NUM
ejpam-6452	34	22	6367145	6367145	NUM
ejpam-6452	34	23	l6,n	l6,n	PROPN
ejpam-6452	34	24	2	2	NUM
ejpam-6452	34	25	6	6	NUM
ejpam-6452	34	26	46	46	NUM
ejpam-6452	34	27	306	306	NUM
ejpam-6452	34	28	2066	2066	NUM
ejpam-6452	34	29	13926	13926	NUM
ejpam-6452	34	30	93886	93886	NUM
ejpam-6452	34	31	632946	632946	NUM
ejpam-6452	34	32	4267106	4267106	NUM
ejpam-6452	34	33	28767366	28767366	NUM
ejpam-6452	34	34	for	for	ADP
ejpam-6452	34	35	k	k	PROPN
ejpam-6452	34	36	=	=	SYM
ejpam-6452	34	37	2	2	NUM
ejpam-6452	34	38	,	,	PUNCT
ejpam-6452	34	39	we	we	PRON
ejpam-6452	34	40	can	can	AUX
ejpam-6452	34	41	see	see	VERB
ejpam-6452	34	42	that	that	SCONJ
ejpam-6452	34	43	the	the	DET
ejpam-6452	34	44	classical	classical	ADJ
ejpam-6452	34	45	pell	pell	NOUN
ejpam-6452	34	46	–	–	PUNCT
ejpam-6452	34	47	lucas	lucas	NOUN
ejpam-6452	34	48	numbers	number	NOUN
ejpam-6452	34	49	are	be	AUX
ejpam-6452	34	50	obtained	obtain	VERB
ejpam-6452	34	51	.	.	PUNCT
ejpam-6452	35	1	moreover	moreover	ADV
ejpam-6452	35	2	,	,	PUNCT
ejpam-6452	35	3	sequences	sequence	NOUN
ejpam-6452	35	4	{	{	PUNCT
ejpam-6452	35	5	l2,n	l2,n	PROPN
ejpam-6452	35	6	}	}	PUNCT
ejpam-6452	35	7	,	,	PUNCT
ejpam-6452	35	8	{	{	PUNCT
ejpam-6452	35	9	l3,n	l3,n	NOUN
ejpam-6452	35	10	}	}	PUNCT
ejpam-6452	35	11	,	,	PUNCT
ejpam-6452	35	12	and	and	CCONJ
ejpam-6452	35	13	{	{	PUNCT
ejpam-6452	35	14	l4,n	l4,n	PROPN
ejpam-6452	35	15	}	}	PUNCT
ejpam-6452	35	16	are	be	AUX
ejpam-6452	35	17	listed	list	VERB
ejpam-6452	35	18	in	in	ADP
ejpam-6452	35	19	the	the	DET
ejpam-6452	35	20	online	online	ADJ
ejpam-6452	35	21	encyclopaedia	encyclopaedia	NOUN
ejpam-6452	35	22	of	of	ADP
ejpam-6452	35	23	integer	integer	NOUN
ejpam-6452	35	24	sequences	sequence	NOUN
ejpam-6452	36	1	[	[	X
ejpam-6452	36	2	8	8	NUM
ejpam-6452	36	3	]	]	PUNCT
ejpam-6452	36	4	under	under	ADP
ejpam-6452	36	5	the	the	DET
ejpam-6452	36	6	symbols	symbol	NOUN
ejpam-6452	36	7	a002203	a002203	NOUN
ejpam-6452	36	8	,	,	PUNCT
ejpam-6452	36	9	a206776	a206776	NOUN
ejpam-6452	36	10	,	,	PUNCT
ejpam-6452	36	11	and	and	CCONJ
ejpam-6452	36	12	a080042	a080042	NOUN
ejpam-6452	36	13	,	,	PUNCT
ejpam-6452	36	14	respectively	respectively	ADV
ejpam-6452	36	15	.	.	PUNCT
ejpam-6452	37	1	in	in	ADP
ejpam-6452	37	2	this	this	DET
ejpam-6452	37	3	paper	paper	NOUN
ejpam-6452	37	4	,	,	PUNCT
ejpam-6452	37	5	we	we	PRON
ejpam-6452	37	6	study	study	VERB
ejpam-6452	37	7	all	all	DET
ejpam-6452	37	8	the	the	DET
ejpam-6452	37	9	third	third	ADJ
ejpam-6452	37	10	kind	kind	NOUN
ejpam-6452	37	11	of	of	ADP
ejpam-6452	37	12	one	one	NUM
ejpam-6452	37	13	-	-	PUNCT
ejpam-6452	37	14	parameter	parameter	NOUN
ejpam-6452	37	15	generalization	generalization	NOUN
ejpam-6452	37	16	of	of	ADP
ejpam-6452	37	17	pell	pell	PROPN
ejpam-6452	37	18	numbers	number	NOUN
ejpam-6452	37	19	that	that	PRON
ejpam-6452	37	20	preserve	preserve	VERB
ejpam-6452	37	21	the	the	DET
ejpam-6452	37	22	recurrence	recurrence	NOUN
ejpam-6452	37	23	relation	relation	NOUN
ejpam-6452	37	24	(	(	PUNCT
ejpam-6452	37	25	1	1	NUM
ejpam-6452	37	26	)	)	PUNCT
ejpam-6452	37	27	with	with	ADP
ejpam-6452	37	28	arbitrary	arbitrary	ADJ
ejpam-6452	37	29	initial	initial	ADJ
ejpam-6452	37	30	conditions	condition	NOUN
ejpam-6452	37	31	.	.	PUNCT
ejpam-6452	38	1	we	we	PRON
ejpam-6452	38	2	see	see	VERB
ejpam-6452	38	3	that	that	SCONJ
ejpam-6452	38	4	the	the	DET
ejpam-6452	38	5	generalized	generalize	VERB
ejpam-6452	38	6	pell	pell	NOUN
ejpam-6452	38	7	-	-	PUNCT
ejpam-6452	38	8	lucas	lucas	NOUN
ejpam-6452	38	9	-	-	PUNCT
ejpam-6452	38	10	like	like	NOUN
ejpam-6452	38	11	is	be	AUX
ejpam-6452	38	12	a	a	DET
ejpam-6452	38	13	simple	simple	ADJ
ejpam-6452	38	14	association	association	NOUN
ejpam-6452	38	15	of	of	ADP
ejpam-6452	38	16	generalized	generalized	ADJ
ejpam-6452	38	17	pell	pell	NOUN
ejpam-6452	38	18	numbers	number	NOUN
ejpam-6452	38	19	.	.	PUNCT
ejpam-6452	39	1	consequently	consequently	ADV
ejpam-6452	39	2	,	,	PUNCT
ejpam-6452	39	3	we	we	PRON
ejpam-6452	39	4	give	give	VERB
ejpam-6452	39	5	some	some	DET
ejpam-6452	39	6	new	new	ADJ
ejpam-6452	39	7	and	and	CCONJ
ejpam-6452	39	8	well	well	ADV
ejpam-6452	39	9	-	-	PUNCT
ejpam-6452	39	10	known	know	VERB
ejpam-6452	39	11	identities	identity	NOUN
ejpam-6452	39	12	.	.	PUNCT
ejpam-6452	40	1	furthermore	furthermore	ADV
ejpam-6452	40	2	,	,	PUNCT
ejpam-6452	40	3	we	we	PRON
ejpam-6452	40	4	propose	propose	VERB
ejpam-6452	40	5	integral	integral	ADJ
ejpam-6452	40	6	representations	representation	NOUN
ejpam-6452	40	7	of	of	ADP
ejpam-6452	40	8	these	these	DET
ejpam-6452	40	9	numbers	number	NOUN
ejpam-6452	40	10	associated	associate	VERB
ejpam-6452	40	11	with	with	ADP
ejpam-6452	40	12	generalized	generalized	ADJ
ejpam-6452	40	13	pell	pell	NOUN
ejpam-6452	40	14	and	and	CCONJ
ejpam-6452	40	15	pell	pell	NOUN
ejpam-6452	40	16	-	-	PUNCT
ejpam-6452	40	17	lucaslike	lucaslike	NOUN
ejpam-6452	40	18	numbers	number	NOUN
ejpam-6452	40	19	.	.	PUNCT
ejpam-6452	41	1	2	2	X
ejpam-6452	41	2	.	.	X
ejpam-6452	41	3	the	the	DET
ejpam-6452	41	4	companion	companion	NOUN
ejpam-6452	41	5	numbers	number	NOUN
ejpam-6452	41	6	of	of	ADP
ejpam-6452	41	7	generalized	generalized	ADJ
ejpam-6452	41	8	pell	pell	NOUN
ejpam-6452	41	9	numbers	number	NOUN
ejpam-6452	41	10	in	in	ADP
ejpam-6452	41	11	this	this	DET
ejpam-6452	41	12	section	section	NOUN
ejpam-6452	41	13	,	,	PUNCT
ejpam-6452	41	14	we	we	PRON
ejpam-6452	41	15	point	point	VERB
ejpam-6452	41	16	out	out	ADP
ejpam-6452	41	17	the	the	DET
ejpam-6452	41	18	third	third	ADJ
ejpam-6452	41	19	kind	kind	NOUN
ejpam-6452	41	20	of	of	ADP
ejpam-6452	41	21	generalized	generalized	ADJ
ejpam-6452	41	22	pell	pell	NOUN
ejpam-6452	41	23	numbers	number	NOUN
ejpam-6452	41	24	to	to	PART
ejpam-6452	41	25	study	study	VERB
ejpam-6452	41	26	a	a	DET
ejpam-6452	41	27	generalization	generalization	NOUN
ejpam-6452	41	28	of	of	ADP
ejpam-6452	41	29	the	the	DET
ejpam-6452	41	30	pell	pell	NOUN
ejpam-6452	41	31	numbers	number	NOUN
ejpam-6452	41	32	with	with	ADP
ejpam-6452	41	33	one	one	NUM
ejpam-6452	41	34	parameter	parameter	NOUN
ejpam-6452	41	35	positive	positive	ADJ
ejpam-6452	41	36	integer	integer	NOUN
ejpam-6452	41	37	k	k	PROPN
ejpam-6452	41	38	≥	≥	NUM
ejpam-6452	41	39	2	2	NUM
ejpam-6452	41	40	which	which	PRON
ejpam-6452	41	41	is	be	AUX
ejpam-6452	41	42	called	call	VERB
ejpam-6452	41	43	the	the	DET
ejpam-6452	41	44	companion	companion	NOUN
ejpam-6452	41	45	generalized	generalize	VERB
ejpam-6452	41	46	pell	pell	NOUN
ejpam-6452	41	47	number	number	NOUN
ejpam-6452	41	48	,	,	PUNCT
ejpam-6452	41	49	denoted	denote	VERB
ejpam-6452	41	50	by	by	ADP
ejpam-6452	41	51	gpk	gpk	PROPN
ejpam-6452	41	52	,	,	PUNCT
ejpam-6452	41	53	n	n	PROPN
ejpam-6452	41	54	=	=	PROPN
ejpam-6452	41	55	gpk	gpk	PROPN
ejpam-6452	41	56	,	,	PUNCT
ejpam-6452	41	57	n(a	n(a	PROPN
ejpam-6452	41	58	,	,	PUNCT
ejpam-6452	41	59	b	b	NOUN
ejpam-6452	41	60	)	)	PUNCT
ejpam-6452	41	61	,	,	PUNCT
ejpam-6452	41	62	defined	define	VERB
ejpam-6452	41	63	by	by	ADP
ejpam-6452	41	64	a	a	DET
ejpam-6452	41	65	recurrence	recurrence	NOUN
ejpam-6452	41	66	relation	relation	NOUN
ejpam-6452	41	67	gk,0	gk,0	PROPN
ejpam-6452	41	68	=	=	PUNCT
ejpam-6452	41	69	a	a	DET
ejpam-6452	41	70	,	,	PUNCT
ejpam-6452	41	71	gpk,1	gpk,1	NOUN
ejpam-6452	41	72	=	=	SYM
ejpam-6452	41	73	b	b	PROPN
ejpam-6452	41	74	,	,	PUNCT
ejpam-6452	41	75	and	and	CCONJ
ejpam-6452	41	76	gpk	gpk	PROPN
ejpam-6452	41	77	,	,	PUNCT
ejpam-6452	41	78	n	n	NOUN
ejpam-6452	41	79	=	=	PUNCT
ejpam-6452	41	80	kgpk	kgpk	NOUN
ejpam-6452	41	81	,	,	PUNCT
ejpam-6452	41	82	n−1	n−1	PROPN
ejpam-6452	41	83	+	+	CCONJ
ejpam-6452	41	84	(	(	PUNCT
ejpam-6452	41	85	k	k	X
ejpam-6452	41	86	−	−	PROPN
ejpam-6452	41	87	1)gpk	1)gpk	NUM
ejpam-6452	41	88	,	,	PUNCT
ejpam-6452	41	89	n−2	n−2	PROPN
ejpam-6452	41	90	,	,	PUNCT
ejpam-6452	41	91	n	n	PRON
ejpam-6452	41	92	≥	≥	NOUN
ejpam-6452	41	93	2	2	NUM
ejpam-6452	41	94	,	,	PUNCT
ejpam-6452	41	95	(	(	PUNCT
ejpam-6452	41	96	2	2	X
ejpam-6452	41	97	)	)	PUNCT
ejpam-6452	41	98	where	where	SCONJ
ejpam-6452	41	99	a	a	PRON
ejpam-6452	41	100	and	and	CCONJ
ejpam-6452	41	101	b	b	NOUN
ejpam-6452	41	102	are	be	AUX
ejpam-6452	41	103	arbitrary	arbitrary	ADJ
ejpam-6452	41	104	non	non	ADJ
ejpam-6452	41	105	-	-	ADJ
ejpam-6452	41	106	negative	negative	ADJ
ejpam-6452	41	107	integers	integer	NOUN
ejpam-6452	41	108	such	such	ADJ
ejpam-6452	41	109	that	that	SCONJ
ejpam-6452	41	110	a	a	DET
ejpam-6452	41	111	+	+	NOUN
ejpam-6452	41	112	b	b	NOUN
ejpam-6452	41	113	̸=	̸=	PROPN
ejpam-6452	41	114	0	0	NUM
ejpam-6452	41	115	.	.	PUNCT
ejpam-6452	42	1	note	note	VERB
ejpam-6452	42	2	that	that	SCONJ
ejpam-6452	42	3	gpk	gpk	NOUN
ejpam-6452	42	4	,	,	PUNCT
ejpam-6452	42	5	n	n	PRON
ejpam-6452	42	6	correspond	correspond	VERB
ejpam-6452	42	7	to	to	ADP
ejpam-6452	42	8	special	special	ADJ
ejpam-6452	42	9	cases	case	NOUN
ejpam-6452	42	10	of	of	ADP
ejpam-6452	42	11	horadam	horadam	NOUN
ejpam-6452	42	12	numbers	number	NOUN
ejpam-6452	42	13	[	[	X
ejpam-6452	42	14	9	9	NUM
ejpam-6452	42	15	]	]	PUNCT
ejpam-6452	42	16	.	.	PUNCT
ejpam-6452	43	1	the	the	DET
ejpam-6452	43	2	first	first	ADJ
ejpam-6452	43	3	terms	term	NOUN
ejpam-6452	43	4	gpk	gpk	PROPN
ejpam-6452	43	5	,	,	PUNCT
ejpam-6452	43	6	n	n	PRON
ejpam-6452	43	7	are	be	AUX
ejpam-6452	43	8	:	:	PUNCT
ejpam-6452	43	9	gpk,0	gpk,0	X
ejpam-6452	43	10	=	=	PUNCT
ejpam-6452	43	11	a	a	DET
ejpam-6452	43	12	gpk,1	gpk,1	NOUN
ejpam-6452	43	13	=	=	SYM
ejpam-6452	43	14	b	b	PROPN
ejpam-6452	43	15	w.	w.	PROPN
ejpam-6452	43	16	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	43	17	,	,	PUNCT
ejpam-6452	43	18	a.	a.	NOUN
ejpam-6452	43	19	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	43	20	/	/	SYM
ejpam-6452	43	21	eur	eur	PROPN
ejpam-6452	43	22	.	.	PUNCT
ejpam-6452	44	1	j.	j.	PROPN
ejpam-6452	44	2	pure	pure	PROPN
ejpam-6452	44	3	appl	appl	PROPN
ejpam-6452	44	4	.	.	PROPN
ejpam-6452	44	5	math	math	PROPN
ejpam-6452	44	6	,	,	PUNCT
ejpam-6452	44	7	18	18	NUM
ejpam-6452	44	8	(	(	PUNCT
ejpam-6452	44	9	3	3	NUM
ejpam-6452	44	10	)	)	PUNCT
ejpam-6452	44	11	(	(	PUNCT
ejpam-6452	44	12	2025	2025	NUM
ejpam-6452	44	13	)	)	PUNCT
ejpam-6452	44	14	,	,	PUNCT
ejpam-6452	44	15	6452	6452	NUM
ejpam-6452	44	16	4	4	NUM
ejpam-6452	44	17	of	of	ADP
ejpam-6452	44	18	18	18	NUM
ejpam-6452	44	19	gpk,2	gpk,2	NOUN
ejpam-6452	44	20	=	=	PUNCT
ejpam-6452	44	21	(	(	PUNCT
ejpam-6452	44	22	a	a	DET
ejpam-6452	44	23	+	+	X
ejpam-6452	44	24	b)k	b)k	ADJ
ejpam-6452	44	25	−	−	ADP
ejpam-6452	44	26	a	a	DET
ejpam-6452	44	27	gpk,3	gpk,3	NOUN
ejpam-6452	44	28	=	=	PUNCT
ejpam-6452	44	29	(	(	PUNCT
ejpam-6452	44	30	a	a	DET
ejpam-6452	44	31	+	+	X
ejpam-6452	44	32	b)k2	b)k2	NOUN
ejpam-6452	44	33	−	−	PROPN
ejpam-6452	44	34	(	(	PUNCT
ejpam-6452	44	35	a−	a−	PROPN
ejpam-6452	44	36	b)k	b)k	NOUN
ejpam-6452	45	1	−	−	PROPN
ejpam-6452	45	2	b	b	X
ejpam-6452	45	3	gpk,4	gpk,4	NOUN
ejpam-6452	45	4	=	=	SYM
ejpam-6452	45	5	(	(	PUNCT
ejpam-6452	45	6	a	a	DET
ejpam-6452	45	7	+	+	NUM
ejpam-6452	45	8	b)k3	b)k3	NOUN
ejpam-6452	45	9	+	+	CCONJ
ejpam-6452	45	10	2bk2	2bk2	NUM
ejpam-6452	45	11	−	−	NUM
ejpam-6452	45	12	2(a	2(a	NUM
ejpam-6452	45	13	+	+	CCONJ
ejpam-6452	45	14	b)k	b)k	NOUN
ejpam-6452	45	15	+	+	CCONJ
ejpam-6452	45	16	a	a	DET
ejpam-6452	45	17	gpk,5	gpk,5	NOUN
ejpam-6452	45	18	=	=	SYM
ejpam-6452	45	19	(	(	PUNCT
ejpam-6452	45	20	a	a	DET
ejpam-6452	45	21	+	+	X
ejpam-6452	45	22	b)k4	b)k4	PROPN
ejpam-6452	45	23	+	+	CCONJ
ejpam-6452	45	24	(	(	PUNCT
ejpam-6452	45	25	a	a	DET
ejpam-6452	45	26	+	+	NOUN
ejpam-6452	45	27	3b)k3	3b)k3	NUM
ejpam-6452	45	28	−	−	PROPN
ejpam-6452	45	29	2(2a	2(2a	NUM
ejpam-6452	45	30	+	+	CCONJ
ejpam-6452	45	31	b)k2	b)k2	NOUN
ejpam-6452	45	32	+	+	CCONJ
ejpam-6452	45	33	(	(	PUNCT
ejpam-6452	45	34	a−	a−	PROPN
ejpam-6452	45	35	2b)k	2b)k	PROPN
ejpam-6452	45	36	+	+	CCONJ
ejpam-6452	45	37	b	b	X
ejpam-6452	45	38	gpk,6	gpk,6	NOUN
ejpam-6452	45	39	=	=	SYM
ejpam-6452	45	40	(	(	PUNCT
ejpam-6452	45	41	a	a	DET
ejpam-6452	45	42	+	+	X
ejpam-6452	45	43	b)k5	b)k5	NOUN
ejpam-6452	45	44	+	+	CCONJ
ejpam-6452	45	45	2(a	2(a	NUM
ejpam-6452	45	46	+	+	CCONJ
ejpam-6452	45	47	2b)k4	2b)k4	NUM
ejpam-6452	45	48	−	−	NOUN
ejpam-6452	45	49	(	(	PUNCT
ejpam-6452	45	50	5a	5a	NUM
ejpam-6452	45	51	+	+	CCONJ
ejpam-6452	45	52	b)k3	b)k3	PROPN
ejpam-6452	45	53	−	−	PROPN
ejpam-6452	45	54	(	(	PUNCT
ejpam-6452	45	55	a	a	DET
ejpam-6452	45	56	+	+	NOUN
ejpam-6452	45	57	6b)k2	6b)k2	NUM
ejpam-6452	45	58	+	+	CCONJ
ejpam-6452	45	59	3(a	3(a	NUM
ejpam-6452	45	60	+	+	NUM
ejpam-6452	45	61	b)k	b)k	ADJ
ejpam-6452	45	62	−	−	CCONJ
ejpam-6452	45	63	a.	a.	NOUN
ejpam-6452	45	64	some	some	DET
ejpam-6452	45	65	particular	particular	ADJ
ejpam-6452	45	66	cases	case	NOUN
ejpam-6452	45	67	of	of	ADP
ejpam-6452	45	68	the	the	DET
ejpam-6452	45	69	previous	previous	ADJ
ejpam-6452	45	70	definition	definition	NOUN
ejpam-6452	45	71	are	be	AUX
ejpam-6452	45	72	(	(	PUNCT
ejpam-6452	45	73	i	i	NOUN
ejpam-6452	45	74	)	)	PUNCT
ejpam-6452	45	75	pk	pk	PROPN
ejpam-6452	45	76	,	,	PUNCT
ejpam-6452	45	77	n	n	PROPN
ejpam-6452	45	78	=	=	PROPN
ejpam-6452	45	79	gpk	gpk	PROPN
ejpam-6452	45	80	,	,	PUNCT
ejpam-6452	45	81	n(0	n(0	PROPN
ejpam-6452	45	82	,	,	PUNCT
ejpam-6452	45	83	1	1	NUM
ejpam-6452	45	84	)	)	PUNCT
ejpam-6452	45	85	,	,	PUNCT
ejpam-6452	45	86	(	(	PUNCT
ejpam-6452	45	87	ii	ii	NOUN
ejpam-6452	45	88	)	)	PUNCT
ejpam-6452	45	89	lk	lk	PROPN
ejpam-6452	45	90	,	,	PUNCT
ejpam-6452	45	91	n	n	PROPN
ejpam-6452	45	92	=	=	PROPN
ejpam-6452	45	93	gpk	gpk	PROPN
ejpam-6452	45	94	,	,	PUNCT
ejpam-6452	45	95	n(2	n(2	PROPN
ejpam-6452	45	96	,	,	PUNCT
ejpam-6452	45	97	k	k	PROPN
ejpam-6452	45	98	)	)	PUNCT
ejpam-6452	45	99	,	,	PUNCT
ejpam-6452	45	100	(	(	PUNCT
ejpam-6452	45	101	iii	iii	X
ejpam-6452	45	102	)	)	PUNCT
ejpam-6452	45	103	qk	qk	NOUN
ejpam-6452	45	104	,	,	PUNCT
ejpam-6452	45	105	n	n	PROPN
ejpam-6452	45	106	=	=	PROPN
ejpam-6452	45	107	gpk	gpk	PROPN
ejpam-6452	45	108	,	,	PUNCT
ejpam-6452	45	109	n(2	n(2	PROPN
ejpam-6452	45	110	,	,	PUNCT
ejpam-6452	45	111	2	2	NUM
ejpam-6452	45	112	)	)	PUNCT
ejpam-6452	45	113	,	,	PUNCT
ejpam-6452	45	114	(	(	PUNCT
ejpam-6452	45	115	iv	iv	X
ejpam-6452	45	116	)	)	PUNCT
ejpam-6452	45	117	qk	qk	NOUN
ejpam-6452	45	118	,	,	PUNCT
ejpam-6452	45	119	n	n	PROPN
ejpam-6452	45	120	=	=	PROPN
ejpam-6452	45	121	gpk	gpk	PROPN
ejpam-6452	45	122	,	,	PUNCT
ejpam-6452	45	123	n(1	n(1	PROPN
ejpam-6452	45	124	,	,	PUNCT
ejpam-6452	45	125	1	1	NUM
ejpam-6452	45	126	)	)	PUNCT
ejpam-6452	45	127	,	,	PUNCT
ejpam-6452	45	128	(	(	PUNCT
ejpam-6452	45	129	v	v	NOUN
ejpam-6452	45	130	)	)	PUNCT
ejpam-6452	45	131	generalized	generalized	ADJ
ejpam-6452	45	132	pell	pell	NOUN
ejpam-6452	45	133	numbers	number	NOUN
ejpam-6452	45	134	introduced	introduce	VERB
ejpam-6452	45	135	in	in	ADP
ejpam-6452	45	136	[	[	X
ejpam-6452	45	137	10	10	NUM
ejpam-6452	45	138	]	]	PUNCT
ejpam-6452	45	139	,	,	PUNCT
ejpam-6452	45	140	ha	ha	INTJ
ejpam-6452	45	141	,	,	PUNCT
ejpam-6452	45	142	b	b	PROPN
ejpam-6452	45	143	n	n	NOUN
ejpam-6452	45	144	=	=	SYM
ejpam-6452	45	145	gp2,n(a	gp2,n(a	PROPN
ejpam-6452	45	146	,	,	PUNCT
ejpam-6452	45	147	b	b	NOUN
ejpam-6452	45	148	)	)	PUNCT
ejpam-6452	45	149	,	,	PUNCT
ejpam-6452	45	150	(	(	PUNCT
ejpam-6452	45	151	vi	vi	NOUN
ejpam-6452	45	152	)	)	PUNCT
ejpam-6452	45	153	pn	pn	NOUN
ejpam-6452	45	154	=	=	SYM
ejpam-6452	45	155	p2,n	p2,n	PROPN
ejpam-6452	45	156	=	=	SYM
ejpam-6452	45	157	gp2,n(0	gp2,n(0	NOUN
ejpam-6452	45	158	,	,	PUNCT
ejpam-6452	45	159	1	1	NUM
ejpam-6452	45	160	)	)	PUNCT
ejpam-6452	45	161	,	,	PUNCT
ejpam-6452	45	162	(	(	PUNCT
ejpam-6452	45	163	vii	vii	PROPN
ejpam-6452	45	164	)	)	PUNCT
ejpam-6452	45	165	qn	qn	NOUN
ejpam-6452	45	166	=	=	PUNCT
ejpam-6452	45	167	l2,n	l2,n	PROPN
ejpam-6452	45	168	=	=	PUNCT
ejpam-6452	45	169	q2,n	q2,n	PROPN
ejpam-6452	45	170	=	=	PUNCT
ejpam-6452	45	171	gp2,n(2	gp2,n(2	X
ejpam-6452	45	172	,	,	PUNCT
ejpam-6452	45	173	2	2	NUM
ejpam-6452	45	174	)	)	PUNCT
ejpam-6452	45	175	,	,	PUNCT
ejpam-6452	45	176	and	and	CCONJ
ejpam-6452	45	177	(	(	PUNCT
ejpam-6452	45	178	viii	viii	NOUN
ejpam-6452	45	179	)	)	PUNCT
ejpam-6452	45	180	qn	qn	PROPN
ejpam-6452	45	181	=	=	PROPN
ejpam-6452	45	182	gp2,n(1	gp2,n(1	PROPN
ejpam-6452	45	183	,	,	PUNCT
ejpam-6452	45	184	1	1	NUM
ejpam-6452	45	185	)	)	PUNCT
ejpam-6452	45	186	.	.	PUNCT
ejpam-6452	46	1	the	the	DET
ejpam-6452	46	2	binet	binet	NOUN
ejpam-6452	46	3	’s	’s	PART
ejpam-6452	46	4	formula	formula	NOUN
ejpam-6452	46	5	for	for	ADP
ejpam-6452	46	6	gpk	gpk	NOUN
ejpam-6452	46	7	,	,	PUNCT
ejpam-6452	46	8	n	n	X
ejpam-6452	46	9	is	be	AUX
ejpam-6452	46	10	given	give	VERB
ejpam-6452	46	11	in	in	ADP
ejpam-6452	46	12	the	the	DET
ejpam-6452	46	13	following	follow	VERB
ejpam-6452	46	14	theorem	theorem	NOUN
ejpam-6452	46	15	.	.	PUNCT
ejpam-6452	47	1	theorem	theorem	ADJ
ejpam-6452	47	2	1	1	NUM
ejpam-6452	47	3	(	(	PUNCT
ejpam-6452	47	4	binet	binet	PROPN
ejpam-6452	47	5	’s	’s	PART
ejpam-6452	47	6	formulas	formula	NOUN
ejpam-6452	47	7	)	)	PUNCT
ejpam-6452	47	8	.	.	PUNCT
ejpam-6452	48	1	let	let	VERB
ejpam-6452	48	2	k	k	NOUN
ejpam-6452	48	3	and	and	CCONJ
ejpam-6452	48	4	n	n	ADV
ejpam-6452	48	5	be	be	VERB
ejpam-6452	48	6	non	non	ADJ
ejpam-6452	48	7	-	-	ADJ
ejpam-6452	48	8	negative	negative	ADJ
ejpam-6452	48	9	integers	integer	NOUN
ejpam-6452	48	10	with	with	ADP
ejpam-6452	48	11	k	k	PROPN
ejpam-6452	48	12	≥	≥	NUM
ejpam-6452	48	13	2	2	NUM
ejpam-6452	48	14	and	and	CCONJ
ejpam-6452	48	15	∆k	∆k	PROPN
ejpam-6452	48	16	=	=	SYM
ejpam-6452	48	17	√	√	PROPN
ejpam-6452	48	18	k2	k2	NOUN
ejpam-6452	49	1	+	+	CCONJ
ejpam-6452	49	2	4k	4k	NOUN
ejpam-6452	49	3	−	−	NOUN
ejpam-6452	50	1	4	4	X
ejpam-6452	50	2	.	.	PUNCT
ejpam-6452	50	3	then	then	ADV
ejpam-6452	50	4	gpk	gpk	PROPN
ejpam-6452	50	5	,	,	PUNCT
ejpam-6452	50	6	n	n	PROPN
ejpam-6452	50	7	=	=	SYM
ejpam-6452	50	8	2b−	2b−	PROPN
ejpam-6452	50	9	ak	ak	PROPN
ejpam-6452	50	10	+	+	CCONJ
ejpam-6452	50	11	a∆k	a∆k	ADJ
ejpam-6452	50	12	2∆k	2∆k	NUM
ejpam-6452	50	13	σn	σn	PROPN
ejpam-6452	51	1	k	k	PROPN
ejpam-6452	51	2	+	+	PROPN
ejpam-6452	51	3	ak	ak	PROPN
ejpam-6452	51	4	−	−	PROPN
ejpam-6452	51	5	2b	2b	NOUN
ejpam-6452	51	6	+	+	CCONJ
ejpam-6452	51	7	a∆k	a∆k	ADJ
ejpam-6452	51	8	2∆k	2∆k	NUM
ejpam-6452	51	9	(	(	PUNCT
ejpam-6452	51	10	1	1	NUM
ejpam-6452	51	11	−	−	NOUN
ejpam-6452	51	12	k)n	k)n	NOUN
ejpam-6452	51	13	σn	σn	PROPN
ejpam-6452	51	14	k	k	PROPN
ejpam-6452	51	15	,	,	PUNCT
ejpam-6452	51	16	(	(	PUNCT
ejpam-6452	51	17	3	3	X
ejpam-6452	51	18	)	)	PUNCT
ejpam-6452	51	19	where	where	SCONJ
ejpam-6452	51	20	σk	σk	PROPN
ejpam-6452	51	21	=	=	SYM
ejpam-6452	51	22	k+∆k	k+∆k	NOUN
ejpam-6452	51	23	2	2	NUM
ejpam-6452	51	24	.	.	PUNCT
ejpam-6452	52	1	proof	proof	NOUN
ejpam-6452	52	2	.	.	PUNCT
ejpam-6452	53	1	the	the	DET
ejpam-6452	53	2	recurrence	recurrence	NOUN
ejpam-6452	53	3	relation	relation	NOUN
ejpam-6452	53	4	(	(	PUNCT
ejpam-6452	53	5	2	2	X
ejpam-6452	53	6	)	)	PUNCT
ejpam-6452	53	7	generates	generate	VERB
ejpam-6452	53	8	a	a	DET
ejpam-6452	53	9	characteristic	characteristic	ADJ
ejpam-6452	53	10	equation	equation	NOUN
ejpam-6452	53	11	of	of	ADP
ejpam-6452	53	12	the	the	DET
ejpam-6452	53	13	form	form	NOUN
ejpam-6452	53	14	r2	r2	PROPN
ejpam-6452	53	15	−	−	PROPN
ejpam-6452	53	16	kr	kr	PROPN
ejpam-6452	54	1	+	+	CCONJ
ejpam-6452	54	2	1	1	NUM
ejpam-6452	54	3	−	−	PROPN
ejpam-6452	54	4	k	k	NOUN
ejpam-6452	54	5	=	=	NOUN
ejpam-6452	54	6	0	0	PROPN
ejpam-6452	54	7	.	.	PUNCT
ejpam-6452	55	1	(	(	PUNCT
ejpam-6452	55	2	4	4	NUM
ejpam-6452	55	3	)	)	PUNCT
ejpam-6452	55	4	since	since	SCONJ
ejpam-6452	55	5	∆2	∆2	X
ejpam-6452	55	6	k	k	PROPN
ejpam-6452	55	7	=	=	PROPN
ejpam-6452	55	8	k2	k2	PROPN
ejpam-6452	55	9	+	+	CCONJ
ejpam-6452	55	10	4k	4k	NOUN
ejpam-6452	55	11	−	−	NOUN
ejpam-6452	55	12	4	4	NUM
ejpam-6452	55	13	=	=	SYM
ejpam-6452	55	14	k2	k2	PROPN
ejpam-6452	55	15	+	+	CCONJ
ejpam-6452	55	16	4(k	4(k	NUM
ejpam-6452	55	17	−	−	NOUN
ejpam-6452	55	18	1	1	NUM
ejpam-6452	55	19	)	)	PUNCT
ejpam-6452	55	20	>	>	X
ejpam-6452	55	21	0	0	PUNCT
ejpam-6452	56	1	for	for	ADP
ejpam-6452	56	2	k	k	PROPN
ejpam-6452	56	3	≥	≥	NUM
ejpam-6452	56	4	2	2	NUM
ejpam-6452	56	5	,	,	PUNCT
ejpam-6452	56	6	this	this	DET
ejpam-6452	56	7	equation	equation	NOUN
ejpam-6452	56	8	has	have	VERB
ejpam-6452	56	9	two	two	NUM
ejpam-6452	56	10	roots	root	NOUN
ejpam-6452	56	11	,	,	PUNCT
ejpam-6452	57	1	r1	r1	NOUN
ejpam-6452	57	2	=	=	PUNCT
ejpam-6452	57	3	k	k	PROPN
ejpam-6452	58	1	+	+	CCONJ
ejpam-6452	58	2	∆k	∆k	PROPN
ejpam-6452	58	3	2	2	NUM
ejpam-6452	58	4	=	=	SYM
ejpam-6452	58	5	σk	σk	NOUN
ejpam-6452	58	6	and	and	CCONJ
ejpam-6452	58	7	r2	r2	PROPN
ejpam-6452	58	8	=	=	SYM
ejpam-6452	59	1	k	k	PROPN
ejpam-6452	60	1	−	−	PROPN
ejpam-6452	60	2	∆k	∆k	PROPN
ejpam-6452	60	3	2	2	NUM
ejpam-6452	60	4	=	=	SYM
ejpam-6452	60	5	(	(	PUNCT
ejpam-6452	60	6	k	k	NOUN
ejpam-6452	60	7	−	−	PROPN
ejpam-6452	60	8	∆k	∆k	PROPN
ejpam-6452	60	9	)	)	PUNCT
ejpam-6452	60	10	(	(	PUNCT
ejpam-6452	60	11	k	k	X
ejpam-6452	60	12	+	+	CCONJ
ejpam-6452	60	13	∆k	∆k	PROPN
ejpam-6452	60	14	)	)	PUNCT
ejpam-6452	60	15	2	2	NUM
ejpam-6452	60	16	(	(	PUNCT
ejpam-6452	60	17	k	k	NOUN
ejpam-6452	60	18	+	+	CCONJ
ejpam-6452	60	19	∆k	∆k	PROPN
ejpam-6452	60	20	)	)	PUNCT
ejpam-6452	60	21	=	=	SYM
ejpam-6452	60	22	2(1	2(1	NUM
ejpam-6452	60	23	−	−	NOUN
ejpam-6452	60	24	k	k	NOUN
ejpam-6452	60	25	)	)	PUNCT
ejpam-6452	60	26	(	(	PUNCT
ejpam-6452	60	27	k	k	X
ejpam-6452	60	28	+	+	CCONJ
ejpam-6452	60	29	∆k	∆k	PROPN
ejpam-6452	60	30	)	)	PUNCT
ejpam-6452	60	31	=	=	SYM
ejpam-6452	60	32	1	1	NUM
ejpam-6452	60	33	−	−	NOUN
ejpam-6452	60	34	k	k	X
ejpam-6452	60	35	σk	σk	PROPN
ejpam-6452	60	36	.	.	PUNCT
ejpam-6452	61	1	w.	w.	PROPN
ejpam-6452	61	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	61	3	,	,	PUNCT
ejpam-6452	61	4	a.	a.	NOUN
ejpam-6452	61	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	61	6	/	/	SYM
ejpam-6452	61	7	eur	eur	PROPN
ejpam-6452	61	8	.	.	PUNCT
ejpam-6452	62	1	j.	j.	PROPN
ejpam-6452	62	2	pure	pure	PROPN
ejpam-6452	62	3	appl	appl	PROPN
ejpam-6452	62	4	.	.	PROPN
ejpam-6452	62	5	math	math	PROPN
ejpam-6452	62	6	,	,	PUNCT
ejpam-6452	62	7	18	18	NUM
ejpam-6452	62	8	(	(	PUNCT
ejpam-6452	62	9	3	3	NUM
ejpam-6452	62	10	)	)	PUNCT
ejpam-6452	62	11	(	(	PUNCT
ejpam-6452	62	12	2025	2025	NUM
ejpam-6452	62	13	)	)	PUNCT
ejpam-6452	62	14	,	,	PUNCT
ejpam-6452	62	15	6452	6452	NUM
ejpam-6452	62	16	5	5	NUM
ejpam-6452	62	17	of	of	ADP
ejpam-6452	62	18	18	18	NUM
ejpam-6452	62	19	note	note	NOUN
ejpam-6452	63	1	that	that	SCONJ
ejpam-6452	63	2	r1	r1	PROPN
ejpam-6452	63	3	+	+	CCONJ
ejpam-6452	63	4	r2	r2	PROPN
ejpam-6452	63	5	=	=	SYM
ejpam-6452	63	6	k	k	PROPN
ejpam-6452	63	7	,	,	PUNCT
ejpam-6452	63	8	r1	r1	NOUN
ejpam-6452	63	9	−	−	PROPN
ejpam-6452	63	10	r2	r2	PROPN
ejpam-6452	63	11	=	=	SYM
ejpam-6452	63	12	∆k	∆k	PROPN
ejpam-6452	63	13	,	,	PUNCT
ejpam-6452	63	14	and	and	CCONJ
ejpam-6452	63	15	r1r2	r1r2	NOUN
ejpam-6452	63	16	=	=	SYM
ejpam-6452	63	17	1−	1−	NUM
ejpam-6452	63	18	k.	k.	NOUN
ejpam-6452	63	19	therefore	therefore	ADV
ejpam-6452	63	20	,	,	PUNCT
ejpam-6452	63	21	the	the	DET
ejpam-6452	63	22	general	general	ADJ
ejpam-6452	63	23	term	term	NOUN
ejpam-6452	63	24	gpk	gpk	PROPN
ejpam-6452	63	25	,	,	PUNCT
ejpam-6452	63	26	n	n	PRON
ejpam-6452	63	27	can	can	AUX
ejpam-6452	63	28	be	be	AUX
ejpam-6452	63	29	expressed	express	VERB
ejpam-6452	63	30	as	as	ADP
ejpam-6452	63	31	gpk	gpk	NOUN
ejpam-6452	63	32	,	,	PUNCT
ejpam-6452	63	33	n	n	NOUN
ejpam-6452	63	34	=	=	PUNCT
ejpam-6452	63	35	αrn1	αrn1	VERB
ejpam-6452	63	36	+	+	CCONJ
ejpam-6452	64	1	βrn2	βrn2	NOUN
ejpam-6452	64	2	=	=	SYM
ejpam-6452	64	3	ασn	ασn	NOUN
ejpam-6452	65	1	k	k	NOUN
ejpam-6452	66	1	+	+	CCONJ
ejpam-6452	66	2	β	β	X
ejpam-6452	66	3	(	(	PUNCT
ejpam-6452	66	4	1	1	NUM
ejpam-6452	66	5	−	−	NOUN
ejpam-6452	66	6	k)n	k)n	NOUN
ejpam-6452	66	7	σn	σn	PROPN
ejpam-6452	66	8	k	k	NOUN
ejpam-6452	66	9	for	for	ADP
ejpam-6452	66	10	some	some	DET
ejpam-6452	66	11	coefficients	coefficient	NOUN
ejpam-6452	66	12	α	α	PROPN
ejpam-6452	66	13	and	and	CCONJ
ejpam-6452	66	14	β	β	X
ejpam-6452	66	15	.	.	PUNCT
ejpam-6452	67	1	since	since	SCONJ
ejpam-6452	67	2	gpk,0	gpk,0	PROPN
ejpam-6452	67	3	=	=	PROPN
ejpam-6452	67	4	a	a	PRON
ejpam-6452	67	5	and	and	CCONJ
ejpam-6452	67	6	gpk,1	gpk,1	NOUN
ejpam-6452	67	7	=	=	SYM
ejpam-6452	67	8	b	b	NOUN
ejpam-6452	67	9	,	,	PUNCT
ejpam-6452	67	10	we	we	PRON
ejpam-6452	67	11	get	get	VERB
ejpam-6452	67	12	α	α	NOUN
ejpam-6452	67	13	+	+	X
ejpam-6452	67	14	β	β	X
ejpam-6452	67	15	=	=	PUNCT
ejpam-6452	67	16	a	a	PROPN
ejpam-6452	67	17	and	and	CCONJ
ejpam-6452	67	18	αr1	αr1	PRON
ejpam-6452	67	19	+	+	CCONJ
ejpam-6452	67	20	βr2	βr2	PROPN
ejpam-6452	68	1	=	=	SYM
ejpam-6452	68	2	b.	b.	PROPN
ejpam-6452	69	1	it	it	PRON
ejpam-6452	69	2	can	can	AUX
ejpam-6452	69	3	be	be	AUX
ejpam-6452	69	4	shown	show	VERB
ejpam-6452	69	5	that	that	SCONJ
ejpam-6452	69	6	α	α	PRON
ejpam-6452	69	7	=	=	SYM
ejpam-6452	69	8	2b−	2b−	PROPN
ejpam-6452	69	9	ak	ak	PROPN
ejpam-6452	69	10	+	+	NUM
ejpam-6452	69	11	a∆k	a∆k	ADJ
ejpam-6452	69	12	2∆k	2∆k	NUM
ejpam-6452	69	13	and	and	CCONJ
ejpam-6452	69	14	β	β	X
ejpam-6452	69	15	=	=	SYM
ejpam-6452	69	16	ak	ak	PROPN
ejpam-6452	69	17	−	−	PROPN
ejpam-6452	69	18	2b	2b	NOUN
ejpam-6452	69	19	+	+	CCONJ
ejpam-6452	69	20	a∆k	a∆k	ADJ
ejpam-6452	69	21	2∆k	2∆k	NUM
ejpam-6452	69	22	.	.	PUNCT
ejpam-6452	70	1	therefore	therefore	ADV
ejpam-6452	70	2	,	,	PUNCT
ejpam-6452	70	3	(	(	PUNCT
ejpam-6452	70	4	3	3	X
ejpam-6452	70	5	)	)	PUNCT
ejpam-6452	70	6	has	have	AUX
ejpam-6452	70	7	been	be	AUX
ejpam-6452	70	8	proved	prove	VERB
ejpam-6452	70	9	.	.	PUNCT
ejpam-6452	71	1	if	if	SCONJ
ejpam-6452	71	2	(	(	PUNCT
ejpam-6452	71	3	a	a	PRON
ejpam-6452	71	4	,	,	PUNCT
ejpam-6452	71	5	b	b	NOUN
ejpam-6452	71	6	)	)	PUNCT
ejpam-6452	71	7	∈	∈	NOUN
ejpam-6452	71	8	{	{	PUNCT
ejpam-6452	71	9	(	(	PUNCT
ejpam-6452	71	10	0	0	NUM
ejpam-6452	71	11	,	,	PUNCT
ejpam-6452	71	12	1	1	NUM
ejpam-6452	71	13	)	)	PUNCT
ejpam-6452	71	14	,	,	PUNCT
ejpam-6452	71	15	(	(	PUNCT
ejpam-6452	71	16	2	2	NUM
ejpam-6452	71	17	,	,	PUNCT
ejpam-6452	71	18	k	k	NOUN
ejpam-6452	71	19	)	)	PUNCT
ejpam-6452	71	20	,	,	PUNCT
ejpam-6452	71	21	(	(	PUNCT
ejpam-6452	71	22	2	2	NUM
ejpam-6452	71	23	,	,	PUNCT
ejpam-6452	71	24	2	2	NUM
ejpam-6452	71	25	)	)	PUNCT
ejpam-6452	71	26	,	,	PUNCT
ejpam-6452	71	27	(	(	PUNCT
ejpam-6452	71	28	1	1	NUM
ejpam-6452	71	29	,	,	PUNCT
ejpam-6452	71	30	1	1	NUM
ejpam-6452	71	31	)	)	PUNCT
ejpam-6452	71	32	}	}	PUNCT
ejpam-6452	71	33	,	,	PUNCT
ejpam-6452	71	34	then	then	ADV
ejpam-6452	71	35	we	we	PRON
ejpam-6452	71	36	have	have	VERB
ejpam-6452	71	37	the	the	DET
ejpam-6452	71	38	following	follow	VERB
ejpam-6452	71	39	:	:	PUNCT
ejpam-6452	71	40	corollary	corollary	ADJ
ejpam-6452	71	41	1	1	X
ejpam-6452	71	42	.	.	PUNCT
ejpam-6452	72	1	let	let	VERB
ejpam-6452	72	2	k	k	NOUN
ejpam-6452	72	3	and	and	CCONJ
ejpam-6452	72	4	n	n	ADV
ejpam-6452	72	5	be	be	VERB
ejpam-6452	72	6	non	non	ADJ
ejpam-6452	72	7	-	-	ADJ
ejpam-6452	72	8	negative	negative	ADJ
ejpam-6452	72	9	integers	integer	NOUN
ejpam-6452	72	10	with	with	ADP
ejpam-6452	72	11	k	k	PROPN
ejpam-6452	72	12	≥	≥	NUM
ejpam-6452	72	13	2	2	NUM
ejpam-6452	72	14	and	and	CCONJ
ejpam-6452	72	15	∆k	∆k	PROPN
ejpam-6452	72	16	=	=	SYM
ejpam-6452	72	17	√	√	PROPN
ejpam-6452	72	18	k2	k2	NOUN
ejpam-6452	73	1	+	+	CCONJ
ejpam-6452	73	2	4k	4k	NOUN
ejpam-6452	73	3	−	−	NOUN
ejpam-6452	74	1	4	4	X
ejpam-6452	74	2	.	.	PUNCT
ejpam-6452	75	1	then	then	ADV
ejpam-6452	75	2	(	(	PUNCT
ejpam-6452	75	3	i	i	NOUN
ejpam-6452	75	4	)	)	PUNCT
ejpam-6452	75	5	pk	pk	PROPN
ejpam-6452	75	6	,	,	PUNCT
ejpam-6452	75	7	n	n	NOUN
ejpam-6452	75	8	=	=	SYM
ejpam-6452	75	9	1	1	NUM
ejpam-6452	75	10	∆k	∆k	PROPN
ejpam-6452	75	11	(	(	PUNCT
ejpam-6452	75	12	σn	σn	NOUN
ejpam-6452	75	13	k	k	PROPN
ejpam-6452	75	14	−	−	PROPN
ejpam-6452	75	15	(	(	PUNCT
ejpam-6452	75	16	1−k)n	1−k)n	NUM
ejpam-6452	75	17	σn	σn	NOUN
ejpam-6452	75	18	k	k	PROPN
ejpam-6452	75	19	)	)	PUNCT
ejpam-6452	75	20	,	,	PUNCT
ejpam-6452	75	21	(	(	PUNCT
ejpam-6452	75	22	ii	ii	NOUN
ejpam-6452	75	23	)	)	PUNCT
ejpam-6452	75	24	lk	lk	PROPN
ejpam-6452	75	25	,	,	PUNCT
ejpam-6452	75	26	n	n	NOUN
ejpam-6452	75	27	=	=	NOUN
ejpam-6452	75	28	σn	σn	X
ejpam-6452	75	29	k	k	PROPN
ejpam-6452	76	1	+	+	CCONJ
ejpam-6452	76	2	(	(	PUNCT
ejpam-6452	76	3	1−k)n	1−k)n	NUM
ejpam-6452	76	4	σn	σn	NOUN
ejpam-6452	76	5	k	k	PROPN
ejpam-6452	76	6	,	,	PUNCT
ejpam-6452	76	7	(	(	PUNCT
ejpam-6452	76	8	iii	iii	NOUN
ejpam-6452	76	9	)	)	PUNCT
ejpam-6452	76	10	qk	qk	NOUN
ejpam-6452	76	11	,	,	PUNCT
ejpam-6452	76	12	n	n	NOUN
ejpam-6452	76	13	=	=	SYM
ejpam-6452	76	14	(	(	PUNCT
ejpam-6452	76	15	2−k+∆k	2−k+∆k	NUM
ejpam-6452	76	16	∆k	∆k	NOUN
ejpam-6452	76	17	)	)	PUNCT
ejpam-6452	76	18	σn	σn	PROPN
ejpam-6452	76	19	k	k	PROPN
ejpam-6452	77	1	+	+	CCONJ
ejpam-6452	77	2	(	(	PUNCT
ejpam-6452	77	3	k−2∆k	k−2∆k	PROPN
ejpam-6452	77	4	∆k	∆k	PROPN
ejpam-6452	77	5	)	)	PUNCT
ejpam-6452	77	6	(	(	PUNCT
ejpam-6452	77	7	1−k)n	1−k)n	NUM
ejpam-6452	77	8	σn	σn	NOUN
ejpam-6452	77	9	k	k	PROPN
ejpam-6452	77	10	,	,	PUNCT
ejpam-6452	77	11	and	and	CCONJ
ejpam-6452	77	12	(	(	PUNCT
ejpam-6452	77	13	iv	iv	X
ejpam-6452	77	14	)	)	PUNCT
ejpam-6452	77	15	qk	qk	NOUN
ejpam-6452	77	16	,	,	PUNCT
ejpam-6452	77	17	n	n	NOUN
ejpam-6452	77	18	=	=	SYM
ejpam-6452	77	19	(	(	PUNCT
ejpam-6452	77	20	2−k+∆k	2−k+∆k	NUM
ejpam-6452	77	21	2∆k	2∆k	NUM
ejpam-6452	77	22	)	)	PUNCT
ejpam-6452	77	23	σn	σn	PROPN
ejpam-6452	77	24	k	k	PROPN
ejpam-6452	78	1	+	+	CCONJ
ejpam-6452	78	2	(	(	PUNCT
ejpam-6452	78	3	k−2∆k	k−2∆k	NOUN
ejpam-6452	78	4	2∆k	2∆k	NUM
ejpam-6452	78	5	)	)	PUNCT
ejpam-6452	78	6	(	(	PUNCT
ejpam-6452	78	7	1−k)n	1−k)n	NUM
ejpam-6452	78	8	σn	σn	PROPN
ejpam-6452	79	1	k	k	PROPN
ejpam-6452	79	2	.	.	PUNCT
ejpam-6452	80	1	remark	remark	PROPN
ejpam-6452	80	2	1	1	NUM
ejpam-6452	80	3	.	.	PUNCT
ejpam-6452	81	1	as	as	ADP
ejpam-6452	81	2	in	in	ADP
ejpam-6452	81	3	corollary	corollary	ADJ
ejpam-6452	81	4	1	1	NUM
ejpam-6452	81	5	,	,	PUNCT
ejpam-6452	81	6	we	we	PRON
ejpam-6452	81	7	get	get	VERB
ejpam-6452	81	8	the	the	DET
ejpam-6452	81	9	following	following	NOUN
ejpam-6452	81	10	:	:	PUNCT
ejpam-6452	81	11	•	•	X
ejpam-6452	81	12	(	(	PUNCT
ejpam-6452	81	13	i	i	NOUN
ejpam-6452	81	14	)	)	PUNCT
ejpam-6452	81	15	and	and	CCONJ
ejpam-6452	81	16	(	(	PUNCT
ejpam-6452	81	17	iii	iii	X
ejpam-6452	81	18	)	)	PUNCT
ejpam-6452	81	19	are	be	AUX
ejpam-6452	81	20	presented	present	VERB
ejpam-6452	81	21	in	in	ADP
ejpam-6452	81	22	[	[	X
ejpam-6452	81	23	5	5	NUM
ejpam-6452	81	24	,	,	PUNCT
ejpam-6452	81	25	corollary	corollary	ADJ
ejpam-6452	81	26	2.3	2.3	NUM
ejpam-6452	81	27	]	]	PUNCT
ejpam-6452	81	28	;	;	PUNCT
ejpam-6452	81	29	•	•	X
ejpam-6452	81	30	(	(	PUNCT
ejpam-6452	81	31	iv	iv	X
ejpam-6452	81	32	)	)	PUNCT
ejpam-6452	81	33	is	be	AUX
ejpam-6452	81	34	presented	present	VERB
ejpam-6452	81	35	in	in	ADP
ejpam-6452	81	36	[	[	X
ejpam-6452	81	37	6	6	NUM
ejpam-6452	81	38	,	,	PUNCT
ejpam-6452	81	39	theorem	theorem	VERB
ejpam-6452	81	40	2.3	2.3	NUM
ejpam-6452	81	41	]	]	PUNCT
ejpam-6452	81	42	;	;	PUNCT
ejpam-6452	81	43	•	•	NUM
ejpam-6452	81	44	the	the	DET
ejpam-6452	81	45	binet	binet	NOUN
ejpam-6452	81	46	’s	’s	PART
ejpam-6452	81	47	formula	formula	NOUN
ejpam-6452	81	48	of	of	ADP
ejpam-6452	81	49	lk	lk	PROPN
ejpam-6452	81	50	,	,	PUNCT
ejpam-6452	81	51	n	n	X
ejpam-6452	81	52	is	be	AUX
ejpam-6452	81	53	simpler	simple	ADJ
ejpam-6452	81	54	than	than	ADP
ejpam-6452	81	55	that	that	PRON
ejpam-6452	81	56	of	of	ADP
ejpam-6452	81	57	qk	qk	PROPN
ejpam-6452	81	58	,	,	PUNCT
ejpam-6452	81	59	n	n	PROPN
ejpam-6452	81	60	and	and	CCONJ
ejpam-6452	81	61	qk	qk	INTJ
ejpam-6452	81	62	,	,	PUNCT
ejpam-6452	81	63	n.	n.	NOUN
ejpam-6452	81	64	if	if	SCONJ
ejpam-6452	81	65	k	k	PROPN
ejpam-6452	81	66	=	=	SYM
ejpam-6452	81	67	2	2	NUM
ejpam-6452	81	68	,	,	PUNCT
ejpam-6452	81	69	then	then	ADV
ejpam-6452	81	70	we	we	PRON
ejpam-6452	81	71	have	have	VERB
ejpam-6452	81	72	the	the	DET
ejpam-6452	81	73	following	follow	VERB
ejpam-6452	81	74	:	:	PUNCT
ejpam-6452	81	75	corollary	corollary	ADJ
ejpam-6452	81	76	2	2	X
ejpam-6452	81	77	.	.	PUNCT
ejpam-6452	82	1	let	let	VERB
ejpam-6452	82	2	n	n	PRON
ejpam-6452	82	3	be	be	AUX
ejpam-6452	82	4	a	a	DET
ejpam-6452	82	5	non	non	ADJ
ejpam-6452	82	6	-	-	ADJ
ejpam-6452	82	7	negative	negative	ADJ
ejpam-6452	82	8	integer	integer	NOUN
ejpam-6452	82	9	.	.	PUNCT
ejpam-6452	83	1	then	then	ADV
ejpam-6452	83	2	ha	ha	INTJ
ejpam-6452	83	3	,	,	PUNCT
ejpam-6452	83	4	b	b	PROPN
ejpam-6452	83	5	n	n	NOUN
ejpam-6452	83	6	=	=	SYM
ejpam-6452	83	7	b−	b−	PROPN
ejpam-6452	83	8	a	a	PRON
ejpam-6452	83	9	+	+	NOUN
ejpam-6452	83	10	√	√	NUM
ejpam-6452	83	11	2a	2a	NUM
ejpam-6452	83	12	2	2	NUM
ejpam-6452	83	13	√	√	NUM
ejpam-6452	83	14	2	2	NUM
ejpam-6452	83	15	(	(	PUNCT
ejpam-6452	83	16	1	1	NUM
ejpam-6452	83	17	+	+	NUM
ejpam-6452	83	18	√	√	NUM
ejpam-6452	83	19	2)n	2)n	NUM
ejpam-6452	83	20	+	+	ADP
ejpam-6452	83	21	a−	a−	PROPN
ejpam-6452	83	22	b−	b−	NOUN
ejpam-6452	83	23	√	√	PROPN
ejpam-6452	83	24	2a	2a	NUM
ejpam-6452	83	25	2	2	NUM
ejpam-6452	83	26	√	√	NUM
ejpam-6452	83	27	2	2	NUM
ejpam-6452	83	28	(	(	PUNCT
ejpam-6452	83	29	1	1	NUM
ejpam-6452	83	30	−	−	NUM
ejpam-6452	83	31	√	√	NUM
ejpam-6452	83	32	2)n	2)n	NUM
ejpam-6452	83	33	.	.	PUNCT
ejpam-6452	84	1	(	(	PUNCT
ejpam-6452	84	2	5	5	X
ejpam-6452	84	3	)	)	PUNCT
ejpam-6452	84	4	proof	proof	NOUN
ejpam-6452	84	5	.	.	PUNCT
ejpam-6452	85	1	notice	notice	VERB
ejpam-6452	85	2	that	that	SCONJ
ejpam-6452	85	3	gp2,n	gp2,n	NOUN
ejpam-6452	85	4	=	=	SYM
ejpam-6452	85	5	ha	ha	INTJ
ejpam-6452	85	6	,	,	PUNCT
ejpam-6452	85	7	b	b	PROPN
ejpam-6452	85	8	n	n	PROPN
ejpam-6452	85	9	,	,	PUNCT
ejpam-6452	85	10	∆2	∆2	X
ejpam-6452	85	11	=	=	SYM
ejpam-6452	85	12	2	2	NUM
ejpam-6452	85	13	√	√	NUM
ejpam-6452	85	14	2	2	NUM
ejpam-6452	85	15	and	and	CCONJ
ejpam-6452	85	16	σ2	σ2	NOUN
ejpam-6452	85	17	=	=	SYM
ejpam-6452	85	18	1	1	NUM
ejpam-6452	85	19	+	+	CCONJ
ejpam-6452	85	20	√	√	NUM
ejpam-6452	85	21	2	2	NUM
ejpam-6452	85	22	.	.	PUNCT
ejpam-6452	85	23	setting	set	VERB
ejpam-6452	85	24	k	k	X
ejpam-6452	85	25	=	=	SYM
ejpam-6452	85	26	2	2	NUM
ejpam-6452	85	27	in	in	ADP
ejpam-6452	85	28	(	(	PUNCT
ejpam-6452	85	29	3	3	NUM
ejpam-6452	85	30	)	)	PUNCT
ejpam-6452	85	31	,	,	PUNCT
ejpam-6452	85	32	we	we	PRON
ejpam-6452	85	33	get	get	VERB
ejpam-6452	85	34	(	(	PUNCT
ejpam-6452	85	35	5	5	NUM
ejpam-6452	85	36	)	)	PUNCT
ejpam-6452	85	37	which	which	PRON
ejpam-6452	85	38	completes	complete	VERB
ejpam-6452	85	39	the	the	DET
ejpam-6452	85	40	proof	proof	NOUN
ejpam-6452	85	41	.	.	PUNCT
ejpam-6452	86	1	theorem	theorem	NOUN
ejpam-6452	86	2	2	2	NUM
ejpam-6452	86	3	.	.	PUNCT
ejpam-6452	87	1	let	let	VERB
ejpam-6452	87	2	k	k	NOUN
ejpam-6452	87	3	and	and	CCONJ
ejpam-6452	87	4	n	n	ADV
ejpam-6452	87	5	be	be	VERB
ejpam-6452	87	6	non	non	ADJ
ejpam-6452	87	7	-	-	ADJ
ejpam-6452	87	8	negative	negative	ADJ
ejpam-6452	87	9	integers	integer	NOUN
ejpam-6452	87	10	with	with	ADP
ejpam-6452	87	11	k	k	PROPN
ejpam-6452	87	12	≥	≥	NUM
ejpam-6452	87	13	2	2	NUM
ejpam-6452	87	14	and	and	CCONJ
ejpam-6452	87	15	∆k	∆k	PROPN
ejpam-6452	87	16	=	=	SYM
ejpam-6452	87	17	√	√	PROPN
ejpam-6452	87	18	k2	k2	NOUN
ejpam-6452	88	1	+	+	CCONJ
ejpam-6452	88	2	4k	4k	NOUN
ejpam-6452	88	3	−	−	NOUN
ejpam-6452	89	1	4	4	X
ejpam-6452	89	2	.	.	PUNCT
ejpam-6452	90	1	then	then	ADV
ejpam-6452	90	2	the	the	DET
ejpam-6452	90	3	following	follow	VERB
ejpam-6452	90	4	hold	hold	NOUN
ejpam-6452	90	5	:	:	PUNCT
ejpam-6452	90	6	w.	w.	NOUN
ejpam-6452	90	7	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	90	8	,	,	PUNCT
ejpam-6452	90	9	a.	a.	NOUN
ejpam-6452	90	10	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	90	11	/	/	SYM
ejpam-6452	90	12	eur	eur	PROPN
ejpam-6452	90	13	.	.	PUNCT
ejpam-6452	91	1	j.	j.	PROPN
ejpam-6452	91	2	pure	pure	PROPN
ejpam-6452	91	3	appl	appl	PROPN
ejpam-6452	91	4	.	.	PROPN
ejpam-6452	91	5	math	math	PROPN
ejpam-6452	91	6	,	,	PUNCT
ejpam-6452	91	7	18	18	NUM
ejpam-6452	91	8	(	(	PUNCT
ejpam-6452	91	9	3	3	NUM
ejpam-6452	91	10	)	)	PUNCT
ejpam-6452	91	11	(	(	PUNCT
ejpam-6452	91	12	2025	2025	NUM
ejpam-6452	91	13	)	)	PUNCT
ejpam-6452	91	14	,	,	PUNCT
ejpam-6452	91	15	6452	6452	NUM
ejpam-6452	91	16	6	6	NUM
ejpam-6452	91	17	of	of	ADP
ejpam-6452	91	18	18	18	NUM
ejpam-6452	91	19	(	(	PUNCT
ejpam-6452	91	20	i	i	NOUN
ejpam-6452	91	21	)	)	PUNCT
ejpam-6452	91	22	lk	lk	PROPN
ejpam-6452	91	23	,	,	PUNCT
ejpam-6452	91	24	n	n	PROPN
ejpam-6452	91	25	+	+	CCONJ
ejpam-6452	91	26	∆k	∆k	PROPN
ejpam-6452	91	27	pk	pk	NOUN
ejpam-6452	91	28	,	,	PUNCT
ejpam-6452	91	29	n	n	NOUN
ejpam-6452	91	30	=	=	SYM
ejpam-6452	91	31	2σn	2σn	PROPN
ejpam-6452	92	1	k	k	X
ejpam-6452	92	2	;	;	PUNCT
ejpam-6452	92	3	(	(	PUNCT
ejpam-6452	92	4	ii	ii	NOUN
ejpam-6452	92	5	)	)	PUNCT
ejpam-6452	92	6	lk	lk	PROPN
ejpam-6452	92	7	,	,	PUNCT
ejpam-6452	92	8	n	n	CCONJ
ejpam-6452	92	9	−	−	PROPN
ejpam-6452	92	10	∆k	∆k	PROPN
ejpam-6452	92	11	pk	pk	NOUN
ejpam-6452	92	12	,	,	PUNCT
ejpam-6452	92	13	n	n	NOUN
ejpam-6452	92	14	=	=	SYM
ejpam-6452	92	15	2	2	NUM
ejpam-6452	92	16	(	(	PUNCT
ejpam-6452	92	17	1−k)n	1−k)n	NUM
ejpam-6452	92	18	σn	σn	NOUN
ejpam-6452	92	19	k	k	X
ejpam-6452	92	20	;	;	PUNCT
ejpam-6452	92	21	(	(	PUNCT
ejpam-6452	92	22	iii	iii	X
ejpam-6452	92	23	)	)	PUNCT
ejpam-6452	92	24	l2	l2	NOUN
ejpam-6452	92	25	k	k	NOUN
ejpam-6452	92	26	,	,	PUNCT
ejpam-6452	92	27	n	n	PROPN
ejpam-6452	92	28	−	−	PROPN
ejpam-6452	92	29	∆2	∆2	NOUN
ejpam-6452	92	30	kp2	kp2	VERB
ejpam-6452	92	31	k	k	PROPN
ejpam-6452	92	32	,	,	PUNCT
ejpam-6452	92	33	n	n	PROPN
ejpam-6452	92	34	=	=	SYM
ejpam-6452	92	35	4(1	4(1	NOUN
ejpam-6452	92	36	−	−	NOUN
ejpam-6452	92	37	k)n	k)n	NOUN
ejpam-6452	92	38	.	.	PUNCT
ejpam-6452	93	1	proof	proof	NOUN
ejpam-6452	93	2	.	.	PUNCT
ejpam-6452	94	1	the	the	DET
ejpam-6452	94	2	conclusions	conclusion	NOUN
ejpam-6452	94	3	follow	follow	VERB
ejpam-6452	94	4	from	from	ADP
ejpam-6452	94	5	(	(	PUNCT
ejpam-6452	94	6	i	i	NOUN
ejpam-6452	94	7	)	)	PUNCT
ejpam-6452	94	8	and	and	CCONJ
ejpam-6452	94	9	(	(	PUNCT
ejpam-6452	94	10	ii	ii	NOUN
ejpam-6452	94	11	)	)	PUNCT
ejpam-6452	94	12	of	of	ADP
ejpam-6452	94	13	corollary	corollary	ADJ
ejpam-6452	94	14	1	1	NUM
ejpam-6452	94	15	.	.	PUNCT
ejpam-6452	95	1	next	next	ADV
ejpam-6452	95	2	,	,	PUNCT
ejpam-6452	95	3	we	we	PRON
ejpam-6452	95	4	present	present	VERB
ejpam-6452	95	5	that	that	SCONJ
ejpam-6452	95	6	the	the	DET
ejpam-6452	95	7	companion	companion	NOUN
ejpam-6452	95	8	generalized	generalize	VERB
ejpam-6452	95	9	pell	pell	NOUN
ejpam-6452	95	10	numbers	number	NOUN
ejpam-6452	95	11	are	be	AUX
ejpam-6452	95	12	associated	associate	VERB
ejpam-6452	95	13	with	with	ADP
ejpam-6452	95	14	the	the	DET
ejpam-6452	95	15	generalized	generalized	ADJ
ejpam-6452	95	16	pell	pell	NOUN
ejpam-6452	95	17	and	and	CCONJ
ejpam-6452	95	18	generalized	generalize	VERB
ejpam-6452	95	19	pell	pell	NOUN
ejpam-6452	95	20	-	-	PUNCT
ejpam-6452	95	21	lucas	lucas	NOUN
ejpam-6452	95	22	-	-	PUNCT
ejpam-6452	95	23	like	like	ADJ
ejpam-6452	95	24	numbers	number	NOUN
ejpam-6452	95	25	in	in	ADP
ejpam-6452	95	26	the	the	DET
ejpam-6452	95	27	following	follow	VERB
ejpam-6452	95	28	results	result	NOUN
ejpam-6452	95	29	.	.	PUNCT
ejpam-6452	96	1	theorem	theorem	NOUN
ejpam-6452	96	2	3	3	X
ejpam-6452	96	3	.	.	PUNCT
ejpam-6452	97	1	let	let	VERB
ejpam-6452	97	2	k	k	PROPN
ejpam-6452	97	3	and	and	CCONJ
ejpam-6452	97	4	n	n	ADV
ejpam-6452	97	5	be	be	VERB
ejpam-6452	97	6	non	non	ADJ
ejpam-6452	97	7	-	-	ADJ
ejpam-6452	97	8	negative	negative	ADJ
ejpam-6452	97	9	integers	integer	NOUN
ejpam-6452	97	10	with	with	ADP
ejpam-6452	97	11	k	k	PROPN
ejpam-6452	97	12	≥	≥	NUM
ejpam-6452	97	13	2	2	NUM
ejpam-6452	97	14	.	.	PUNCT
ejpam-6452	98	1	then	then	ADV
ejpam-6452	98	2	gpk	gpk	PROPN
ejpam-6452	98	3	,	,	PUNCT
ejpam-6452	98	4	n	n	PROPN
ejpam-6452	98	5	=	=	PUNCT
ejpam-6452	98	6	a	a	DET
ejpam-6452	98	7	2	2	NUM
ejpam-6452	98	8	lk	lk	NOUN
ejpam-6452	98	9	,	,	PUNCT
ejpam-6452	98	10	n	n	PROPN
ejpam-6452	98	11	+	+	CCONJ
ejpam-6452	98	12	2b−	2b−	PROPN
ejpam-6452	98	13	ak	ak	PROPN
ejpam-6452	98	14	2	2	NUM
ejpam-6452	98	15	pk	pk	NOUN
ejpam-6452	98	16	,	,	PUNCT
ejpam-6452	98	17	n.	n.	NOUN
ejpam-6452	98	18	proof	proof	NOUN
ejpam-6452	98	19	.	.	PUNCT
ejpam-6452	99	1	it	it	PRON
ejpam-6452	99	2	follows	follow	VERB
ejpam-6452	99	3	from	from	ADP
ejpam-6452	99	4	(	(	PUNCT
ejpam-6452	99	5	3	3	NUM
ejpam-6452	99	6	)	)	PUNCT
ejpam-6452	99	7	,	,	PUNCT
ejpam-6452	99	8	(	(	PUNCT
ejpam-6452	99	9	i	i	NOUN
ejpam-6452	99	10	)	)	PUNCT
ejpam-6452	99	11	and	and	CCONJ
ejpam-6452	99	12	(	(	PUNCT
ejpam-6452	99	13	ii	ii	NOUN
ejpam-6452	99	14	)	)	PUNCT
ejpam-6452	99	15	of	of	ADP
ejpam-6452	99	16	theorem	theorem	NOUN
ejpam-6452	99	17	2	2	NUM
ejpam-6452	99	18	that	that	SCONJ
ejpam-6452	99	19	gpk	gpk	NOUN
ejpam-6452	99	20	,	,	PUNCT
ejpam-6452	99	21	n	n	NOUN
ejpam-6452	99	22	=	=	SYM
ejpam-6452	99	23	(	(	PUNCT
ejpam-6452	99	24	2b−	2b−	PROPN
ejpam-6452	99	25	ak	ak	PROPN
ejpam-6452	99	26	+	+	CCONJ
ejpam-6452	99	27	a∆k	a∆k	ADJ
ejpam-6452	99	28	2∆k	2∆k	NUM
ejpam-6452	99	29	)	)	PUNCT
ejpam-6452	99	30	σn	σn	PROPN
ejpam-6452	100	1	k	k	PROPN
ejpam-6452	101	1	+	+	CCONJ
ejpam-6452	101	2	(	(	PUNCT
ejpam-6452	101	3	ak	ak	PROPN
ejpam-6452	101	4	−	−	PROPN
ejpam-6452	101	5	2b	2b	NOUN
ejpam-6452	101	6	+	+	CCONJ
ejpam-6452	101	7	a∆k	a∆k	ADJ
ejpam-6452	101	8	2∆k	2∆k	NUM
ejpam-6452	101	9	)	)	PUNCT
ejpam-6452	101	10	(	(	PUNCT
ejpam-6452	101	11	1	1	NUM
ejpam-6452	101	12	−	−	NOUN
ejpam-6452	101	13	k)n	k)n	NOUN
ejpam-6452	101	14	σn	σn	NOUN
ejpam-6452	101	15	k	k	NOUN
ejpam-6452	102	1	=	=	PRON
ejpam-6452	102	2	(	(	PUNCT
ejpam-6452	102	3	2b−	2b−	PROPN
ejpam-6452	102	4	ak	ak	PROPN
ejpam-6452	102	5	+	+	NUM
ejpam-6452	102	6	a∆k	a∆k	ADJ
ejpam-6452	102	7	2∆k	2∆k	NUM
ejpam-6452	102	8	)	)	PUNCT
ejpam-6452	102	9	(	(	PUNCT
ejpam-6452	102	10	lk	lk	PROPN
ejpam-6452	102	11	,	,	PUNCT
ejpam-6452	102	12	n	n	PROPN
ejpam-6452	102	13	+	+	CCONJ
ejpam-6452	102	14	∆k	∆k	PROPN
ejpam-6452	102	15	pk	pk	NOUN
ejpam-6452	102	16	,	,	PUNCT
ejpam-6452	102	17	n	n	PRON
ejpam-6452	102	18	2	2	NUM
ejpam-6452	102	19	)	)	PUNCT
ejpam-6452	102	20	+	+	CCONJ
ejpam-6452	102	21	(	(	PUNCT
ejpam-6452	102	22	ak	ak	PROPN
ejpam-6452	102	23	−	−	PROPN
ejpam-6452	102	24	2b	2b	NOUN
ejpam-6452	102	25	+	+	CCONJ
ejpam-6452	102	26	a∆k	a∆k	ADJ
ejpam-6452	102	27	2∆k	2∆k	NUM
ejpam-6452	102	28	)	)	PUNCT
ejpam-6452	102	29	(	(	PUNCT
ejpam-6452	102	30	lk	lk	PROPN
ejpam-6452	102	31	,	,	PUNCT
ejpam-6452	102	32	n	n	CCONJ
ejpam-6452	102	33	−	−	PROPN
ejpam-6452	102	34	∆k	∆k	PROPN
ejpam-6452	102	35	pk	pk	NOUN
ejpam-6452	102	36	,	,	PUNCT
ejpam-6452	102	37	n	n	X
ejpam-6452	102	38	2	2	NUM
ejpam-6452	102	39	)	)	PUNCT
ejpam-6452	102	40	=	=	PUNCT
ejpam-6452	102	41	a	a	DET
ejpam-6452	102	42	2	2	NUM
ejpam-6452	102	43	lk	lk	NOUN
ejpam-6452	102	44	,	,	PUNCT
ejpam-6452	102	45	n	n	PROPN
ejpam-6452	102	46	+	+	CCONJ
ejpam-6452	102	47	2b−	2b−	PROPN
ejpam-6452	102	48	ak	ak	PROPN
ejpam-6452	102	49	2	2	NUM
ejpam-6452	102	50	pk	pk	NOUN
ejpam-6452	102	51	,	,	PUNCT
ejpam-6452	102	52	n.	n.	NOUN
ejpam-6452	102	53	this	this	PRON
ejpam-6452	102	54	completes	complete	VERB
ejpam-6452	102	55	the	the	DET
ejpam-6452	102	56	proof	proof	NOUN
ejpam-6452	102	57	.	.	PUNCT
ejpam-6452	103	1	if	if	SCONJ
ejpam-6452	103	2	(	(	PUNCT
ejpam-6452	103	3	a	a	PRON
ejpam-6452	103	4	,	,	PUNCT
ejpam-6452	103	5	b	b	NOUN
ejpam-6452	103	6	)	)	PUNCT
ejpam-6452	103	7	∈	∈	NOUN
ejpam-6452	103	8	{	{	PUNCT
ejpam-6452	103	9	(	(	PUNCT
ejpam-6452	103	10	2	2	NUM
ejpam-6452	103	11	,	,	PUNCT
ejpam-6452	103	12	2	2	NUM
ejpam-6452	103	13	)	)	PUNCT
ejpam-6452	103	14	,	,	PUNCT
ejpam-6452	103	15	(	(	PUNCT
ejpam-6452	103	16	1	1	NUM
ejpam-6452	103	17	,	,	PUNCT
ejpam-6452	103	18	1	1	NUM
ejpam-6452	103	19	)	)	PUNCT
ejpam-6452	103	20	}	}	PUNCT
ejpam-6452	103	21	,	,	PUNCT
ejpam-6452	103	22	then	then	ADV
ejpam-6452	103	23	we	we	PRON
ejpam-6452	103	24	have	have	VERB
ejpam-6452	103	25	the	the	DET
ejpam-6452	103	26	following	follow	VERB
ejpam-6452	103	27	:	:	PUNCT
ejpam-6452	103	28	corollary	corollary	ADJ
ejpam-6452	103	29	3	3	X
ejpam-6452	103	30	.	.	PUNCT
ejpam-6452	104	1	let	let	VERB
ejpam-6452	104	2	k	k	NOUN
ejpam-6452	104	3	and	and	CCONJ
ejpam-6452	104	4	n	n	ADV
ejpam-6452	104	5	be	be	VERB
ejpam-6452	104	6	non	non	ADJ
ejpam-6452	104	7	-	-	ADJ
ejpam-6452	104	8	negative	negative	ADJ
ejpam-6452	104	9	integers	integer	NOUN
ejpam-6452	104	10	with	with	ADP
ejpam-6452	104	11	k	k	PROPN
ejpam-6452	104	12	≥	≥	NUM
ejpam-6452	104	13	2	2	NUM
ejpam-6452	104	14	.	.	PUNCT
ejpam-6452	105	1	then	then	ADV
ejpam-6452	105	2	(	(	PUNCT
ejpam-6452	105	3	i	i	NOUN
ejpam-6452	105	4	)	)	PUNCT
ejpam-6452	105	5	qk	qk	PROPN
ejpam-6452	105	6	,	,	PUNCT
ejpam-6452	105	7	n	n	PROPN
ejpam-6452	105	8	=	=	SYM
ejpam-6452	105	9	lk	lk	PROPN
ejpam-6452	105	10	,	,	PUNCT
ejpam-6452	105	11	n	n	NOUN
ejpam-6452	105	12	−	−	PROPN
ejpam-6452	106	1	(	(	PUNCT
ejpam-6452	106	2	k	k	PROPN
ejpam-6452	106	3	−	−	PROPN
ejpam-6452	107	1	2)pk	2)pk	PROPN
ejpam-6452	107	2	,	,	PUNCT
ejpam-6452	107	3	n	n	CCONJ
ejpam-6452	107	4	;	;	PUNCT
ejpam-6452	107	5	(	(	PUNCT
ejpam-6452	107	6	ii	ii	NOUN
ejpam-6452	107	7	)	)	PUNCT
ejpam-6452	107	8	qk	qk	PROPN
ejpam-6452	107	9	,	,	PUNCT
ejpam-6452	107	10	n	n	NOUN
ejpam-6452	107	11	=	=	SYM
ejpam-6452	107	12	1	1	NUM
ejpam-6452	107	13	2lk	2lk	NOUN
ejpam-6452	107	14	,	,	PUNCT
ejpam-6452	107	15	n	n	CCONJ
ejpam-6452	107	16	−	−	PROPN
ejpam-6452	107	17	(	(	PUNCT
ejpam-6452	107	18	k−2	k−2	PROPN
ejpam-6452	107	19	)	)	PUNCT
ejpam-6452	107	20	2	2	NUM
ejpam-6452	107	21	pk	pk	NOUN
ejpam-6452	107	22	,	,	PUNCT
ejpam-6452	107	23	n.	n.	NOUN
ejpam-6452	107	24	if	if	SCONJ
ejpam-6452	107	25	k	k	PROPN
ejpam-6452	107	26	=	=	SYM
ejpam-6452	107	27	2	2	NUM
ejpam-6452	107	28	,	,	PUNCT
ejpam-6452	107	29	then	then	ADV
ejpam-6452	107	30	we	we	PRON
ejpam-6452	107	31	have	have	VERB
ejpam-6452	107	32	the	the	DET
ejpam-6452	107	33	following	follow	VERB
ejpam-6452	107	34	:	:	PUNCT
ejpam-6452	107	35	corollary	corollary	ADJ
ejpam-6452	107	36	4	4	X
ejpam-6452	107	37	.	.	PUNCT
ejpam-6452	108	1	let	let	VERB
ejpam-6452	108	2	n	n	PRON
ejpam-6452	108	3	be	be	AUX
ejpam-6452	108	4	a	a	DET
ejpam-6452	108	5	non	non	ADJ
ejpam-6452	108	6	-	-	ADJ
ejpam-6452	108	7	negative	negative	ADJ
ejpam-6452	108	8	integer	integer	NOUN
ejpam-6452	108	9	.	.	PUNCT
ejpam-6452	109	1	then	then	ADV
ejpam-6452	109	2	ha	ha	INTJ
ejpam-6452	109	3	,	,	PUNCT
ejpam-6452	109	4	b	b	PROPN
ejpam-6452	109	5	n	n	NOUN
ejpam-6452	109	6	=	=	PUNCT
ejpam-6452	109	7	a	a	DET
ejpam-6452	109	8	2qn	2qn	ADJ
ejpam-6452	109	9	+	+	CCONJ
ejpam-6452	109	10	(	(	PUNCT
ejpam-6452	109	11	b−	b−	PROPN
ejpam-6452	109	12	a)pn	a)pn	PROPN
ejpam-6452	109	13	.	.	PUNCT
ejpam-6452	110	1	from	from	ADP
ejpam-6452	110	2	theorem	theorem	ADJ
ejpam-6452	110	3	2	2	NUM
ejpam-6452	110	4	and	and	CCONJ
ejpam-6452	110	5	corollary	corollary	ADJ
ejpam-6452	110	6	3	3	NUM
ejpam-6452	110	7	,	,	PUNCT
ejpam-6452	110	8	we	we	PRON
ejpam-6452	110	9	have	have	VERB
ejpam-6452	110	10	the	the	DET
ejpam-6452	110	11	following	follow	VERB
ejpam-6452	110	12	:	:	PUNCT
ejpam-6452	110	13	corollary	corollary	ADJ
ejpam-6452	110	14	5	5	NUM
ejpam-6452	110	15	(	(	PUNCT
ejpam-6452	110	16	[	[	X
ejpam-6452	110	17	11	11	NUM
ejpam-6452	110	18	,	,	PUNCT
ejpam-6452	110	19	lemma	lemma	PROPN
ejpam-6452	110	20	2.1	2.1	NUM
ejpam-6452	110	21	]	]	PUNCT
ejpam-6452	110	22	)	)	PUNCT
ejpam-6452	110	23	.	.	PUNCT
ejpam-6452	111	1	let	let	VERB
ejpam-6452	111	2	k	k	NOUN
ejpam-6452	111	3	and	and	CCONJ
ejpam-6452	111	4	n	n	ADV
ejpam-6452	111	5	be	be	VERB
ejpam-6452	111	6	non	non	ADJ
ejpam-6452	111	7	-	-	ADJ
ejpam-6452	111	8	negative	negative	ADJ
ejpam-6452	111	9	integers	integer	NOUN
ejpam-6452	111	10	with	with	ADP
ejpam-6452	111	11	k	k	PROPN
ejpam-6452	111	12	≥	≥	NUM
ejpam-6452	111	13	2	2	NUM
ejpam-6452	111	14	and	and	CCONJ
ejpam-6452	111	15	∆k	∆k	PROPN
ejpam-6452	111	16	=	=	SYM
ejpam-6452	111	17	√	√	PROPN
ejpam-6452	111	18	k2	k2	NOUN
ejpam-6452	112	1	+	+	CCONJ
ejpam-6452	112	2	4k	4k	NOUN
ejpam-6452	112	3	−	−	NOUN
ejpam-6452	113	1	4	4	X
ejpam-6452	113	2	.	.	PUNCT
ejpam-6452	114	1	then	then	ADV
ejpam-6452	114	2	the	the	DET
ejpam-6452	114	3	following	follow	VERB
ejpam-6452	114	4	hold	hold	NOUN
ejpam-6452	114	5	:	:	PUNCT
ejpam-6452	114	6	(	(	PUNCT
ejpam-6452	114	7	i	i	NOUN
ejpam-6452	114	8	)	)	PUNCT
ejpam-6452	114	9	qk	qk	PROPN
ejpam-6452	114	10	,	,	PUNCT
ejpam-6452	114	11	n	n	PROPN
ejpam-6452	115	1	+	+	CCONJ
ejpam-6452	116	1	(	(	PUNCT
ejpam-6452	116	2	k	k	NOUN
ejpam-6452	116	3	−	−	PROPN
ejpam-6452	116	4	2	2	NUM
ejpam-6452	116	5	+	+	CCONJ
ejpam-6452	116	6	∆k	∆k	PROPN
ejpam-6452	116	7	)	)	PUNCT
ejpam-6452	116	8	pk	pk	NOUN
ejpam-6452	116	9	,	,	PUNCT
ejpam-6452	116	10	n	n	NOUN
ejpam-6452	116	11	=	=	SYM
ejpam-6452	116	12	2σn	2σn	PROPN
ejpam-6452	117	1	k	k	X
ejpam-6452	117	2	;	;	PUNCT
ejpam-6452	117	3	(	(	PUNCT
ejpam-6452	117	4	ii	ii	NOUN
ejpam-6452	117	5	)	)	PUNCT
ejpam-6452	117	6	qk	qk	PROPN
ejpam-6452	117	7	,	,	PUNCT
ejpam-6452	117	8	n	n	PROPN
ejpam-6452	117	9	+	+	CCONJ
ejpam-6452	117	10	(	(	PUNCT
ejpam-6452	117	11	k	k	NOUN
ejpam-6452	117	12	−	−	PROPN
ejpam-6452	117	13	2	2	NUM
ejpam-6452	117	14	−	−	PROPN
ejpam-6452	117	15	∆k	∆k	PROPN
ejpam-6452	117	16	)	)	PUNCT
ejpam-6452	117	17	pk	pk	NOUN
ejpam-6452	117	18	,	,	PUNCT
ejpam-6452	117	19	n	n	NOUN
ejpam-6452	117	20	=	=	SYM
ejpam-6452	117	21	2	2	NUM
ejpam-6452	117	22	(	(	PUNCT
ejpam-6452	117	23	1−k)n	1−k)n	NUM
ejpam-6452	117	24	σn	σn	NOUN
ejpam-6452	117	25	k	k	X
ejpam-6452	117	26	;	;	PUNCT
ejpam-6452	117	27	(	(	PUNCT
ejpam-6452	117	28	iii	iii	X
ejpam-6452	117	29	)	)	PUNCT
ejpam-6452	117	30	(	(	PUNCT
ejpam-6452	117	31	qk	qk	NOUN
ejpam-6452	117	32	,	,	PUNCT
ejpam-6452	117	33	n	n	PROPN
ejpam-6452	117	34	+	+	CCONJ
ejpam-6452	117	35	(	(	PUNCT
ejpam-6452	117	36	k	k	PROPN
ejpam-6452	117	37	−	−	PROPN
ejpam-6452	117	38	2)pk	2)pk	NUM
ejpam-6452	117	39	,	,	PUNCT
ejpam-6452	117	40	n)2	n)2	NOUN
ejpam-6452	117	41	−	−	PROPN
ejpam-6452	117	42	∆kp2	∆kp2	PROPN
ejpam-6452	117	43	k	k	NOUN
ejpam-6452	117	44	,	,	PUNCT
ejpam-6452	117	45	n	n	NOUN
ejpam-6452	117	46	=	=	SYM
ejpam-6452	117	47	4(1	4(1	NOUN
ejpam-6452	117	48	−	−	PROPN
ejpam-6452	117	49	k)n	k)n	NOUN
ejpam-6452	117	50	.	.	PUNCT
ejpam-6452	118	1	w.	w.	PROPN
ejpam-6452	118	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	118	3	,	,	PUNCT
ejpam-6452	118	4	a.	a.	NOUN
ejpam-6452	118	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	118	6	/	/	SYM
ejpam-6452	118	7	eur	eur	PROPN
ejpam-6452	118	8	.	.	PUNCT
ejpam-6452	119	1	j.	j.	PROPN
ejpam-6452	119	2	pure	pure	PROPN
ejpam-6452	119	3	appl	appl	PROPN
ejpam-6452	119	4	.	.	PROPN
ejpam-6452	119	5	math	math	PROPN
ejpam-6452	119	6	,	,	PUNCT
ejpam-6452	119	7	18	18	NUM
ejpam-6452	119	8	(	(	PUNCT
ejpam-6452	119	9	3	3	NUM
ejpam-6452	119	10	)	)	PUNCT
ejpam-6452	119	11	(	(	PUNCT
ejpam-6452	119	12	2025	2025	NUM
ejpam-6452	119	13	)	)	PUNCT
ejpam-6452	119	14	,	,	PUNCT
ejpam-6452	119	15	6452	6452	NUM
ejpam-6452	119	16	7	7	NUM
ejpam-6452	119	17	of	of	ADP
ejpam-6452	119	18	18	18	NUM
ejpam-6452	119	19	theorem	theorem	NOUN
ejpam-6452	119	20	4	4	NUM
ejpam-6452	119	21	.	.	PUNCT
ejpam-6452	120	1	let	let	VERB
ejpam-6452	120	2	k	k	NOUN
ejpam-6452	120	3	,	,	PUNCT
ejpam-6452	120	4	m	m	VERB
ejpam-6452	120	5	and	and	CCONJ
ejpam-6452	120	6	n	n	ADV
ejpam-6452	120	7	be	be	VERB
ejpam-6452	120	8	non	non	ADJ
ejpam-6452	120	9	-	-	ADJ
ejpam-6452	120	10	negative	negative	ADJ
ejpam-6452	120	11	integers	integer	NOUN
ejpam-6452	120	12	with	with	ADP
ejpam-6452	120	13	k	k	PROPN
ejpam-6452	120	14	≥	≥	NUM
ejpam-6452	120	15	2	2	NUM
ejpam-6452	120	16	and	and	CCONJ
ejpam-6452	120	17	∆k	∆k	PROPN
ejpam-6452	120	18	=	=	SYM
ejpam-6452	120	19	√	√	PROPN
ejpam-6452	120	20	k2	k2	NOUN
ejpam-6452	121	1	+	+	CCONJ
ejpam-6452	121	2	4k	4k	NOUN
ejpam-6452	121	3	−	−	NOUN
ejpam-6452	122	1	4	4	X
ejpam-6452	122	2	.	.	PUNCT
ejpam-6452	123	1	then	then	ADV
ejpam-6452	123	2	the	the	DET
ejpam-6452	123	3	following	follow	VERB
ejpam-6452	123	4	hold	hold	NOUN
ejpam-6452	123	5	:	:	PUNCT
ejpam-6452	123	6	(	(	PUNCT
ejpam-6452	123	7	i	i	NOUN
ejpam-6452	123	8	)	)	PUNCT
ejpam-6452	123	9	2pk	2pk	NOUN
ejpam-6452	123	10	,	,	PUNCT
ejpam-6452	123	11	m+n	m+n	NOUN
ejpam-6452	123	12	=	=	SYM
ejpam-6452	123	13	pk	pk	PROPN
ejpam-6452	123	14	,	,	PUNCT
ejpam-6452	123	15	mlk	mlk	PROPN
ejpam-6452	123	16	,	,	PUNCT
ejpam-6452	123	17	n	n	PROPN
ejpam-6452	123	18	+	+	CCONJ
ejpam-6452	123	19	pk	pk	NOUN
ejpam-6452	123	20	,	,	PUNCT
ejpam-6452	123	21	nlk	nlk	NOUN
ejpam-6452	123	22	,	,	PUNCT
ejpam-6452	123	23	m	m	PROPN
ejpam-6452	123	24	;	;	PUNCT
ejpam-6452	123	25	(	(	PUNCT
ejpam-6452	123	26	ii	ii	NOUN
ejpam-6452	123	27	)	)	PUNCT
ejpam-6452	123	28	2lk	2lk	NOUN
ejpam-6452	123	29	,	,	PUNCT
ejpam-6452	123	30	m+n	m+n	NOUN
ejpam-6452	123	31	=	=	SYM
ejpam-6452	123	32	lk	lk	PROPN
ejpam-6452	123	33	,	,	PUNCT
ejpam-6452	123	34	mlk	mlk	PROPN
ejpam-6452	123	35	,	,	PUNCT
ejpam-6452	123	36	n	n	PROPN
ejpam-6452	123	37	+	+	CCONJ
ejpam-6452	123	38	∆2	∆2	PROPN
ejpam-6452	123	39	kpk	kpk	PROPN
ejpam-6452	123	40	,	,	PUNCT
ejpam-6452	123	41	mpk	mpk	NOUN
ejpam-6452	123	42	,	,	PUNCT
ejpam-6452	123	43	n.	n.	NOUN
ejpam-6452	123	44	proof	proof	NOUN
ejpam-6452	123	45	.	.	PUNCT
ejpam-6452	124	1	(	(	PUNCT
ejpam-6452	124	2	i	i	NOUN
ejpam-6452	124	3	)	)	PUNCT
ejpam-6452	124	4	using	use	VERB
ejpam-6452	124	5	(	(	PUNCT
ejpam-6452	124	6	i	i	NOUN
ejpam-6452	124	7	)	)	PUNCT
ejpam-6452	124	8	and	and	CCONJ
ejpam-6452	124	9	(	(	PUNCT
ejpam-6452	124	10	ii	ii	NOUN
ejpam-6452	124	11	)	)	PUNCT
ejpam-6452	124	12	of	of	ADP
ejpam-6452	124	13	corollary	corollary	ADJ
ejpam-6452	124	14	1	1	NUM
ejpam-6452	124	15	,	,	PUNCT
ejpam-6452	124	16	we	we	PRON
ejpam-6452	124	17	have	have	VERB
ejpam-6452	124	18	pk	pk	NOUN
ejpam-6452	124	19	,	,	PUNCT
ejpam-6452	124	20	mlk	mlk	PROPN
ejpam-6452	124	21	,	,	PUNCT
ejpam-6452	124	22	n	n	NOUN
ejpam-6452	124	23	=	=	PUNCT
ejpam-6452	124	24	[	[	PUNCT
ejpam-6452	124	25	1	1	NUM
ejpam-6452	124	26	∆k	∆k	NOUN
ejpam-6452	124	27	(	(	PUNCT
ejpam-6452	124	28	σm	σm	NOUN
ejpam-6452	124	29	k	k	PROPN
ejpam-6452	125	1	−	−	PROPN
ejpam-6452	125	2	(	(	PUNCT
ejpam-6452	125	3	1	1	NUM
ejpam-6452	125	4	−	−	PROPN
ejpam-6452	125	5	k)m	k)m	NOUN
ejpam-6452	125	6	σm	σm	X
ejpam-6452	125	7	k	k	PROPN
ejpam-6452	125	8	)	)	PUNCT
ejpam-6452	125	9	]	]	X
ejpam-6452	125	10	(	(	PUNCT
ejpam-6452	125	11	σn	σn	X
ejpam-6452	125	12	k	k	PROPN
ejpam-6452	125	13	+	+	CCONJ
ejpam-6452	125	14	(	(	PUNCT
ejpam-6452	125	15	1	1	NUM
ejpam-6452	125	16	−	−	NOUN
ejpam-6452	125	17	k)n	k)n	NOUN
ejpam-6452	125	18	σn	σn	PROPN
ejpam-6452	125	19	k	k	NOUN
ejpam-6452	125	20	)	)	PUNCT
ejpam-6452	125	21	=	=	SYM
ejpam-6452	126	1	1	1	NUM
ejpam-6452	126	2	∆k	∆k	PROPN
ejpam-6452	126	3	(	(	PUNCT
ejpam-6452	126	4	σm+n	σm+n	NOUN
ejpam-6452	126	5	k	k	PROPN
ejpam-6452	126	6	+	+	PUNCT
ejpam-6452	126	7	(	(	PUNCT
ejpam-6452	126	8	1	1	NUM
ejpam-6452	126	9	−	−	PROPN
ejpam-6452	126	10	k)nσm	k)nσm	PROPN
ejpam-6452	126	11	k	k	PROPN
ejpam-6452	126	12	σn	σn	PROPN
ejpam-6452	126	13	k	k	PROPN
ejpam-6452	126	14	−	−	PROPN
ejpam-6452	126	15	(	(	PUNCT
ejpam-6452	126	16	1	1	NUM
ejpam-6452	126	17	−	−	NOUN
ejpam-6452	126	18	k)mσn	k)mσn	PROPN
ejpam-6452	126	19	k	k	PROPN
ejpam-6452	126	20	σm	σm	INTJ
ejpam-6452	126	21	k	k	NOUN
ejpam-6452	126	22	−	−	PROPN
ejpam-6452	126	23	(	(	PUNCT
ejpam-6452	126	24	1	1	NUM
ejpam-6452	126	25	−	−	PROPN
ejpam-6452	126	26	k)m+n	k)m+n	PROPN
ejpam-6452	126	27	σm+n	σm+n	PROPN
ejpam-6452	126	28	k	k	PROPN
ejpam-6452	126	29	)	)	PUNCT
ejpam-6452	126	30	and	and	CCONJ
ejpam-6452	126	31	pk	pk	NOUN
ejpam-6452	126	32	,	,	PUNCT
ejpam-6452	126	33	nlk	nlk	NOUN
ejpam-6452	126	34	,	,	PUNCT
ejpam-6452	126	35	m	m	VERB
ejpam-6452	126	36	=	=	PUNCT
ejpam-6452	126	37	[	[	PUNCT
ejpam-6452	126	38	1	1	NUM
ejpam-6452	126	39	∆k	∆k	PROPN
ejpam-6452	126	40	(	(	PUNCT
ejpam-6452	126	41	σn	σn	NOUN
ejpam-6452	126	42	k	k	PROPN
ejpam-6452	126	43	−	−	PROPN
ejpam-6452	126	44	(	(	PUNCT
ejpam-6452	126	45	1	1	NUM
ejpam-6452	126	46	−	−	NOUN
ejpam-6452	126	47	k)n	k)n	NOUN
ejpam-6452	126	48	σn	σn	PROPN
ejpam-6452	126	49	k	k	PROPN
ejpam-6452	126	50	)	)	PUNCT
ejpam-6452	126	51	]	]	PUNCT
ejpam-6452	126	52	(	(	PUNCT
ejpam-6452	126	53	σm	σm	X
ejpam-6452	126	54	k	k	PROPN
ejpam-6452	127	1	+	+	CCONJ
ejpam-6452	127	2	(	(	PUNCT
ejpam-6452	127	3	1	1	NUM
ejpam-6452	127	4	−	−	PROPN
ejpam-6452	127	5	k)m	k)m	NOUN
ejpam-6452	127	6	σm	σm	X
ejpam-6452	128	1	k	k	X
ejpam-6452	128	2	)	)	PUNCT
ejpam-6452	129	1	=	=	SYM
ejpam-6452	129	2	1	1	NUM
ejpam-6452	129	3	∆k	∆k	PROPN
ejpam-6452	129	4	(	(	PUNCT
ejpam-6452	129	5	σm+n	σm+n	NOUN
ejpam-6452	129	6	k	k	PROPN
ejpam-6452	130	1	+	+	PUNCT
ejpam-6452	130	2	(	(	PUNCT
ejpam-6452	130	3	1	1	NUM
ejpam-6452	130	4	−	−	PROPN
ejpam-6452	130	5	k)mσn	k)mσn	PROPN
ejpam-6452	130	6	k	k	PROPN
ejpam-6452	130	7	σn	σn	PROPN
ejpam-6452	131	1	k	k	PROPN
ejpam-6452	131	2	−	−	PROPN
ejpam-6452	131	3	(	(	PUNCT
ejpam-6452	131	4	1	1	NUM
ejpam-6452	131	5	−	−	PROPN
ejpam-6452	131	6	k)nσm	k)nσm	PROPN
ejpam-6452	131	7	k	k	PROPN
ejpam-6452	131	8	σn	σn	PROPN
ejpam-6452	132	1	k	k	PROPN
ejpam-6452	133	1	−	−	PROPN
ejpam-6452	134	1	(	(	PUNCT
ejpam-6452	134	2	1	1	NUM
ejpam-6452	134	3	−	−	PROPN
ejpam-6452	134	4	k)m+n	k)m+n	PROPN
ejpam-6452	134	5	σm+n	σm+n	PROPN
ejpam-6452	134	6	k	k	PROPN
ejpam-6452	134	7	)	)	PUNCT
ejpam-6452	134	8	.	.	PUNCT
ejpam-6452	135	1	so	so	ADV
ejpam-6452	135	2	,	,	PUNCT
ejpam-6452	135	3	we	we	PRON
ejpam-6452	135	4	get	get	VERB
ejpam-6452	135	5	pk	pk	NOUN
ejpam-6452	135	6	,	,	PUNCT
ejpam-6452	135	7	mlk	mlk	PROPN
ejpam-6452	135	8	,	,	PUNCT
ejpam-6452	135	9	n	n	PROPN
ejpam-6452	135	10	+	+	CCONJ
ejpam-6452	135	11	pk	pk	NOUN
ejpam-6452	135	12	,	,	PUNCT
ejpam-6452	135	13	nlk	nlk	NOUN
ejpam-6452	135	14	,	,	PUNCT
ejpam-6452	135	15	m	m	VERB
ejpam-6452	135	16	=	=	SYM
ejpam-6452	135	17	2	2	NUM
ejpam-6452	135	18	∆k	∆k	PROPN
ejpam-6452	135	19	(	(	PUNCT
ejpam-6452	135	20	σm+n	σm+n	NOUN
ejpam-6452	135	21	k	k	NOUN
ejpam-6452	136	1	−	−	PROPN
ejpam-6452	137	1	(	(	PUNCT
ejpam-6452	137	2	1	1	NUM
ejpam-6452	137	3	−	−	PROPN
ejpam-6452	137	4	k)m+n	k)m+n	PROPN
ejpam-6452	137	5	σm+n	σm+n	PROPN
ejpam-6452	137	6	k	k	PROPN
ejpam-6452	137	7	)	)	PUNCT
ejpam-6452	138	1	=	=	SYM
ejpam-6452	138	2	2pk	2pk	NOUN
ejpam-6452	138	3	,	,	PUNCT
ejpam-6452	138	4	m+n	m+n	PROPN
ejpam-6452	138	5	.	.	PUNCT
ejpam-6452	138	6	(	(	PUNCT
ejpam-6452	138	7	ii	ii	NOUN
ejpam-6452	138	8	)	)	PUNCT
ejpam-6452	138	9	using	use	VERB
ejpam-6452	138	10	(	(	PUNCT
ejpam-6452	138	11	i	i	NOUN
ejpam-6452	138	12	)	)	PUNCT
ejpam-6452	138	13	of	of	ADP
ejpam-6452	138	14	corollary	corollary	ADJ
ejpam-6452	138	15	1	1	NUM
ejpam-6452	138	16	,	,	PUNCT
ejpam-6452	138	17	we	we	PRON
ejpam-6452	138	18	have	have	VERB
ejpam-6452	138	19	pk	pk	NOUN
ejpam-6452	138	20	,	,	PUNCT
ejpam-6452	138	21	mpk	mpk	NOUN
ejpam-6452	138	22	,	,	PUNCT
ejpam-6452	138	23	n	n	NOUN
ejpam-6452	138	24	=	=	PUNCT
ejpam-6452	138	25	[	[	PUNCT
ejpam-6452	138	26	1	1	NUM
ejpam-6452	138	27	∆k	∆k	NOUN
ejpam-6452	138	28	(	(	PUNCT
ejpam-6452	138	29	σm	σm	NOUN
ejpam-6452	138	30	k	k	PROPN
ejpam-6452	139	1	−	−	PROPN
ejpam-6452	139	2	(	(	PUNCT
ejpam-6452	139	3	1	1	NUM
ejpam-6452	139	4	−	−	PROPN
ejpam-6452	139	5	k)m	k)m	NOUN
ejpam-6452	139	6	σm	σm	X
ejpam-6452	139	7	k	k	PROPN
ejpam-6452	139	8	)	)	PUNCT
ejpam-6452	140	1	]	]	PUNCT
ejpam-6452	140	2	[	[	PUNCT
ejpam-6452	140	3	1	1	NUM
ejpam-6452	140	4	∆k	∆k	PROPN
ejpam-6452	140	5	(	(	PUNCT
ejpam-6452	140	6	σn	σn	NOUN
ejpam-6452	140	7	k	k	PROPN
ejpam-6452	140	8	−	−	PROPN
ejpam-6452	140	9	(	(	PUNCT
ejpam-6452	140	10	1	1	NUM
ejpam-6452	140	11	−	−	NOUN
ejpam-6452	140	12	k)n	k)n	NOUN
ejpam-6452	140	13	σn	σn	PROPN
ejpam-6452	140	14	k	k	PROPN
ejpam-6452	140	15	)	)	PUNCT
ejpam-6452	140	16	]	]	PUNCT
ejpam-6452	140	17	=	=	PUNCT
ejpam-6452	141	1	1	1	NUM
ejpam-6452	141	2	∆2	∆2	PART
ejpam-6452	141	3	k	k	PROPN
ejpam-6452	141	4	(	(	PUNCT
ejpam-6452	141	5	σm+n	σm+n	PROPN
ejpam-6452	141	6	k	k	NOUN
ejpam-6452	141	7	−	−	PROPN
ejpam-6452	141	8	(	(	PUNCT
ejpam-6452	141	9	1	1	NUM
ejpam-6452	141	10	−	−	PROPN
ejpam-6452	141	11	k)nσm	k)nσm	PROPN
ejpam-6452	141	12	k	k	PROPN
ejpam-6452	141	13	σn	σn	PROPN
ejpam-6452	141	14	k	k	PROPN
ejpam-6452	141	15	−	−	PROPN
ejpam-6452	141	16	(	(	PUNCT
ejpam-6452	141	17	1	1	NUM
ejpam-6452	141	18	−	−	NOUN
ejpam-6452	141	19	k)mσn	k)mσn	PROPN
ejpam-6452	141	20	k	k	PROPN
ejpam-6452	141	21	σm	σm	X
ejpam-6452	141	22	k	k	PROPN
ejpam-6452	142	1	+	+	CCONJ
ejpam-6452	142	2	(	(	PUNCT
ejpam-6452	142	3	1	1	NUM
ejpam-6452	142	4	−	−	PROPN
ejpam-6452	143	1	k)m+n	k)m+n	PROPN
ejpam-6452	143	2	σm+n	σm+n	PROPN
ejpam-6452	143	3	k	k	PROPN
ejpam-6452	143	4	)	)	PUNCT
ejpam-6452	143	5	.	.	PUNCT
ejpam-6452	144	1	by	by	ADP
ejpam-6452	144	2	using	use	VERB
ejpam-6452	144	3	(	(	PUNCT
ejpam-6452	144	4	ii	ii	NOUN
ejpam-6452	144	5	)	)	PUNCT
ejpam-6452	144	6	of	of	ADP
ejpam-6452	144	7	corollary	corollary	ADJ
ejpam-6452	144	8	1	1	NUM
ejpam-6452	144	9	,	,	PUNCT
ejpam-6452	144	10	we	we	PRON
ejpam-6452	144	11	have	have	VERB
ejpam-6452	144	12	lk	lk	PROPN
ejpam-6452	144	13	,	,	PUNCT
ejpam-6452	144	14	mlk	mlk	PROPN
ejpam-6452	144	15	,	,	PUNCT
ejpam-6452	144	16	n	n	NOUN
ejpam-6452	144	17	=	=	PUNCT
ejpam-6452	144	18	(	(	PUNCT
ejpam-6452	144	19	σm	σm	X
ejpam-6452	144	20	k	k	PROPN
ejpam-6452	145	1	+	+	CCONJ
ejpam-6452	145	2	(	(	PUNCT
ejpam-6452	145	3	1	1	NUM
ejpam-6452	145	4	−	−	PROPN
ejpam-6452	145	5	k)m	k)m	NOUN
ejpam-6452	145	6	σm	σm	X
ejpam-6452	145	7	k	k	PROPN
ejpam-6452	145	8	)	)	PUNCT
ejpam-6452	145	9	(	(	PUNCT
ejpam-6452	145	10	σn	σn	PROPN
ejpam-6452	145	11	k	k	X
ejpam-6452	145	12	+	+	CCONJ
ejpam-6452	145	13	(	(	PUNCT
ejpam-6452	145	14	1	1	NUM
ejpam-6452	145	15	−	−	NOUN
ejpam-6452	145	16	k)n	k)n	NOUN
ejpam-6452	145	17	σn	σn	PROPN
ejpam-6452	145	18	k	k	NOUN
ejpam-6452	145	19	)	)	PUNCT
ejpam-6452	146	1	=	=	PUNCT
ejpam-6452	147	1	σm+n	σm+n	PROPN
ejpam-6452	147	2	k	k	X
ejpam-6452	148	1	+	+	PUNCT
ejpam-6452	148	2	(	(	PUNCT
ejpam-6452	148	3	1	1	NUM
ejpam-6452	148	4	−	−	PROPN
ejpam-6452	148	5	k)nσm	k)nσm	PROPN
ejpam-6452	148	6	k	k	PROPN
ejpam-6452	148	7	σn	σn	PROPN
ejpam-6452	148	8	k	k	PROPN
ejpam-6452	148	9	+	+	CCONJ
ejpam-6452	148	10	(	(	PUNCT
ejpam-6452	148	11	1	1	NUM
ejpam-6452	148	12	−	−	NOUN
ejpam-6452	149	1	k)mσn	k)mσn	PROPN
ejpam-6452	149	2	k	k	PROPN
ejpam-6452	149	3	σm	σm	X
ejpam-6452	150	1	k	k	PROPN
ejpam-6452	151	1	+	+	CCONJ
ejpam-6452	151	2	(	(	PUNCT
ejpam-6452	151	3	1	1	NUM
ejpam-6452	151	4	−	−	PROPN
ejpam-6452	152	1	k)m+n	k)m+n	PROPN
ejpam-6452	152	2	σm+n	σm+n	PROPN
ejpam-6452	152	3	k	k	PROPN
ejpam-6452	152	4	.	.	PUNCT
ejpam-6452	153	1	this	this	PRON
ejpam-6452	153	2	implies	imply	VERB
ejpam-6452	153	3	that	that	SCONJ
ejpam-6452	153	4	lk	lk	PROPN
ejpam-6452	153	5	,	,	PUNCT
ejpam-6452	153	6	mlk	mlk	PROPN
ejpam-6452	153	7	,	,	PUNCT
ejpam-6452	153	8	n	n	NOUN
ejpam-6452	153	9	+	+	CCONJ
ejpam-6452	153	10	(	(	PUNCT
ejpam-6452	153	11	k2	k2	X
ejpam-6452	153	12	+	+	CCONJ
ejpam-6452	153	13	4k	4k	NOUN
ejpam-6452	153	14	−	−	PROPN
ejpam-6452	153	15	4)pk	4)pk	NUM
ejpam-6452	153	16	,	,	PUNCT
ejpam-6452	153	17	mpk	mpk	NOUN
ejpam-6452	153	18	,	,	PUNCT
ejpam-6452	153	19	n	n	NOUN
ejpam-6452	153	20	=	=	SYM
ejpam-6452	153	21	2	2	NUM
ejpam-6452	153	22	(	(	PUNCT
ejpam-6452	153	23	σm+n	σm+n	NOUN
ejpam-6452	153	24	k	k	PROPN
ejpam-6452	154	1	+	+	PUNCT
ejpam-6452	154	2	(	(	PUNCT
ejpam-6452	154	3	1	1	NUM
ejpam-6452	154	4	−	−	PROPN
ejpam-6452	154	5	k)m+n	k)m+n	PROPN
ejpam-6452	154	6	σm+n	σm+n	PROPN
ejpam-6452	154	7	k	k	PROPN
ejpam-6452	154	8	)	)	PUNCT
ejpam-6452	155	1	=	=	SYM
ejpam-6452	155	2	2lk	2lk	PROPN
ejpam-6452	155	3	,	,	PUNCT
ejpam-6452	155	4	m+n	m+n	PROPN
ejpam-6452	155	5	.	.	PUNCT
ejpam-6452	155	6	w.	w.	PROPN
ejpam-6452	155	7	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	155	8	,	,	PUNCT
ejpam-6452	155	9	a.	a.	NOUN
ejpam-6452	155	10	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	155	11	/	/	SYM
ejpam-6452	155	12	eur	eur	PROPN
ejpam-6452	155	13	.	.	PUNCT
ejpam-6452	156	1	j.	j.	PROPN
ejpam-6452	156	2	pure	pure	PROPN
ejpam-6452	156	3	appl	appl	PROPN
ejpam-6452	156	4	.	.	PROPN
ejpam-6452	156	5	math	math	PROPN
ejpam-6452	156	6	,	,	PUNCT
ejpam-6452	156	7	18	18	NUM
ejpam-6452	156	8	(	(	PUNCT
ejpam-6452	156	9	3	3	NUM
ejpam-6452	156	10	)	)	PUNCT
ejpam-6452	156	11	(	(	PUNCT
ejpam-6452	156	12	2025	2025	NUM
ejpam-6452	156	13	)	)	PUNCT
ejpam-6452	156	14	,	,	PUNCT
ejpam-6452	156	15	6452	6452	NUM
ejpam-6452	156	16	8	8	NUM
ejpam-6452	156	17	of	of	ADP
ejpam-6452	156	18	18	18	NUM
ejpam-6452	156	19	hence	hence	ADV
ejpam-6452	156	20	,	,	PUNCT
ejpam-6452	156	21	(	(	PUNCT
ejpam-6452	156	22	i	i	NOUN
ejpam-6452	156	23	)	)	PUNCT
ejpam-6452	156	24	and	and	CCONJ
ejpam-6452	156	25	(	(	PUNCT
ejpam-6452	156	26	ii	ii	NOUN
ejpam-6452	156	27	)	)	PUNCT
ejpam-6452	156	28	complete	complete	VERB
ejpam-6452	156	29	the	the	DET
ejpam-6452	156	30	proof	proof	NOUN
ejpam-6452	156	31	.	.	PUNCT
ejpam-6452	157	1	corollary	corollary	ADJ
ejpam-6452	157	2	6	6	NUM
ejpam-6452	157	3	(	(	PUNCT
ejpam-6452	157	4	[	[	X
ejpam-6452	157	5	11	11	NUM
ejpam-6452	157	6	,	,	PUNCT
ejpam-6452	157	7	lemma	lemma	PROPN
ejpam-6452	157	8	2.2	2.2	NUM
ejpam-6452	157	9	]	]	PUNCT
ejpam-6452	157	10	)	)	PUNCT
ejpam-6452	157	11	.	.	PUNCT
ejpam-6452	158	1	let	let	VERB
ejpam-6452	158	2	k	k	NOUN
ejpam-6452	158	3	,	,	PUNCT
ejpam-6452	158	4	m	m	VERB
ejpam-6452	158	5	and	and	CCONJ
ejpam-6452	158	6	n	n	ADV
ejpam-6452	158	7	be	be	VERB
ejpam-6452	158	8	non	non	ADJ
ejpam-6452	158	9	-	-	ADJ
ejpam-6452	158	10	negative	negative	ADJ
ejpam-6452	158	11	integers	integer	NOUN
ejpam-6452	158	12	with	with	ADP
ejpam-6452	158	13	k	k	PROPN
ejpam-6452	158	14	≥	≥	NUM
ejpam-6452	158	15	2	2	NUM
ejpam-6452	158	16	and	and	CCONJ
ejpam-6452	158	17	∆k	∆k	PROPN
ejpam-6452	158	18	=	=	SYM
ejpam-6452	158	19	√	√	PROPN
ejpam-6452	158	20	k2	k2	NOUN
ejpam-6452	158	21	+	+	CCONJ
ejpam-6452	158	22	4k	4k	NOUN
ejpam-6452	158	23	−	−	NOUN
ejpam-6452	159	1	4	4	X
ejpam-6452	159	2	.	.	PUNCT
ejpam-6452	160	1	then	then	ADV
ejpam-6452	160	2	the	the	DET
ejpam-6452	160	3	following	follow	VERB
ejpam-6452	160	4	hold	hold	NOUN
ejpam-6452	160	5	:	:	PUNCT
ejpam-6452	160	6	(	(	PUNCT
ejpam-6452	160	7	i	i	NOUN
ejpam-6452	160	8	)	)	PUNCT
ejpam-6452	160	9	2pk	2pk	NOUN
ejpam-6452	160	10	,	,	PUNCT
ejpam-6452	160	11	m+n	m+n	NOUN
ejpam-6452	160	12	=	=	SYM
ejpam-6452	160	13	pk	pk	PROPN
ejpam-6452	160	14	,	,	PUNCT
ejpam-6452	160	15	mqk	mqk	NOUN
ejpam-6452	160	16	,	,	PUNCT
ejpam-6452	160	17	n	n	PROPN
ejpam-6452	160	18	+	+	CCONJ
ejpam-6452	160	19	pk	pk	NOUN
ejpam-6452	160	20	,	,	PUNCT
ejpam-6452	160	21	nqk	nqk	NOUN
ejpam-6452	160	22	,	,	PUNCT
ejpam-6452	160	23	m	m	VERB
ejpam-6452	160	24	+	+	NOUN
ejpam-6452	160	25	2(k	2(k	NUM
ejpam-6452	161	1	−	−	NOUN
ejpam-6452	162	1	2)pk	2)pk	NUM
ejpam-6452	162	2	,	,	PUNCT
ejpam-6452	162	3	mpk	mpk	NOUN
ejpam-6452	162	4	,	,	PUNCT
ejpam-6452	162	5	n	n	CCONJ
ejpam-6452	162	6	;	;	PUNCT
ejpam-6452	162	7	(	(	PUNCT
ejpam-6452	162	8	ii	ii	NOUN
ejpam-6452	162	9	)	)	PUNCT
ejpam-6452	162	10	2qk	2qk	NOUN
ejpam-6452	162	11	,	,	PUNCT
ejpam-6452	162	12	m+n	m+n	NOUN
ejpam-6452	162	13	=	=	SYM
ejpam-6452	162	14	qk	qk	PROPN
ejpam-6452	162	15	,	,	PUNCT
ejpam-6452	162	16	mqk	mqk	NOUN
ejpam-6452	162	17	,	,	PUNCT
ejpam-6452	162	18	n	n	PROPN
ejpam-6452	162	19	+	+	CCONJ
ejpam-6452	162	20	8(k	8(k	NUM
ejpam-6452	162	21	−	−	NOUN
ejpam-6452	162	22	1)pk	1)pk	NUM
ejpam-6452	162	23	,	,	PUNCT
ejpam-6452	162	24	mpk	mpk	NOUN
ejpam-6452	162	25	,	,	PUNCT
ejpam-6452	162	26	n.	n.	NOUN
ejpam-6452	162	27	proof	proof	NOUN
ejpam-6452	162	28	.	.	PUNCT
ejpam-6452	163	1	it	it	PRON
ejpam-6452	163	2	follows	follow	VERB
ejpam-6452	163	3	from	from	ADP
ejpam-6452	163	4	(	(	PUNCT
ejpam-6452	163	5	i	i	NOUN
ejpam-6452	163	6	)	)	PUNCT
ejpam-6452	163	7	of	of	ADP
ejpam-6452	163	8	theorem	theorem	NOUN
ejpam-6452	163	9	4	4	NUM
ejpam-6452	163	10	and	and	CCONJ
ejpam-6452	163	11	(	(	PUNCT
ejpam-6452	163	12	i	i	NOUN
ejpam-6452	163	13	)	)	PUNCT
ejpam-6452	163	14	of	of	ADP
ejpam-6452	163	15	corollary	corollary	ADJ
ejpam-6452	163	16	3	3	NUM
ejpam-6452	163	17	that	that	DET
ejpam-6452	163	18	2pk	2pk	NOUN
ejpam-6452	163	19	,	,	PUNCT
ejpam-6452	163	20	m+n	m+n	NOUN
ejpam-6452	163	21	=	=	SYM
ejpam-6452	163	22	pk	pk	PROPN
ejpam-6452	163	23	,	,	PUNCT
ejpam-6452	163	24	mlk	mlk	PROPN
ejpam-6452	163	25	,	,	PUNCT
ejpam-6452	163	26	n	n	PROPN
ejpam-6452	163	27	+	+	CCONJ
ejpam-6452	163	28	pk	pk	NOUN
ejpam-6452	163	29	,	,	PUNCT
ejpam-6452	163	30	nlk	nlk	NOUN
ejpam-6452	163	31	,	,	PUNCT
ejpam-6452	163	32	m	m	NOUN
ejpam-6452	163	33	=	=	SYM
ejpam-6452	163	34	pk	pk	PROPN
ejpam-6452	163	35	,	,	PUNCT
ejpam-6452	163	36	m	m	PROPN
ejpam-6452	163	37	(	(	PUNCT
ejpam-6452	163	38	qk	qk	NOUN
ejpam-6452	163	39	,	,	PUNCT
ejpam-6452	163	40	n	n	PROPN
ejpam-6452	163	41	+	+	CCONJ
ejpam-6452	163	42	(	(	PUNCT
ejpam-6452	163	43	k	k	PROPN
ejpam-6452	163	44	−	−	PROPN
ejpam-6452	163	45	2)pk	2)pk	PROPN
ejpam-6452	163	46	,	,	PUNCT
ejpam-6452	163	47	n	n	CCONJ
ejpam-6452	163	48	)	)	PUNCT
ejpam-6452	164	1	+	+	CCONJ
ejpam-6452	164	2	pk	pk	NOUN
ejpam-6452	164	3	,	,	PUNCT
ejpam-6452	164	4	n	n	CCONJ
ejpam-6452	164	5	(	(	PUNCT
ejpam-6452	164	6	qlk	qlk	PROPN
ejpam-6452	164	7	,	,	PUNCT
ejpam-6452	164	8	m	m	VERB
ejpam-6452	164	9	+	+	X
ejpam-6452	164	10	(	(	PUNCT
ejpam-6452	164	11	k	k	PROPN
ejpam-6452	164	12	−	−	PROPN
ejpam-6452	164	13	2)pk	2)pk	PROPN
ejpam-6452	164	14	,	,	PUNCT
ejpam-6452	164	15	m	m	NOUN
ejpam-6452	164	16	)	)	PUNCT
ejpam-6452	164	17	=	=	SYM
ejpam-6452	164	18	pk	pk	NOUN
ejpam-6452	164	19	,	,	PUNCT
ejpam-6452	164	20	mqk	mqk	NOUN
ejpam-6452	164	21	,	,	PUNCT
ejpam-6452	164	22	n	n	PROPN
ejpam-6452	164	23	+	+	CCONJ
ejpam-6452	164	24	pk	pk	NOUN
ejpam-6452	164	25	,	,	PUNCT
ejpam-6452	164	26	nqk	nqk	NOUN
ejpam-6452	164	27	,	,	PUNCT
ejpam-6452	164	28	m	m	VERB
ejpam-6452	164	29	+	+	NOUN
ejpam-6452	164	30	2(k	2(k	NUM
ejpam-6452	165	1	−	−	NOUN
ejpam-6452	165	2	2)pk	2)pk	NUM
ejpam-6452	165	3	,	,	PUNCT
ejpam-6452	165	4	mpk	mpk	NOUN
ejpam-6452	165	5	,	,	PUNCT
ejpam-6452	165	6	n.	n.	NOUN
ejpam-6452	165	7	since	since	SCONJ
ejpam-6452	165	8	lk	lk	PROPN
ejpam-6452	165	9	,	,	PUNCT
ejpam-6452	165	10	mlk	mlk	PROPN
ejpam-6452	165	11	,	,	PUNCT
ejpam-6452	165	12	n	n	NOUN
ejpam-6452	165	13	=	=	SYM
ejpam-6452	165	14	(	(	PUNCT
ejpam-6452	165	15	qk	qk	INTJ
ejpam-6452	165	16	,	,	PUNCT
ejpam-6452	165	17	m	m	VERB
ejpam-6452	165	18	+	+	X
ejpam-6452	165	19	(	(	PUNCT
ejpam-6452	165	20	k	k	PROPN
ejpam-6452	165	21	−	−	PROPN
ejpam-6452	165	22	2)pk	2)pk	PROPN
ejpam-6452	165	23	,	,	PUNCT
ejpam-6452	165	24	m	m	PRON
ejpam-6452	165	25	)	)	PUNCT
ejpam-6452	165	26	(	(	PUNCT
ejpam-6452	165	27	qk	qk	NOUN
ejpam-6452	165	28	,	,	PUNCT
ejpam-6452	165	29	n	n	PROPN
ejpam-6452	165	30	+	+	CCONJ
ejpam-6452	165	31	(	(	PUNCT
ejpam-6452	165	32	k	k	PROPN
ejpam-6452	165	33	−	−	PROPN
ejpam-6452	165	34	2)pk	2)pk	PROPN
ejpam-6452	165	35	,	,	PUNCT
ejpam-6452	165	36	n	n	CCONJ
ejpam-6452	165	37	)	)	PUNCT
ejpam-6452	165	38	=	=	SYM
ejpam-6452	165	39	qk	qk	PROPN
ejpam-6452	165	40	,	,	PUNCT
ejpam-6452	165	41	mqk	mqk	NOUN
ejpam-6452	165	42	,	,	PUNCT
ejpam-6452	165	43	n	n	PROPN
ejpam-6452	165	44	+	+	CCONJ
ejpam-6452	165	45	(	(	PUNCT
ejpam-6452	165	46	k	k	PROPN
ejpam-6452	165	47	−	−	PROPN
ejpam-6452	165	48	2)pk	2)pk	PROPN
ejpam-6452	165	49	,	,	PUNCT
ejpam-6452	165	50	mqk	mqk	NOUN
ejpam-6452	165	51	,	,	PUNCT
ejpam-6452	165	52	n	n	PROPN
ejpam-6452	165	53	+	+	CCONJ
ejpam-6452	165	54	(	(	PUNCT
ejpam-6452	165	55	k	k	PROPN
ejpam-6452	165	56	−	−	PROPN
ejpam-6452	165	57	2)pk	2)pk	PROPN
ejpam-6452	165	58	,	,	PUNCT
ejpam-6452	165	59	nqk	nqk	VERB
ejpam-6452	165	60	,	,	PUNCT
ejpam-6452	165	61	m	m	VERB
ejpam-6452	165	62	+	+	X
ejpam-6452	165	63	(	(	PUNCT
ejpam-6452	165	64	k	k	X
ejpam-6452	165	65	−	−	PROPN
ejpam-6452	165	66	2)2pk	2)2pk	NUM
ejpam-6452	165	67	,	,	PUNCT
ejpam-6452	165	68	mpk	mpk	NOUN
ejpam-6452	165	69	,	,	PUNCT
ejpam-6452	165	70	n	n	NOUN
ejpam-6452	165	71	=	=	SYM
ejpam-6452	165	72	qk	qk	PROPN
ejpam-6452	165	73	,	,	PUNCT
ejpam-6452	165	74	mqk	mqk	NOUN
ejpam-6452	165	75	,	,	PUNCT
ejpam-6452	165	76	n	n	PROPN
ejpam-6452	165	77	+	+	CCONJ
ejpam-6452	165	78	(	(	PUNCT
ejpam-6452	165	79	k	k	NOUN
ejpam-6452	165	80	−	−	PROPN
ejpam-6452	165	81	2	2	NUM
ejpam-6452	165	82	)	)	PUNCT
ejpam-6452	165	83	[	[	X
ejpam-6452	165	84	pk	pk	NOUN
ejpam-6452	165	85	,	,	PUNCT
ejpam-6452	165	86	mqk	mqk	NOUN
ejpam-6452	165	87	,	,	PUNCT
ejpam-6452	165	88	n	n	PROPN
ejpam-6452	165	89	+	+	CCONJ
ejpam-6452	165	90	pk	pk	NOUN
ejpam-6452	165	91	,	,	PUNCT
ejpam-6452	165	92	nqk	nqk	NOUN
ejpam-6452	165	93	,	,	PUNCT
ejpam-6452	165	94	m	m	VERB
ejpam-6452	165	95	+	+	NOUN
ejpam-6452	165	96	2(k	2(k	NUM
ejpam-6452	165	97	−	−	NOUN
ejpam-6452	166	1	2)pk	2)pk	NUM
ejpam-6452	166	2	,	,	PUNCT
ejpam-6452	166	3	mpk	mpk	NOUN
ejpam-6452	166	4	,	,	PUNCT
ejpam-6452	166	5	n	n	CCONJ
ejpam-6452	166	6	]	]	PUNCT
ejpam-6452	166	7	−	−	PROPN
ejpam-6452	166	8	(	(	PUNCT
ejpam-6452	166	9	k	k	X
ejpam-6452	166	10	−	−	PROPN
ejpam-6452	166	11	2)2pk	2)2pk	NUM
ejpam-6452	166	12	,	,	PUNCT
ejpam-6452	166	13	mpk	mpk	NOUN
ejpam-6452	166	14	,	,	PUNCT
ejpam-6452	166	15	n	n	NOUN
ejpam-6452	166	16	=	=	SYM
ejpam-6452	166	17	qk	qk	PROPN
ejpam-6452	166	18	,	,	PUNCT
ejpam-6452	166	19	mqk	mqk	NOUN
ejpam-6452	166	20	,	,	PUNCT
ejpam-6452	166	21	n	n	PROPN
ejpam-6452	166	22	+	+	NOUN
ejpam-6452	166	23	2(k	2(k	NUM
ejpam-6452	166	24	−	−	NOUN
ejpam-6452	167	1	2)pk	2)pk	NUM
ejpam-6452	167	2	,	,	PUNCT
ejpam-6452	167	3	m+n	m+n	NUM
ejpam-6452	167	4	−	−	PROPN
ejpam-6452	167	5	(	(	PUNCT
ejpam-6452	167	6	k	k	NOUN
ejpam-6452	167	7	−	−	PROPN
ejpam-6452	167	8	2)2pk	2)2pk	NUM
ejpam-6452	167	9	,	,	PUNCT
ejpam-6452	167	10	mpk	mpk	NOUN
ejpam-6452	167	11	,	,	PUNCT
ejpam-6452	167	12	n	n	CCONJ
ejpam-6452	167	13	,	,	PUNCT
ejpam-6452	167	14	we	we	PRON
ejpam-6452	167	15	get	get	VERB
ejpam-6452	167	16	2qk	2qk	ADJ
ejpam-6452	167	17	,	,	PUNCT
ejpam-6452	167	18	m+n	m+n	NOUN
ejpam-6452	167	19	=	=	SYM
ejpam-6452	167	20	2lk	2lk	PROPN
ejpam-6452	167	21	,	,	PUNCT
ejpam-6452	167	22	m+n	m+n	NOUN
ejpam-6452	167	23	−	−	PROPN
ejpam-6452	167	24	2(k	2(k	NUM
ejpam-6452	168	1	−	−	NOUN
ejpam-6452	169	1	2)pk	2)pk	NUM
ejpam-6452	169	2	,	,	PUNCT
ejpam-6452	169	3	m+n	m+n	NOUN
ejpam-6452	169	4	=	=	SYM
ejpam-6452	169	5	lk	lk	PROPN
ejpam-6452	169	6	,	,	PUNCT
ejpam-6452	169	7	mlk	mlk	PROPN
ejpam-6452	169	8	,	,	PUNCT
ejpam-6452	169	9	n	n	PROPN
ejpam-6452	169	10	+	+	CCONJ
ejpam-6452	169	11	(	(	PUNCT
ejpam-6452	169	12	k2	k2	PROPN
ejpam-6452	169	13	+	+	CCONJ
ejpam-6452	169	14	4k	4k	NOUN
ejpam-6452	169	15	−	−	NOUN
ejpam-6452	169	16	4	4	NUM
ejpam-6452	169	17	)	)	PUNCT
ejpam-6452	169	18	pk	pk	NOUN
ejpam-6452	169	19	,	,	PUNCT
ejpam-6452	169	20	mpk	mpk	NOUN
ejpam-6452	169	21	,	,	PUNCT
ejpam-6452	169	22	n	n	PRON
ejpam-6452	169	23	−	−	PROPN
ejpam-6452	169	24	2(k	2(k	NUM
ejpam-6452	169	25	−	−	NOUN
ejpam-6452	170	1	2)pk	2)pk	NUM
ejpam-6452	170	2	,	,	PUNCT
ejpam-6452	170	3	m+n	m+n	NOUN
ejpam-6452	170	4	=	=	SYM
ejpam-6452	170	5	qk	qk	PROPN
ejpam-6452	170	6	,	,	PUNCT
ejpam-6452	170	7	mqk	mqk	NOUN
ejpam-6452	170	8	,	,	PUNCT
ejpam-6452	170	9	n	n	CCONJ
ejpam-6452	170	10	−	−	PROPN
ejpam-6452	170	11	(	(	PUNCT
ejpam-6452	170	12	k	k	NOUN
ejpam-6452	170	13	−	−	PROPN
ejpam-6452	170	14	2)2pk	2)2pk	NUM
ejpam-6452	170	15	,	,	PUNCT
ejpam-6452	170	16	mpk	mpk	NOUN
ejpam-6452	170	17	,	,	PUNCT
ejpam-6452	170	18	n	n	PROPN
ejpam-6452	170	19	+	+	CCONJ
ejpam-6452	170	20	(	(	PUNCT
ejpam-6452	170	21	k2	k2	X
ejpam-6452	170	22	+	+	CCONJ
ejpam-6452	170	23	4k	4k	NOUN
ejpam-6452	170	24	−	−	PROPN
ejpam-6452	170	25	4)pk	4)pk	NUM
ejpam-6452	170	26	,	,	PUNCT
ejpam-6452	170	27	mpk	mpk	NOUN
ejpam-6452	170	28	,	,	PUNCT
ejpam-6452	170	29	n	n	NOUN
ejpam-6452	170	30	=	=	SYM
ejpam-6452	170	31	qk	qk	PROPN
ejpam-6452	170	32	,	,	PUNCT
ejpam-6452	170	33	mqk	mqk	NOUN
ejpam-6452	170	34	,	,	PUNCT
ejpam-6452	170	35	n	n	PROPN
ejpam-6452	170	36	+	+	CCONJ
ejpam-6452	170	37	8(k	8(k	NUM
ejpam-6452	170	38	−	−	NOUN
ejpam-6452	170	39	1)pk	1)pk	NUM
ejpam-6452	170	40	,	,	PUNCT
ejpam-6452	170	41	mpk	mpk	NOUN
ejpam-6452	170	42	,	,	PUNCT
ejpam-6452	170	43	n.	n.	PROPN
ejpam-6452	170	44	hence	hence	ADV
ejpam-6452	170	45	,	,	PUNCT
ejpam-6452	170	46	(	(	PUNCT
ejpam-6452	170	47	i	i	NOUN
ejpam-6452	170	48	)	)	PUNCT
ejpam-6452	170	49	and	and	CCONJ
ejpam-6452	170	50	(	(	PUNCT
ejpam-6452	170	51	ii	ii	NOUN
ejpam-6452	170	52	)	)	PUNCT
ejpam-6452	170	53	complete	complete	VERB
ejpam-6452	170	54	the	the	DET
ejpam-6452	170	55	proof	proof	NOUN
ejpam-6452	170	56	.	.	PUNCT
ejpam-6452	171	1	theorem	theorem	ADJ
ejpam-6452	171	2	5	5	NUM
ejpam-6452	171	3	(	(	PUNCT
ejpam-6452	171	4	asymptotic	asymptotic	ADJ
ejpam-6452	171	5	behavior	behavior	NOUN
ejpam-6452	171	6	)	)	PUNCT
ejpam-6452	171	7	.	.	PUNCT
ejpam-6452	172	1	let	let	VERB
ejpam-6452	172	2	k	k	PRON
ejpam-6452	172	3	be	be	AUX
ejpam-6452	172	4	a	a	DET
ejpam-6452	172	5	positive	positive	ADJ
ejpam-6452	172	6	integer	integer	NOUN
ejpam-6452	172	7	with	with	ADP
ejpam-6452	172	8	k	k	PROPN
ejpam-6452	172	9	≥	≥	NUM
ejpam-6452	172	10	2	2	NUM
ejpam-6452	172	11	.	.	PUNCT
ejpam-6452	173	1	then	then	ADV
ejpam-6452	173	2	lim	lim	PROPN
ejpam-6452	173	3	n→∞	n→∞	NUM
ejpam-6452	173	4	gpk	gpk	PROPN
ejpam-6452	173	5	,	,	PUNCT
ejpam-6452	173	6	n+1	n+1	PROPN
ejpam-6452	173	7	gpk	gpk	PROPN
ejpam-6452	173	8	,	,	PUNCT
ejpam-6452	173	9	n	n	NOUN
ejpam-6452	173	10	=	=	PUNCT
ejpam-6452	173	11	σk	σk	PROPN
ejpam-6452	173	12	.	.	NOUN
ejpam-6452	173	13	proof	proof	NOUN
ejpam-6452	173	14	.	.	PUNCT
ejpam-6452	174	1	by	by	ADP
ejpam-6452	174	2	using	use	VERB
ejpam-6452	174	3	(	(	PUNCT
ejpam-6452	174	4	3	3	NUM
ejpam-6452	174	5	)	)	PUNCT
ejpam-6452	174	6	,	,	PUNCT
ejpam-6452	174	7	we	we	PRON
ejpam-6452	174	8	have	have	VERB
ejpam-6452	174	9	lim	lim	PROPN
ejpam-6452	174	10	n→∞	n→∞	NUM
ejpam-6452	174	11	gpk	gpk	PROPN
ejpam-6452	174	12	,	,	PUNCT
ejpam-6452	174	13	n+1	n+1	PROPN
ejpam-6452	174	14	gpk	gpk	PROPN
ejpam-6452	174	15	,	,	PUNCT
ejpam-6452	174	16	n	n	PROPN
ejpam-6452	174	17	=	=	PROPN
ejpam-6452	174	18	lim	lim	PROPN
ejpam-6452	174	19	n→∞	n→∞	X
ejpam-6452	175	1	(	(	PUNCT
ejpam-6452	175	2	2b−ak+a∆k	2b−ak+a∆k	NUM
ejpam-6452	175	3	2∆k	2∆k	NUM
ejpam-6452	175	4	)	)	PUNCT
ejpam-6452	175	5	σn+1	σn+1	PROPN
ejpam-6452	176	1	k	k	PROPN
ejpam-6452	176	2	+	+	CCONJ
ejpam-6452	176	3	(	(	PUNCT
ejpam-6452	176	4	ak−2b+a∆k	ak−2b+a∆k	X
ejpam-6452	176	5	2∆k	2∆k	NUM
ejpam-6452	176	6	)	)	PUNCT
ejpam-6452	177	1	(	(	PUNCT
ejpam-6452	177	2	1−k)n+1	1−k)n+1	NUM
ejpam-6452	177	3	σn+1	σn+1	NOUN
ejpam-6452	177	4	k	k	NOUN
ejpam-6452	177	5	(	(	PUNCT
ejpam-6452	177	6	2b−ak+a∆k	2b−ak+a∆k	NUM
ejpam-6452	177	7	2∆k	2∆k	NUM
ejpam-6452	177	8	)	)	PUNCT
ejpam-6452	177	9	σn	σn	PROPN
ejpam-6452	178	1	k	k	PROPN
ejpam-6452	178	2	+	+	CCONJ
ejpam-6452	178	3	(	(	PUNCT
ejpam-6452	178	4	ak−2b+a∆k	ak−2b+a∆k	X
ejpam-6452	178	5	2∆k	2∆k	NUM
ejpam-6452	178	6	)	)	PUNCT
ejpam-6452	179	1	(	(	PUNCT
ejpam-6452	179	2	1−k)n	1−k)n	NUM
ejpam-6452	179	3	σn	σn	NOUN
ejpam-6452	179	4	k	k	PROPN
ejpam-6452	179	5	=	=	PROPN
ejpam-6452	179	6	lim	lim	PROPN
ejpam-6452	179	7	n→∞	n→∞	X
ejpam-6452	180	1	(	(	PUNCT
ejpam-6452	180	2	2b−ak+a∆k	2b−ak+a∆k	NUM
ejpam-6452	180	3	2∆k	2∆k	NUM
ejpam-6452	180	4	)	)	PUNCT
ejpam-6452	181	1	σk	σk	CCONJ
ejpam-6452	181	2	+	+	CCONJ
ejpam-6452	181	3	(	(	PUNCT
ejpam-6452	181	4	ak−2b+a∆k	ak−2b+a∆k	X
ejpam-6452	181	5	2∆k	2∆k	NUM
ejpam-6452	181	6	)	)	PUNCT
ejpam-6452	181	7	(	(	PUNCT
ejpam-6452	181	8	1−k)n	1−k)n	NUM
ejpam-6452	181	9	σ2n	σ2n	X
ejpam-6452	181	10	k	k	X
ejpam-6452	181	11	·	·	PUNCT
ejpam-6452	181	12	(	(	PUNCT
ejpam-6452	181	13	1−k	1−k	NUM
ejpam-6452	181	14	)	)	PUNCT
ejpam-6452	181	15	σk	σk	PROPN
ejpam-6452	181	16	(	(	PUNCT
ejpam-6452	181	17	2b−ak+a∆k	2b−ak+a∆k	NUM
ejpam-6452	181	18	2∆k	2∆k	NUM
ejpam-6452	181	19	)	)	PUNCT
ejpam-6452	182	1	+	+	CCONJ
ejpam-6452	182	2	(	(	PUNCT
ejpam-6452	182	3	ak−2b+a∆k	ak−2b+a∆k	X
ejpam-6452	182	4	2∆k	2∆k	NUM
ejpam-6452	182	5	)	)	PUNCT
ejpam-6452	182	6	(	(	PUNCT
ejpam-6452	182	7	1−k)n	1−k)n	NUM
ejpam-6452	182	8	σ2n	σ2n	X
ejpam-6452	182	9	k	k	X
ejpam-6452	182	10	.	.	PUNCT
ejpam-6452	183	1	(	(	PUNCT
ejpam-6452	183	2	6	6	X
ejpam-6452	183	3	)	)	PUNCT
ejpam-6452	183	4	w.	w.	NOUN
ejpam-6452	183	5	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	183	6	,	,	PUNCT
ejpam-6452	183	7	a.	a.	NOUN
ejpam-6452	183	8	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	183	9	/	/	SYM
ejpam-6452	183	10	eur	eur	PROPN
ejpam-6452	183	11	.	.	PUNCT
ejpam-6452	184	1	j.	j.	PROPN
ejpam-6452	184	2	pure	pure	PROPN
ejpam-6452	184	3	appl	appl	PROPN
ejpam-6452	184	4	.	.	PROPN
ejpam-6452	184	5	math	math	PROPN
ejpam-6452	184	6	,	,	PUNCT
ejpam-6452	184	7	18	18	NUM
ejpam-6452	184	8	(	(	PUNCT
ejpam-6452	184	9	3	3	NUM
ejpam-6452	184	10	)	)	PUNCT
ejpam-6452	184	11	(	(	PUNCT
ejpam-6452	184	12	2025	2025	NUM
ejpam-6452	184	13	)	)	PUNCT
ejpam-6452	184	14	,	,	PUNCT
ejpam-6452	184	15	6452	6452	NUM
ejpam-6452	184	16	9	9	NUM
ejpam-6452	184	17	of	of	ADP
ejpam-6452	184	18	18	18	NUM
ejpam-6452	184	19	since	since	SCONJ
ejpam-6452	184	20	σk	σk	ADV
ejpam-6452	184	21	is	be	AUX
ejpam-6452	184	22	the	the	DET
ejpam-6452	184	23	root	root	NOUN
ejpam-6452	184	24	of	of	ADP
ejpam-6452	184	25	(	(	PUNCT
ejpam-6452	184	26	4	4	NUM
ejpam-6452	184	27	)	)	PUNCT
ejpam-6452	185	1	,	,	PUNCT
ejpam-6452	185	2	we	we	PRON
ejpam-6452	185	3	have	have	VERB
ejpam-6452	185	4	σ2	σ2	PROPN
ejpam-6452	185	5	k	k	PROPN
ejpam-6452	185	6	=	=	PUNCT
ejpam-6452	185	7	kσk	kσk	PROPN
ejpam-6452	185	8	+	+	CCONJ
ejpam-6452	186	1	(	(	PUNCT
ejpam-6452	186	2	k	k	NOUN
ejpam-6452	186	3	−	−	PROPN
ejpam-6452	186	4	1	1	NUM
ejpam-6452	186	5	)	)	PUNCT
ejpam-6452	186	6	>	>	X
ejpam-6452	187	1	k	k	X
ejpam-6452	188	1	−	−	NOUN
ejpam-6452	188	2	1	1	NUM
ejpam-6452	188	3	and	and	CCONJ
ejpam-6452	188	4	so	so	ADV
ejpam-6452	188	5	∣∣∣1−k	∣∣∣1−k	PROPN
ejpam-6452	188	6	σ2	σ2	PROPN
ejpam-6452	188	7	k	k	PROPN
ejpam-6452	188	8	∣∣∣	∣∣∣	X
ejpam-6452	188	9	<	<	X
ejpam-6452	188	10	1	1	NUM
ejpam-6452	188	11	.	.	PUNCT
ejpam-6452	188	12	then	then	ADV
ejpam-6452	188	13	lim	lim	PROPN
ejpam-6452	188	14	n→∞	n→∞	X
ejpam-6452	188	15	(	(	PUNCT
ejpam-6452	188	16	1	1	NUM
ejpam-6452	188	17	−	−	NOUN
ejpam-6452	189	1	k)n	k)n	NOUN
ejpam-6452	189	2	σ2n	σ2n	X
ejpam-6452	189	3	k	k	X
ejpam-6452	190	1	=	=	SYM
ejpam-6452	190	2	lim	lim	PROPN
ejpam-6452	190	3	n→∞	n→∞	X
ejpam-6452	190	4	(	(	PUNCT
ejpam-6452	190	5	1	1	NUM
ejpam-6452	190	6	−	−	PROPN
ejpam-6452	190	7	k	k	PROPN
ejpam-6452	190	8	σ2	σ2	PROPN
ejpam-6452	190	9	k	k	PROPN
ejpam-6452	190	10	)	)	PUNCT
ejpam-6452	190	11	n	n	PROPN
ejpam-6452	190	12	=	=	SYM
ejpam-6452	190	13	0	0	PROPN
ejpam-6452	190	14	.	.	PUNCT
ejpam-6452	191	1	this	this	PRON
ejpam-6452	191	2	together	together	ADV
ejpam-6452	191	3	with	with	ADP
ejpam-6452	191	4	(	(	PUNCT
ejpam-6452	191	5	6	6	NUM
ejpam-6452	191	6	)	)	PUNCT
ejpam-6452	191	7	gives	give	VERB
ejpam-6452	191	8	lim	lim	PROPN
ejpam-6452	191	9	n→∞	n→∞	NUM
ejpam-6452	191	10	gpk	gpk	PROPN
ejpam-6452	191	11	,	,	PUNCT
ejpam-6452	191	12	n+1	n+1	PROPN
ejpam-6452	191	13	gpk	gpk	PROPN
ejpam-6452	191	14	,	,	PUNCT
ejpam-6452	191	15	n	n	NOUN
ejpam-6452	191	16	=	=	PUNCT
ejpam-6452	191	17	σk	σk	PROPN
ejpam-6452	191	18	.	.	PUNCT
ejpam-6452	192	1	this	this	PRON
ejpam-6452	192	2	completes	complete	VERB
ejpam-6452	192	3	the	the	DET
ejpam-6452	192	4	proof	proof	NOUN
ejpam-6452	192	5	.	.	PUNCT
ejpam-6452	193	1	if	if	SCONJ
ejpam-6452	193	2	(	(	PUNCT
ejpam-6452	193	3	a	a	PRON
ejpam-6452	193	4	,	,	PUNCT
ejpam-6452	193	5	b	b	NOUN
ejpam-6452	193	6	)	)	PUNCT
ejpam-6452	193	7	∈	∈	NOUN
ejpam-6452	193	8	{	{	PUNCT
ejpam-6452	193	9	(	(	PUNCT
ejpam-6452	193	10	2	2	NUM
ejpam-6452	193	11	,	,	PUNCT
ejpam-6452	193	12	k	k	NOUN
ejpam-6452	193	13	)	)	PUNCT
ejpam-6452	193	14	,	,	PUNCT
ejpam-6452	193	15	(	(	PUNCT
ejpam-6452	193	16	0	0	NUM
ejpam-6452	193	17	,	,	PUNCT
ejpam-6452	193	18	1	1	NUM
ejpam-6452	193	19	)	)	PUNCT
ejpam-6452	193	20	,	,	PUNCT
ejpam-6452	193	21	(	(	PUNCT
ejpam-6452	193	22	2	2	NUM
ejpam-6452	193	23	,	,	PUNCT
ejpam-6452	193	24	2	2	NUM
ejpam-6452	193	25	)	)	PUNCT
ejpam-6452	193	26	,	,	PUNCT
ejpam-6452	193	27	(	(	PUNCT
ejpam-6452	193	28	1	1	NUM
ejpam-6452	193	29	,	,	PUNCT
ejpam-6452	193	30	1	1	NUM
ejpam-6452	193	31	)	)	PUNCT
ejpam-6452	193	32	}	}	PUNCT
ejpam-6452	193	33	,	,	PUNCT
ejpam-6452	193	34	then	then	ADV
ejpam-6452	193	35	we	we	PRON
ejpam-6452	193	36	have	have	VERB
ejpam-6452	193	37	the	the	DET
ejpam-6452	193	38	following	follow	VERB
ejpam-6452	193	39	:	:	PUNCT
ejpam-6452	193	40	corollary	corollary	ADJ
ejpam-6452	193	41	7	7	NUM
ejpam-6452	193	42	(	(	PUNCT
ejpam-6452	193	43	[	[	X
ejpam-6452	193	44	5	5	NUM
ejpam-6452	193	45	,	,	PUNCT
ejpam-6452	193	46	lemma	lemma	PROPN
ejpam-6452	193	47	2.4	2.4	NUM
ejpam-6452	193	48	]	]	PUNCT
ejpam-6452	193	49	)	)	PUNCT
ejpam-6452	193	50	.	.	PUNCT
ejpam-6452	194	1	let	let	VERB
ejpam-6452	194	2	k	k	PRON
ejpam-6452	194	3	be	be	AUX
ejpam-6452	194	4	a	a	DET
ejpam-6452	194	5	positive	positive	ADJ
ejpam-6452	194	6	integer	integer	NOUN
ejpam-6452	194	7	with	with	ADP
ejpam-6452	194	8	k	k	PROPN
ejpam-6452	194	9	≥	≥	NUM
ejpam-6452	194	10	2	2	NUM
ejpam-6452	194	11	.	.	PUNCT
ejpam-6452	195	1	then	then	ADV
ejpam-6452	195	2	lim	lim	PROPN
ejpam-6452	195	3	n→∞	n→∞	NUM
ejpam-6452	195	4	lk	lk	PROPN
ejpam-6452	195	5	,	,	PUNCT
ejpam-6452	195	6	n+1	n+1	PROPN
ejpam-6452	195	7	lk	lk	PROPN
ejpam-6452	195	8	,	,	PUNCT
ejpam-6452	195	9	n	n	NOUN
ejpam-6452	195	10	=	=	PROPN
ejpam-6452	195	11	lim	lim	PROPN
ejpam-6452	195	12	n→∞	n→∞	NUM
ejpam-6452	195	13	pk	pk	PROPN
ejpam-6452	195	14	,	,	PUNCT
ejpam-6452	195	15	n+1	n+1	PROPN
ejpam-6452	195	16	pk	pk	NOUN
ejpam-6452	195	17	,	,	PUNCT
ejpam-6452	195	18	n	n	PROPN
ejpam-6452	195	19	=	=	PROPN
ejpam-6452	195	20	lim	lim	PROPN
ejpam-6452	195	21	n→∞	n→∞	NUM
ejpam-6452	195	22	qk	qk	PROPN
ejpam-6452	195	23	,	,	PUNCT
ejpam-6452	195	24	n+1	n+1	PROPN
ejpam-6452	195	25	qk	qk	PROPN
ejpam-6452	195	26	,	,	PUNCT
ejpam-6452	195	27	n	n	PROPN
ejpam-6452	195	28	=	=	PROPN
ejpam-6452	195	29	lim	lim	PROPN
ejpam-6452	195	30	n→∞	n→∞	NUM
ejpam-6452	195	31	qk	qk	PROPN
ejpam-6452	195	32	,	,	PUNCT
ejpam-6452	195	33	n+1	n+1	PROPN
ejpam-6452	195	34	qk	qk	PROPN
ejpam-6452	195	35	,	,	PUNCT
ejpam-6452	195	36	n	n	NOUN
ejpam-6452	195	37	=	=	SYM
ejpam-6452	195	38	σk	σk	PROPN
ejpam-6452	195	39	.	.	PUNCT
ejpam-6452	196	1	if	if	SCONJ
ejpam-6452	196	2	k	k	PROPN
ejpam-6452	196	3	=	=	SYM
ejpam-6452	196	4	2	2	NUM
ejpam-6452	196	5	,	,	PUNCT
ejpam-6452	196	6	then	then	ADV
ejpam-6452	196	7	we	we	PRON
ejpam-6452	196	8	have	have	VERB
ejpam-6452	196	9	the	the	DET
ejpam-6452	196	10	following	follow	VERB
ejpam-6452	196	11	:	:	PUNCT
ejpam-6452	196	12	corollary	corollary	ADJ
ejpam-6452	196	13	8	8	NUM
ejpam-6452	196	14	.	.	PUNCT
ejpam-6452	197	1	limn→∞	limn→∞	PROPN
ejpam-6452	197	2	ha	ha	INTJ
ejpam-6452	197	3	,	,	PUNCT
ejpam-6452	197	4	b	b	PROPN
ejpam-6452	197	5	n+1	n+1	PROPN
ejpam-6452	197	6	ha	ha	INTJ
ejpam-6452	197	7	,	,	PUNCT
ejpam-6452	197	8	b	b	PROPN
ejpam-6452	197	9	n	n	NOUN
ejpam-6452	197	10	=	=	SYM
ejpam-6452	197	11	1	1	NUM
ejpam-6452	197	12	+	+	CCONJ
ejpam-6452	197	13	√	√	NUM
ejpam-6452	197	14	2	2	NUM
ejpam-6452	197	15	.	.	PUNCT
ejpam-6452	197	16	theorem	theorem	NOUN
ejpam-6452	197	17	6	6	NUM
ejpam-6452	197	18	(	(	PUNCT
ejpam-6452	197	19	catalan	catalan	NOUN
ejpam-6452	197	20	’s	’s	PART
ejpam-6452	197	21	identities	identity	NOUN
ejpam-6452	197	22	)	)	PUNCT
ejpam-6452	197	23	.	.	PUNCT
ejpam-6452	198	1	let	let	VERB
ejpam-6452	198	2	k	k	NOUN
ejpam-6452	198	3	,	,	PUNCT
ejpam-6452	198	4	n	n	CCONJ
ejpam-6452	198	5	,	,	PUNCT
ejpam-6452	198	6	and	and	CCONJ
ejpam-6452	198	7	r	r	NOUN
ejpam-6452	198	8	be	be	VERB
ejpam-6452	198	9	non	non	ADJ
ejpam-6452	198	10	-	-	ADJ
ejpam-6452	198	11	negative	negative	ADJ
ejpam-6452	198	12	integers	integer	NOUN
ejpam-6452	198	13	with	with	ADP
ejpam-6452	198	14	k	k	PROPN
ejpam-6452	198	15	≥	≥	NUM
ejpam-6452	198	16	2	2	NUM
ejpam-6452	198	17	,	,	PUNCT
ejpam-6452	198	18	∆k	∆k	PROPN
ejpam-6452	198	19	=	=	SYM
ejpam-6452	198	20	√	√	PROPN
ejpam-6452	198	21	k2	k2	NOUN
ejpam-6452	198	22	+	+	CCONJ
ejpam-6452	198	23	4k	4k	NOUN
ejpam-6452	198	24	−	−	PROPN
ejpam-6452	198	25	4	4	NUM
ejpam-6452	198	26	,	,	PUNCT
ejpam-6452	198	27	and	and	CCONJ
ejpam-6452	198	28	n	n	PRON
ejpam-6452	198	29	≥	≥	PROPN
ejpam-6452	198	30	r.	r.	PROPN
ejpam-6452	198	31	then	then	ADV
ejpam-6452	198	32	gpk	gpk	PROPN
ejpam-6452	198	33	,	,	PUNCT
ejpam-6452	198	34	n−rgpk	n−rgpk	PROPN
ejpam-6452	198	35	,	,	PUNCT
ejpam-6452	198	36	n+r	n+r	PROPN
ejpam-6452	198	37	−	−	PROPN
ejpam-6452	198	38	gp2	gp2	PROPN
ejpam-6452	198	39	k	k	PROPN
ejpam-6452	198	40	,	,	PUNCT
ejpam-6452	198	41	n	n	PROPN
ejpam-6452	198	42	=	=	SYM
ejpam-6452	198	43	1	1	NUM
ejpam-6452	198	44	4	4	NUM
ejpam-6452	198	45	(	(	PUNCT
ejpam-6452	198	46	2b−	2b−	PROPN
ejpam-6452	198	47	ak	ak	NOUN
ejpam-6452	198	48	+	+	NOUN
ejpam-6452	198	49	a∆k)(ak	a∆k)(ak	NUM
ejpam-6452	198	50	−	−	NOUN
ejpam-6452	198	51	2b	2b	NOUN
ejpam-6452	199	1	+	+	CCONJ
ejpam-6452	199	2	a∆k)(1	a∆k)(1	ADP
ejpam-6452	199	3	−	−	PROPN
ejpam-6452	199	4	k)n−rp2	k)n−rp2	PROPN
ejpam-6452	199	5	k	k	PROPN
ejpam-6452	199	6	,	,	PUNCT
ejpam-6452	199	7	r.	r.	PROPN
ejpam-6452	199	8	proof	proof	NOUN
ejpam-6452	199	9	.	.	PUNCT
ejpam-6452	200	1	let	let	VERB
ejpam-6452	200	2	α	α	NOUN
ejpam-6452	200	3	=	=	SYM
ejpam-6452	200	4	2b−ak+a∆k	2b−ak+a∆k	NUM
ejpam-6452	200	5	2∆k	2∆k	NUM
ejpam-6452	200	6	and	and	CCONJ
ejpam-6452	200	7	β	β	X
ejpam-6452	200	8	=	=	SYM
ejpam-6452	200	9	ak−2b+a∆k	ak−2b+a∆k	X
ejpam-6452	200	10	2∆k	2∆k	NUM
ejpam-6452	200	11	.	.	PUNCT
ejpam-6452	201	1	by	by	ADP
ejpam-6452	201	2	using	use	VERB
ejpam-6452	201	3	(	(	PUNCT
ejpam-6452	201	4	3	3	NUM
ejpam-6452	201	5	)	)	PUNCT
ejpam-6452	201	6	,	,	PUNCT
ejpam-6452	201	7	we	we	PRON
ejpam-6452	201	8	have	have	VERB
ejpam-6452	201	9	gpk	gpk	PROPN
ejpam-6452	201	10	,	,	PUNCT
ejpam-6452	201	11	n−rgpk	n−rgpk	PROPN
ejpam-6452	201	12	,	,	PUNCT
ejpam-6452	201	13	n+r	n+r	X
ejpam-6452	201	14	=	=	SYM
ejpam-6452	201	15	(	(	PUNCT
ejpam-6452	201	16	ασn−r	ασn−r	NOUN
ejpam-6452	201	17	k	k	PROPN
ejpam-6452	202	1	+	+	CCONJ
ejpam-6452	202	2	β	β	X
ejpam-6452	202	3	(	(	PUNCT
ejpam-6452	202	4	1	1	NUM
ejpam-6452	202	5	−	−	PROPN
ejpam-6452	202	6	k)n−r	k)n−r	NOUN
ejpam-6452	202	7	σn−r	σn−r	NOUN
ejpam-6452	202	8	k	k	PROPN
ejpam-6452	202	9	)	)	PUNCT
ejpam-6452	202	10	(	(	PUNCT
ejpam-6452	203	1	ασn+r	ασn+r	PROPN
ejpam-6452	203	2	k	k	NOUN
ejpam-6452	204	1	+	+	NUM
ejpam-6452	204	2	β	β	X
ejpam-6452	204	3	(	(	PUNCT
ejpam-6452	204	4	1	1	NUM
ejpam-6452	204	5	−	−	PROPN
ejpam-6452	204	6	k)n+r	k)n+r	PROPN
ejpam-6452	204	7	σn+r	σn+r	PROPN
ejpam-6452	204	8	k	k	NOUN
ejpam-6452	204	9	)	)	PUNCT
ejpam-6452	204	10	=	=	SYM
ejpam-6452	204	11	α2σ2n	α2σ2n	NUM
ejpam-6452	205	1	k	k	X
ejpam-6452	205	2	+	+	CCONJ
ejpam-6452	205	3	β2	β2	NOUN
ejpam-6452	205	4	(	(	PUNCT
ejpam-6452	205	5	1	1	NUM
ejpam-6452	205	6	−	−	PROPN
ejpam-6452	205	7	k)2n	k)2n	ADJ
ejpam-6452	205	8	σ2n	σ2n	NOUN
ejpam-6452	205	9	k	k	PROPN
ejpam-6452	206	1	+	+	CCONJ
ejpam-6452	206	2	αβ(1	αβ(1	PROPN
ejpam-6452	206	3	−	−	PROPN
ejpam-6452	206	4	k)n−r	k)n−r	NOUN
ejpam-6452	206	5	(	(	PUNCT
ejpam-6452	206	6	σ2r	σ2r	PROPN
ejpam-6452	206	7	k	k	X
ejpam-6452	206	8	+	+	CCONJ
ejpam-6452	206	9	(	(	PUNCT
ejpam-6452	206	10	1	1	NUM
ejpam-6452	206	11	−	−	NOUN
ejpam-6452	206	12	k)2r	k)2r	NOUN
ejpam-6452	206	13	σ2r	σ2r	VERB
ejpam-6452	206	14	k	k	PROPN
ejpam-6452	206	15	)	)	PUNCT
ejpam-6452	206	16	and	and	CCONJ
ejpam-6452	206	17	gp2	gp2	PROPN
ejpam-6452	206	18	k	k	PROPN
ejpam-6452	206	19	,	,	PUNCT
ejpam-6452	206	20	n	n	PROPN
ejpam-6452	206	21	=	=	PUNCT
ejpam-6452	206	22	(	(	PUNCT
ejpam-6452	206	23	ασn	ασn	NOUN
ejpam-6452	207	1	k	k	NOUN
ejpam-6452	208	1	+	+	CCONJ
ejpam-6452	208	2	β	β	X
ejpam-6452	208	3	(	(	PUNCT
ejpam-6452	208	4	1	1	NUM
ejpam-6452	208	5	−	−	NOUN
ejpam-6452	208	6	k)n	k)n	NOUN
ejpam-6452	208	7	σn	σn	PROPN
ejpam-6452	208	8	k	k	PROPN
ejpam-6452	208	9	)	)	PUNCT
ejpam-6452	208	10	2	2	NUM
ejpam-6452	208	11	=	=	SYM
ejpam-6452	208	12	α2σ2n	α2σ2n	NUM
ejpam-6452	208	13	k	k	PROPN
ejpam-6452	209	1	+	+	CCONJ
ejpam-6452	209	2	β2	β2	NOUN
ejpam-6452	209	3	(	(	PUNCT
ejpam-6452	209	4	1	1	NUM
ejpam-6452	209	5	−	−	PROPN
ejpam-6452	209	6	k)2n	k)2n	ADJ
ejpam-6452	209	7	σ2n	σ2n	NOUN
ejpam-6452	209	8	k	k	PROPN
ejpam-6452	210	1	+	+	CCONJ
ejpam-6452	210	2	αβ(1	αβ(1	PROPN
ejpam-6452	210	3	−	−	PROPN
ejpam-6452	210	4	k)n−r	k)n−r	NOUN
ejpam-6452	210	5	(	(	PUNCT
ejpam-6452	210	6	2(1	2(1	NUM
ejpam-6452	210	7	−	−	NOUN
ejpam-6452	210	8	k)r	k)r	NOUN
ejpam-6452	210	9	)	)	PUNCT
ejpam-6452	210	10	.	.	PUNCT
ejpam-6452	211	1	then	then	ADV
ejpam-6452	211	2	gpk	gpk	PROPN
ejpam-6452	211	3	,	,	PUNCT
ejpam-6452	211	4	n−rgpk	n−rgpk	PROPN
ejpam-6452	211	5	,	,	PUNCT
ejpam-6452	211	6	n+r	n+r	PROPN
ejpam-6452	211	7	−	−	PROPN
ejpam-6452	211	8	gp2	gp2	PROPN
ejpam-6452	211	9	k	k	PROPN
ejpam-6452	211	10	,	,	PUNCT
ejpam-6452	211	11	n	n	PROPN
ejpam-6452	211	12	=	=	SYM
ejpam-6452	211	13	αβ(1	αβ(1	PROPN
ejpam-6452	211	14	−	−	PROPN
ejpam-6452	211	15	k)n−r	k)n−r	NOUN
ejpam-6452	211	16	(	(	PUNCT
ejpam-6452	211	17	σ2r	σ2r	PROPN
ejpam-6452	211	18	k	k	X
ejpam-6452	211	19	+	+	CCONJ
ejpam-6452	211	20	(	(	PUNCT
ejpam-6452	211	21	1	1	NUM
ejpam-6452	211	22	−	−	NOUN
ejpam-6452	211	23	k)2r	k)2r	NOUN
ejpam-6452	211	24	σ2r	σ2r	VERB
ejpam-6452	211	25	k	k	PROPN
ejpam-6452	211	26	−	−	PROPN
ejpam-6452	211	27	2(1	2(1	NUM
ejpam-6452	211	28	−	−	NOUN
ejpam-6452	211	29	k)r	k)r	NOUN
ejpam-6452	211	30	)	)	PUNCT
ejpam-6452	212	1	=	=	PUNCT
ejpam-6452	212	2	(	(	PUNCT
ejpam-6452	212	3	2b−	2b−	PROPN
ejpam-6452	212	4	ak	ak	AUX
ejpam-6452	212	5	+	+	NUM
ejpam-6452	212	6	a∆k	a∆k	ADJ
ejpam-6452	212	7	2∆k	2∆k	NUM
ejpam-6452	212	8	)	)	PUNCT
ejpam-6452	212	9	(	(	PUNCT
ejpam-6452	212	10	ak	ak	PROPN
ejpam-6452	212	11	−	−	PROPN
ejpam-6452	212	12	2b	2b	NOUN
ejpam-6452	212	13	+	+	CCONJ
ejpam-6452	212	14	a∆k	a∆k	ADJ
ejpam-6452	212	15	2∆k	2∆k	NUM
ejpam-6452	212	16	)	)	PUNCT
ejpam-6452	212	17	(	(	PUNCT
ejpam-6452	212	18	1	1	NUM
ejpam-6452	212	19	−	−	NOUN
ejpam-6452	212	20	k)n−r	k)n−r	NOUN
ejpam-6452	212	21	(	(	PUNCT
ejpam-6452	212	22	σr	σr	PROPN
ejpam-6452	212	23	k	k	PROPN
ejpam-6452	212	24	−	−	PROPN
ejpam-6452	212	25	(	(	PUNCT
ejpam-6452	212	26	1	1	NUM
ejpam-6452	212	27	−	−	NOUN
ejpam-6452	212	28	k)r	k)r	X
ejpam-6452	212	29	σr	σr	PROPN
ejpam-6452	212	30	k	k	PROPN
ejpam-6452	212	31	)	)	PUNCT
ejpam-6452	212	32	2	2	PROPN
ejpam-6452	212	33	w.	w.	NOUN
ejpam-6452	212	34	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	212	35	,	,	PUNCT
ejpam-6452	212	36	a.	a.	NOUN
ejpam-6452	212	37	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	212	38	/	/	SYM
ejpam-6452	212	39	eur	eur	PROPN
ejpam-6452	212	40	.	.	PUNCT
ejpam-6452	213	1	j.	j.	PROPN
ejpam-6452	213	2	pure	pure	PROPN
ejpam-6452	213	3	appl	appl	PROPN
ejpam-6452	213	4	.	.	PROPN
ejpam-6452	213	5	math	math	PROPN
ejpam-6452	213	6	,	,	PUNCT
ejpam-6452	213	7	18	18	NUM
ejpam-6452	213	8	(	(	PUNCT
ejpam-6452	213	9	3	3	NUM
ejpam-6452	213	10	)	)	PUNCT
ejpam-6452	213	11	(	(	PUNCT
ejpam-6452	213	12	2025	2025	NUM
ejpam-6452	213	13	)	)	PUNCT
ejpam-6452	213	14	,	,	PUNCT
ejpam-6452	213	15	6452	6452	NUM
ejpam-6452	213	16	10	10	NUM
ejpam-6452	213	17	of	of	ADP
ejpam-6452	213	18	18	18	NUM
ejpam-6452	213	19	=	=	SYM
ejpam-6452	213	20	1	1	NUM
ejpam-6452	213	21	4	4	NUM
ejpam-6452	213	22	(	(	PUNCT
ejpam-6452	213	23	2b−	2b−	PROPN
ejpam-6452	213	24	ak	ak	NOUN
ejpam-6452	213	25	+	+	NOUN
ejpam-6452	213	26	a∆k)(ak	a∆k)(ak	NUM
ejpam-6452	213	27	−	−	NOUN
ejpam-6452	213	28	2b	2b	NOUN
ejpam-6452	213	29	+	+	CCONJ
ejpam-6452	213	30	a∆k	a∆k	ADJ
ejpam-6452	213	31	)	)	PUNCT
ejpam-6452	213	32	(	(	PUNCT
ejpam-6452	213	33	1	1	NUM
ejpam-6452	213	34	−	−	NOUN
ejpam-6452	213	35	k)n−r	k)n−r	NOUN
ejpam-6452	213	36	[	[	PUNCT
ejpam-6452	213	37	1	1	NUM
ejpam-6452	213	38	∆k	∆k	PROPN
ejpam-6452	213	39	(	(	PUNCT
ejpam-6452	213	40	σr	σr	PROPN
ejpam-6452	213	41	k	k	PROPN
ejpam-6452	214	1	+	+	CCONJ
ejpam-6452	214	2	(	(	PUNCT
ejpam-6452	214	3	1	1	NUM
ejpam-6452	214	4	−	−	NOUN
ejpam-6452	214	5	k)r	k)r	X
ejpam-6452	214	6	σr	σr	PROPN
ejpam-6452	214	7	k	k	PROPN
ejpam-6452	214	8	)	)	PUNCT
ejpam-6452	214	9	]	]	SYM
ejpam-6452	214	10	2	2	NUM
ejpam-6452	214	11	=	=	SYM
ejpam-6452	214	12	1	1	NUM
ejpam-6452	214	13	4	4	NUM
ejpam-6452	214	14	(	(	PUNCT
ejpam-6452	214	15	2b−	2b−	PROPN
ejpam-6452	214	16	ak	ak	NOUN
ejpam-6452	214	17	+	+	NOUN
ejpam-6452	214	18	a∆k)(ak	a∆k)(ak	NUM
ejpam-6452	214	19	−	−	NOUN
ejpam-6452	214	20	2b	2b	NOUN
ejpam-6452	214	21	+	+	CCONJ
ejpam-6452	214	22	a∆k)(1	a∆k)(1	ADP
ejpam-6452	214	23	−	−	PROPN
ejpam-6452	214	24	k)n−rp2	k)n−rp2	PROPN
ejpam-6452	214	25	k	k	PROPN
ejpam-6452	214	26	,	,	PUNCT
ejpam-6452	214	27	r.	r.	PROPN
ejpam-6452	214	28	this	this	PRON
ejpam-6452	214	29	completes	complete	VERB
ejpam-6452	214	30	the	the	DET
ejpam-6452	214	31	proof	proof	NOUN
ejpam-6452	214	32	.	.	PUNCT
ejpam-6452	215	1	if	if	SCONJ
ejpam-6452	215	2	(	(	PUNCT
ejpam-6452	215	3	a	a	PRON
ejpam-6452	215	4	,	,	PUNCT
ejpam-6452	215	5	b	b	NOUN
ejpam-6452	215	6	)	)	PUNCT
ejpam-6452	215	7	∈	∈	NOUN
ejpam-6452	215	8	{	{	PUNCT
ejpam-6452	215	9	(	(	PUNCT
ejpam-6452	215	10	2	2	NUM
ejpam-6452	215	11	,	,	PUNCT
ejpam-6452	215	12	k	k	NOUN
ejpam-6452	215	13	)	)	PUNCT
ejpam-6452	215	14	,	,	PUNCT
ejpam-6452	215	15	(	(	PUNCT
ejpam-6452	215	16	0	0	NUM
ejpam-6452	215	17	,	,	PUNCT
ejpam-6452	215	18	1	1	NUM
ejpam-6452	215	19	)	)	PUNCT
ejpam-6452	215	20	,	,	PUNCT
ejpam-6452	215	21	(	(	PUNCT
ejpam-6452	215	22	2	2	NUM
ejpam-6452	215	23	,	,	PUNCT
ejpam-6452	215	24	2	2	NUM
ejpam-6452	215	25	)	)	PUNCT
ejpam-6452	215	26	,	,	PUNCT
ejpam-6452	215	27	(	(	PUNCT
ejpam-6452	215	28	1	1	NUM
ejpam-6452	215	29	,	,	PUNCT
ejpam-6452	215	30	1	1	NUM
ejpam-6452	215	31	)	)	PUNCT
ejpam-6452	215	32	}	}	PUNCT
ejpam-6452	215	33	,	,	PUNCT
ejpam-6452	215	34	then	then	ADV
ejpam-6452	215	35	we	we	PRON
ejpam-6452	215	36	have	have	VERB
ejpam-6452	215	37	the	the	DET
ejpam-6452	215	38	following	follow	VERB
ejpam-6452	215	39	:	:	PUNCT
ejpam-6452	215	40	corollary	corollary	ADJ
ejpam-6452	215	41	9	9	NUM
ejpam-6452	215	42	(	(	PUNCT
ejpam-6452	215	43	[	[	X
ejpam-6452	215	44	5	5	NUM
ejpam-6452	215	45	,	,	PUNCT
ejpam-6452	215	46	theorem	theorem	VERB
ejpam-6452	215	47	2.6	2.6	NUM
ejpam-6452	215	48	]	]	PUNCT
ejpam-6452	215	49	)	)	PUNCT
ejpam-6452	215	50	.	.	PUNCT
ejpam-6452	216	1	let	let	VERB
ejpam-6452	216	2	k	k	NOUN
ejpam-6452	216	3	,	,	PUNCT
ejpam-6452	216	4	n	n	CCONJ
ejpam-6452	216	5	,	,	PUNCT
ejpam-6452	216	6	and	and	CCONJ
ejpam-6452	216	7	r	r	NOUN
ejpam-6452	216	8	be	be	VERB
ejpam-6452	216	9	non	non	ADJ
ejpam-6452	216	10	-	-	ADJ
ejpam-6452	216	11	negative	negative	ADJ
ejpam-6452	216	12	integers	integer	NOUN
ejpam-6452	216	13	with	with	ADP
ejpam-6452	216	14	k	k	PROPN
ejpam-6452	216	15	≥	≥	NUM
ejpam-6452	216	16	2	2	NUM
ejpam-6452	216	17	,	,	PUNCT
ejpam-6452	216	18	∆k	∆k	PROPN
ejpam-6452	216	19	=	=	SYM
ejpam-6452	216	20	√	√	PROPN
ejpam-6452	216	21	k2	k2	NOUN
ejpam-6452	216	22	+	+	CCONJ
ejpam-6452	216	23	4k	4k	NOUN
ejpam-6452	216	24	−	−	PROPN
ejpam-6452	216	25	4	4	NUM
ejpam-6452	216	26	,	,	PUNCT
ejpam-6452	216	27	and	and	CCONJ
ejpam-6452	216	28	n	n	PRON
ejpam-6452	216	29	≥	≥	NOUN
ejpam-6452	216	30	r.	r.	PROPN
ejpam-6452	216	31	then	then	ADV
ejpam-6452	216	32	(	(	PUNCT
ejpam-6452	216	33	i	i	NOUN
ejpam-6452	216	34	)	)	PUNCT
ejpam-6452	216	35	lk	lk	PROPN
ejpam-6452	216	36	,	,	PUNCT
ejpam-6452	216	37	n−rlk	n−rlk	NOUN
ejpam-6452	216	38	,	,	PUNCT
ejpam-6452	216	39	n+r	n+r	PROPN
ejpam-6452	216	40	−	−	PROPN
ejpam-6452	216	41	l2	l2	PROPN
ejpam-6452	216	42	k	k	PROPN
ejpam-6452	216	43	,	,	PUNCT
ejpam-6452	216	44	n	n	PROPN
ejpam-6452	216	45	=	=	SYM
ejpam-6452	216	46	∆2	∆2	PROPN
ejpam-6452	216	47	k(1	k(1	PROPN
ejpam-6452	216	48	−	−	PROPN
ejpam-6452	216	49	k)n−rp2	k)n−rp2	PROPN
ejpam-6452	216	50	k	k	PROPN
ejpam-6452	216	51	,	,	PUNCT
ejpam-6452	216	52	r	r	NOUN
ejpam-6452	216	53	;	;	PUNCT
ejpam-6452	216	54	(	(	PUNCT
ejpam-6452	216	55	ii	ii	NOUN
ejpam-6452	216	56	)	)	PUNCT
ejpam-6452	216	57	pk	pk	NOUN
ejpam-6452	216	58	,	,	PUNCT
ejpam-6452	216	59	n−rpk	n−rpk	ADJ
ejpam-6452	216	60	,	,	PUNCT
ejpam-6452	216	61	n+r	n+r	PROPN
ejpam-6452	216	62	−	−	PROPN
ejpam-6452	216	63	p2	p2	PROPN
ejpam-6452	216	64	k	k	PROPN
ejpam-6452	216	65	,	,	PUNCT
ejpam-6452	216	66	n	n	NOUN
ejpam-6452	216	67	=	=	SYM
ejpam-6452	216	68	−(1	−(1	NOUN
ejpam-6452	216	69	−	−	PROPN
ejpam-6452	217	1	k)n−rp2	k)n−rp2	PROPN
ejpam-6452	217	2	k	k	PROPN
ejpam-6452	217	3	,	,	PUNCT
ejpam-6452	217	4	r	r	NOUN
ejpam-6452	217	5	;	;	PUNCT
ejpam-6452	217	6	(	(	PUNCT
ejpam-6452	217	7	iii	iii	X
ejpam-6452	217	8	)	)	PUNCT
ejpam-6452	217	9	qk	qk	NOUN
ejpam-6452	217	10	,	,	PUNCT
ejpam-6452	217	11	n−rqk	n−rqk	NOUN
ejpam-6452	217	12	,	,	PUNCT
ejpam-6452	217	13	n+r	n+r	PROPN
ejpam-6452	217	14	−q2	−q2	PROPN
ejpam-6452	217	15	k	k	PROPN
ejpam-6452	217	16	,	,	PUNCT
ejpam-6452	217	17	n	n	NOUN
ejpam-6452	217	18	=	=	SYM
ejpam-6452	217	19	−8(1	−8(1	NOUN
ejpam-6452	217	20	−	−	PROPN
ejpam-6452	217	21	k)n−r+1p2	k)n−r+1p2	VERB
ejpam-6452	217	22	k	k	PROPN
ejpam-6452	217	23	,	,	PUNCT
ejpam-6452	217	24	r	r	NOUN
ejpam-6452	217	25	;	;	PUNCT
ejpam-6452	217	26	(	(	PUNCT
ejpam-6452	217	27	iv	iv	X
ejpam-6452	217	28	)	)	PUNCT
ejpam-6452	217	29	qk	qk	NOUN
ejpam-6452	217	30	,	,	PUNCT
ejpam-6452	217	31	n−rqk	n−rqk	NOUN
ejpam-6452	217	32	,	,	PUNCT
ejpam-6452	217	33	n+r	n+r	PROPN
ejpam-6452	217	34	−	−	PROPN
ejpam-6452	217	35	q2k	q2k	PROPN
ejpam-6452	217	36	,	,	PUNCT
ejpam-6452	217	37	n	n	NOUN
ejpam-6452	217	38	=	=	PUNCT
ejpam-6452	217	39	−2(1	−2(1	NOUN
ejpam-6452	217	40	−	−	NOUN
ejpam-6452	217	41	k)n−r+1p2	k)n−r+1p2	VERB
ejpam-6452	217	42	k	k	PROPN
ejpam-6452	217	43	,	,	PUNCT
ejpam-6452	217	44	r.	r.	PROPN
ejpam-6452	217	45	if	if	SCONJ
ejpam-6452	217	46	k	k	PROPN
ejpam-6452	217	47	=	=	SYM
ejpam-6452	217	48	2	2	NUM
ejpam-6452	217	49	,	,	PUNCT
ejpam-6452	217	50	then	then	ADV
ejpam-6452	217	51	we	we	PRON
ejpam-6452	217	52	have	have	VERB
ejpam-6452	217	53	the	the	DET
ejpam-6452	217	54	following	follow	VERB
ejpam-6452	217	55	:	:	PUNCT
ejpam-6452	217	56	corollary	corollary	ADJ
ejpam-6452	217	57	10	10	NUM
ejpam-6452	217	58	.	.	PUNCT
ejpam-6452	218	1	let	let	VERB
ejpam-6452	218	2	n	n	NOUN
ejpam-6452	218	3	and	and	CCONJ
ejpam-6452	218	4	r	r	NOUN
ejpam-6452	218	5	be	be	VERB
ejpam-6452	218	6	non	non	ADJ
ejpam-6452	218	7	-	-	ADJ
ejpam-6452	218	8	negative	negative	ADJ
ejpam-6452	218	9	integers	integer	NOUN
ejpam-6452	218	10	with	with	ADP
ejpam-6452	218	11	n	n	PROPN
ejpam-6452	218	12	≥	≥	PROPN
ejpam-6452	218	13	r.	r.	PROPN
ejpam-6452	218	14	then	then	ADV
ejpam-6452	218	15	ha	ha	INTJ
ejpam-6452	218	16	,	,	PUNCT
ejpam-6452	218	17	b	b	PROPN
ejpam-6452	218	18	n−rh	n−rh	PROPN
ejpam-6452	218	19	a	a	PRON
ejpam-6452	218	20	,	,	PUNCT
ejpam-6452	218	21	b	b	NOUN
ejpam-6452	218	22	n+r	n+r	NUM
ejpam-6452	219	1	−	−	PROPN
ejpam-6452	219	2	(	(	PUNCT
ejpam-6452	219	3	ha	ha	INTJ
ejpam-6452	219	4	,	,	PUNCT
ejpam-6452	219	5	b	b	PROPN
ejpam-6452	219	6	n	n	CCONJ
ejpam-6452	219	7	)	)	SYM
ejpam-6452	219	8	2	2	NUM
ejpam-6452	219	9	=	=	SYM
ejpam-6452	219	10	(	(	PUNCT
ejpam-6452	219	11	b−	b−	NOUN
ejpam-6452	219	12	a	a	PROPN
ejpam-6452	219	13	+	+	NOUN
ejpam-6452	219	14	√	√	ADJ
ejpam-6452	219	15	2a)(a−	2a)(a−	NUM
ejpam-6452	219	16	b−	b−	NOUN
ejpam-6452	219	17	√	√	VERB
ejpam-6452	219	18	2a)(−1)n−rp	2a)(−1)n−rp	NUM
ejpam-6452	219	19	2	2	NUM
ejpam-6452	219	20	r	r	NOUN
ejpam-6452	219	21	.	.	PUNCT
ejpam-6452	220	1	note	note	VERB
ejpam-6452	220	2	that	that	SCONJ
ejpam-6452	220	3	r	r	NOUN
ejpam-6452	220	4	=	=	SYM
ejpam-6452	220	5	1	1	NUM
ejpam-6452	220	6	in	in	ADP
ejpam-6452	220	7	theorem	theorem	NOUN
ejpam-6452	220	8	6	6	NUM
ejpam-6452	220	9	,	,	PUNCT
ejpam-6452	220	10	the	the	DET
ejpam-6452	220	11	catalan	catalan	NOUN
ejpam-6452	220	12	’s	’s	PART
ejpam-6452	220	13	identities	identity	NOUN
ejpam-6452	220	14	give	give	VERB
ejpam-6452	220	15	cassini	cassini	PROPN
ejpam-6452	220	16	’s	’s	PART
ejpam-6452	220	17	identities	identity	NOUN
ejpam-6452	220	18	as	as	SCONJ
ejpam-6452	220	19	follows	follow	VERB
ejpam-6452	220	20	:	:	PUNCT
ejpam-6452	220	21	theorem	theorem	ADJ
ejpam-6452	220	22	7	7	NUM
ejpam-6452	220	23	(	(	PUNCT
ejpam-6452	220	24	cassini	cassini	PROPN
ejpam-6452	220	25	’s	’s	PART
ejpam-6452	220	26	identities	identity	NOUN
ejpam-6452	220	27	)	)	PUNCT
ejpam-6452	220	28	.	.	PUNCT
ejpam-6452	221	1	let	let	VERB
ejpam-6452	221	2	k	k	NOUN
ejpam-6452	221	3	,	,	PUNCT
ejpam-6452	221	4	n	n	CCONJ
ejpam-6452	221	5	,	,	PUNCT
ejpam-6452	221	6	and	and	CCONJ
ejpam-6452	221	7	r	r	NOUN
ejpam-6452	221	8	be	be	VERB
ejpam-6452	221	9	non	non	ADJ
ejpam-6452	221	10	-	-	ADJ
ejpam-6452	221	11	negative	negative	ADJ
ejpam-6452	221	12	integers	integer	NOUN
ejpam-6452	221	13	with	with	ADP
ejpam-6452	221	14	k	k	PROPN
ejpam-6452	221	15	≥	≥	NUM
ejpam-6452	221	16	2	2	NUM
ejpam-6452	221	17	,	,	PUNCT
ejpam-6452	221	18	∆k	∆k	PROPN
ejpam-6452	221	19	=	=	SYM
ejpam-6452	221	20	√	√	PROPN
ejpam-6452	221	21	k2	k2	NOUN
ejpam-6452	221	22	+	+	CCONJ
ejpam-6452	221	23	4k	4k	NOUN
ejpam-6452	221	24	−	−	PROPN
ejpam-6452	221	25	4	4	NUM
ejpam-6452	221	26	,	,	PUNCT
ejpam-6452	221	27	and	and	CCONJ
ejpam-6452	221	28	n	n	PRON
ejpam-6452	221	29	≥	≥	PROPN
ejpam-6452	221	30	r.	r.	PROPN
ejpam-6452	221	31	then	then	ADV
ejpam-6452	221	32	gpk	gpk	PROPN
ejpam-6452	221	33	,	,	PUNCT
ejpam-6452	221	34	n−1gpk	n−1gpk	PROPN
ejpam-6452	221	35	,	,	PUNCT
ejpam-6452	221	36	n+1	n+1	PROPN
ejpam-6452	221	37	−	−	PROPN
ejpam-6452	221	38	gp2	gp2	PROPN
ejpam-6452	221	39	k	k	PROPN
ejpam-6452	221	40	,	,	PUNCT
ejpam-6452	221	41	n	n	PROPN
ejpam-6452	221	42	=	=	SYM
ejpam-6452	221	43	1	1	NUM
ejpam-6452	221	44	4	4	NUM
ejpam-6452	221	45	(	(	PUNCT
ejpam-6452	221	46	2b−	2b−	PROPN
ejpam-6452	221	47	ak	ak	NOUN
ejpam-6452	221	48	+	+	NOUN
ejpam-6452	221	49	a∆k)(ak	a∆k)(ak	NUM
ejpam-6452	221	50	−	−	NOUN
ejpam-6452	221	51	2b	2b	NOUN
ejpam-6452	222	1	+	+	CCONJ
ejpam-6452	222	2	a∆k)(1	a∆k)(1	ADP
ejpam-6452	222	3	−	−	NOUN
ejpam-6452	223	1	k)n−1	k)n−1	PROPN
ejpam-6452	223	2	.	.	PUNCT
ejpam-6452	224	1	if	if	SCONJ
ejpam-6452	224	2	(	(	PUNCT
ejpam-6452	224	3	a	a	PRON
ejpam-6452	224	4	,	,	PUNCT
ejpam-6452	224	5	b	b	NOUN
ejpam-6452	224	6	)	)	PUNCT
ejpam-6452	224	7	∈	∈	NOUN
ejpam-6452	224	8	{	{	PUNCT
ejpam-6452	224	9	(	(	PUNCT
ejpam-6452	224	10	2	2	NUM
ejpam-6452	224	11	,	,	PUNCT
ejpam-6452	224	12	k	k	NOUN
ejpam-6452	224	13	)	)	PUNCT
ejpam-6452	224	14	,	,	PUNCT
ejpam-6452	224	15	(	(	PUNCT
ejpam-6452	224	16	0	0	NUM
ejpam-6452	224	17	,	,	PUNCT
ejpam-6452	224	18	1	1	NUM
ejpam-6452	224	19	)	)	PUNCT
ejpam-6452	224	20	,	,	PUNCT
ejpam-6452	224	21	(	(	PUNCT
ejpam-6452	224	22	2	2	NUM
ejpam-6452	224	23	,	,	PUNCT
ejpam-6452	224	24	2	2	NUM
ejpam-6452	224	25	)	)	PUNCT
ejpam-6452	224	26	,	,	PUNCT
ejpam-6452	224	27	(	(	PUNCT
ejpam-6452	224	28	1	1	NUM
ejpam-6452	224	29	,	,	PUNCT
ejpam-6452	224	30	1	1	NUM
ejpam-6452	224	31	)	)	PUNCT
ejpam-6452	224	32	}	}	PUNCT
ejpam-6452	224	33	,	,	PUNCT
ejpam-6452	224	34	then	then	ADV
ejpam-6452	224	35	we	we	PRON
ejpam-6452	224	36	have	have	VERB
ejpam-6452	224	37	the	the	DET
ejpam-6452	224	38	following	follow	VERB
ejpam-6452	224	39	:	:	PUNCT
ejpam-6452	224	40	corollary	corollary	ADJ
ejpam-6452	224	41	11	11	NUM
ejpam-6452	224	42	.	.	PUNCT
ejpam-6452	225	1	let	let	VERB
ejpam-6452	225	2	k	k	NOUN
ejpam-6452	225	3	,	,	PUNCT
ejpam-6452	225	4	n	n	CCONJ
ejpam-6452	225	5	,	,	PUNCT
ejpam-6452	225	6	and	and	CCONJ
ejpam-6452	225	7	r	r	NOUN
ejpam-6452	225	8	be	be	VERB
ejpam-6452	225	9	non	non	ADJ
ejpam-6452	225	10	-	-	ADJ
ejpam-6452	225	11	negative	negative	ADJ
ejpam-6452	225	12	integers	integer	NOUN
ejpam-6452	225	13	with	with	ADP
ejpam-6452	225	14	k	k	PROPN
ejpam-6452	225	15	≥	≥	NUM
ejpam-6452	225	16	2	2	NUM
ejpam-6452	225	17	,	,	PUNCT
ejpam-6452	225	18	∆k	∆k	PROPN
ejpam-6452	225	19	=	=	SYM
ejpam-6452	225	20	√	√	PROPN
ejpam-6452	225	21	k2	k2	NOUN
ejpam-6452	225	22	+	+	CCONJ
ejpam-6452	225	23	4k	4k	NOUN
ejpam-6452	225	24	−	−	PROPN
ejpam-6452	225	25	4	4	NUM
ejpam-6452	225	26	,	,	PUNCT
ejpam-6452	225	27	and	and	CCONJ
ejpam-6452	225	28	n	n	PRON
ejpam-6452	225	29	≥	≥	NOUN
ejpam-6452	225	30	r.	r.	PROPN
ejpam-6452	225	31	then	then	ADV
ejpam-6452	225	32	(	(	PUNCT
ejpam-6452	225	33	i	i	NOUN
ejpam-6452	225	34	)	)	PUNCT
ejpam-6452	225	35	lk	lk	PROPN
ejpam-6452	225	36	,	,	PUNCT
ejpam-6452	225	37	n−1lk	n−1lk	PROPN
ejpam-6452	225	38	,	,	PUNCT
ejpam-6452	225	39	n+1	n+1	PROPN
ejpam-6452	225	40	−	−	PROPN
ejpam-6452	225	41	l2	l2	NOUN
ejpam-6452	225	42	k	k	NOUN
ejpam-6452	225	43	,	,	PUNCT
ejpam-6452	225	44	n	n	PROPN
ejpam-6452	225	45	=	=	SYM
ejpam-6452	225	46	∆2	∆2	PROPN
ejpam-6452	225	47	k(1	k(1	PROPN
ejpam-6452	225	48	−	−	PROPN
ejpam-6452	226	1	k)n−1	k)n−1	PROPN
ejpam-6452	226	2	;	;	PUNCT
ejpam-6452	226	3	(	(	PUNCT
ejpam-6452	226	4	ii	ii	NOUN
ejpam-6452	226	5	)	)	PUNCT
ejpam-6452	226	6	pk	pk	NOUN
ejpam-6452	226	7	,	,	PUNCT
ejpam-6452	226	8	n−1pk	n−1pk	ADJ
ejpam-6452	226	9	,	,	PUNCT
ejpam-6452	226	10	n+1	n+1	PROPN
ejpam-6452	226	11	−	−	PROPN
ejpam-6452	226	12	p2	p2	PROPN
ejpam-6452	226	13	k	k	PROPN
ejpam-6452	226	14	,	,	PUNCT
ejpam-6452	226	15	n	n	NOUN
ejpam-6452	226	16	=	=	SYM
ejpam-6452	226	17	−(1	−(1	NOUN
ejpam-6452	226	18	−	−	PROPN
ejpam-6452	226	19	k)n−1	k)n−1	NOUN
ejpam-6452	226	20	;	;	PUNCT
ejpam-6452	226	21	(	(	PUNCT
ejpam-6452	226	22	iii	iii	X
ejpam-6452	226	23	)	)	PUNCT
ejpam-6452	226	24	qk	qk	PROPN
ejpam-6452	226	25	,	,	PUNCT
ejpam-6452	226	26	n−1qk	n−1qk	NOUN
ejpam-6452	226	27	,	,	PUNCT
ejpam-6452	226	28	n+1	n+1	PROPN
ejpam-6452	226	29	−q2	−q2	PROPN
ejpam-6452	226	30	k	k	PROPN
ejpam-6452	226	31	,	,	PUNCT
ejpam-6452	226	32	n	n	NOUN
ejpam-6452	226	33	=	=	NOUN
ejpam-6452	226	34	−8(1	−8(1	NOUN
ejpam-6452	226	35	−	−	NOUN
ejpam-6452	226	36	k)n	k)n	NOUN
ejpam-6452	226	37	;	;	PUNCT
ejpam-6452	226	38	(	(	PUNCT
ejpam-6452	226	39	iv	iv	X
ejpam-6452	226	40	)	)	PUNCT
ejpam-6452	226	41	qk	qk	PROPN
ejpam-6452	226	42	,	,	PUNCT
ejpam-6452	226	43	n−1qk	n−1qk	NOUN
ejpam-6452	226	44	,	,	PUNCT
ejpam-6452	226	45	n+1	n+1	PROPN
ejpam-6452	226	46	−	−	PROPN
ejpam-6452	226	47	q2k	q2k	PROPN
ejpam-6452	226	48	,	,	PUNCT
ejpam-6452	226	49	n	n	NOUN
ejpam-6452	226	50	=	=	PUNCT
ejpam-6452	226	51	−2(1	−2(1	NOUN
ejpam-6452	226	52	−	−	NOUN
ejpam-6452	226	53	k)n	k)n	NOUN
ejpam-6452	226	54	.	.	PUNCT
ejpam-6452	227	1	if	if	SCONJ
ejpam-6452	227	2	k	k	PROPN
ejpam-6452	227	3	=	=	SYM
ejpam-6452	227	4	2	2	NUM
ejpam-6452	227	5	,	,	PUNCT
ejpam-6452	227	6	then	then	ADV
ejpam-6452	227	7	we	we	PRON
ejpam-6452	227	8	have	have	VERB
ejpam-6452	227	9	the	the	DET
ejpam-6452	227	10	following	follow	VERB
ejpam-6452	227	11	:	:	PUNCT
ejpam-6452	227	12	corollary	corollary	ADJ
ejpam-6452	227	13	12	12	NUM
ejpam-6452	227	14	.	.	PUNCT
ejpam-6452	228	1	let	let	VERB
ejpam-6452	228	2	n	n	PRON
ejpam-6452	228	3	be	be	AUX
ejpam-6452	228	4	a	a	DET
ejpam-6452	228	5	non	non	ADJ
ejpam-6452	228	6	-	-	ADJ
ejpam-6452	228	7	negative	negative	ADJ
ejpam-6452	228	8	integer	integer	NOUN
ejpam-6452	228	9	.	.	PUNCT
ejpam-6452	229	1	then	then	ADV
ejpam-6452	229	2	ha	ha	INTJ
ejpam-6452	229	3	,	,	PUNCT
ejpam-6452	229	4	b	b	PROPN
ejpam-6452	229	5	n−1h	n−1h	PROPN
ejpam-6452	229	6	a	a	PROPN
ejpam-6452	229	7	,	,	PUNCT
ejpam-6452	229	8	b	b	PROPN
ejpam-6452	229	9	n+1	n+1	PROPN
ejpam-6452	229	10	−	−	PROPN
ejpam-6452	229	11	(	(	PUNCT
ejpam-6452	229	12	ha	ha	INTJ
ejpam-6452	229	13	,	,	PUNCT
ejpam-6452	229	14	b	b	PROPN
ejpam-6452	229	15	n	n	CCONJ
ejpam-6452	229	16	)	)	SYM
ejpam-6452	229	17	2	2	NUM
ejpam-6452	229	18	=	=	SYM
ejpam-6452	229	19	(	(	PUNCT
ejpam-6452	229	20	b−	b−	NOUN
ejpam-6452	229	21	a	a	PROPN
ejpam-6452	229	22	+	+	NOUN
ejpam-6452	229	23	√	√	ADJ
ejpam-6452	229	24	2a)(a−	2a)(a−	NUM
ejpam-6452	229	25	b−	b−	NOUN
ejpam-6452	229	26	√	√	ADP
ejpam-6452	229	27	2a)(−1)n−1	2a)(−1)n−1	NUM
ejpam-6452	229	28	.	.	PUNCT
ejpam-6452	230	1	w.	w.	PROPN
ejpam-6452	230	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	230	3	,	,	PUNCT
ejpam-6452	230	4	a.	a.	NOUN
ejpam-6452	230	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	230	6	/	/	SYM
ejpam-6452	230	7	eur	eur	PROPN
ejpam-6452	230	8	.	.	PUNCT
ejpam-6452	231	1	j.	j.	PROPN
ejpam-6452	231	2	pure	pure	PROPN
ejpam-6452	231	3	appl	appl	PROPN
ejpam-6452	231	4	.	.	PROPN
ejpam-6452	231	5	math	math	PROPN
ejpam-6452	231	6	,	,	PUNCT
ejpam-6452	231	7	18	18	NUM
ejpam-6452	231	8	(	(	PUNCT
ejpam-6452	231	9	3	3	NUM
ejpam-6452	231	10	)	)	PUNCT
ejpam-6452	231	11	(	(	PUNCT
ejpam-6452	231	12	2025	2025	NUM
ejpam-6452	231	13	)	)	PUNCT
ejpam-6452	231	14	,	,	PUNCT
ejpam-6452	231	15	6452	6452	NUM
ejpam-6452	231	16	11	11	NUM
ejpam-6452	231	17	of	of	ADP
ejpam-6452	231	18	18	18	NUM
ejpam-6452	231	19	3	3	NUM
ejpam-6452	231	20	.	.	PUNCT
ejpam-6452	232	1	the	the	DET
ejpam-6452	232	2	integral	integral	ADJ
ejpam-6452	232	3	representation	representation	NOUN
ejpam-6452	232	4	of	of	ADP
ejpam-6452	232	5	the	the	DET
ejpam-6452	232	6	generalized	generalize	VERB
ejpam-6452	232	7	pell	pell	NOUN
ejpam-6452	232	8	numbers	number	NOUN
ejpam-6452	232	9	there	there	PRON
ejpam-6452	232	10	are	be	VERB
ejpam-6452	232	11	several	several	ADJ
ejpam-6452	232	12	ways	way	NOUN
ejpam-6452	232	13	to	to	PART
ejpam-6452	232	14	represent	represent	VERB
ejpam-6452	232	15	the	the	DET
ejpam-6452	232	16	special	special	ADJ
ejpam-6452	232	17	numbers	number	NOUN
ejpam-6452	232	18	.	.	PUNCT
ejpam-6452	233	1	one	one	NUM
ejpam-6452	233	2	of	of	ADP
ejpam-6452	233	3	them	they	PRON
ejpam-6452	233	4	is	be	AUX
ejpam-6452	233	5	the	the	DET
ejpam-6452	233	6	integral	integral	ADJ
ejpam-6452	233	7	representation	representation	NOUN
ejpam-6452	233	8	;	;	PUNCT
ejpam-6452	233	9	see	see	VERB
ejpam-6452	233	10	,	,	PUNCT
ejpam-6452	233	11	for	for	ADP
ejpam-6452	233	12	example	example	NOUN
ejpam-6452	233	13	,	,	PUNCT
ejpam-6452	233	14	[	[	X
ejpam-6452	233	15	11–22	11–22	NUM
ejpam-6452	233	16	]	]	PUNCT
ejpam-6452	233	17	.	.	PUNCT
ejpam-6452	234	1	the	the	DET
ejpam-6452	234	2	integral	integral	ADJ
ejpam-6452	234	3	representations	representation	NOUN
ejpam-6452	234	4	for	for	ADP
ejpam-6452	234	5	pell	pell	NOUN
ejpam-6452	234	6	and	and	CCONJ
ejpam-6452	234	7	pelllucas	pelllucas	ADJ
ejpam-6452	234	8	numbers	number	NOUN
ejpam-6452	234	9	are	be	AUX
ejpam-6452	234	10	studied	study	VERB
ejpam-6452	234	11	by	by	ADP
ejpam-6452	234	12	the	the	DET
ejpam-6452	234	13	second	second	ADJ
ejpam-6452	234	14	author	author	NOUN
ejpam-6452	234	15	in	in	ADP
ejpam-6452	234	16	[	[	X
ejpam-6452	234	17	17	17	NUM
ejpam-6452	234	18	]	]	PUNCT
ejpam-6452	234	19	as	as	SCONJ
ejpam-6452	234	20	follows	follow	VERB
ejpam-6452	234	21	:	:	PUNCT
ejpam-6452	235	1	pℓn	pℓn	INTJ
ejpam-6452	235	2	=	=	PUNCT
ejpam-6452	235	3	npℓ	npℓ	ADJ
ejpam-6452	236	1	2n	2n	NUM
ejpam-6452	236	2	∫	∫	PROPN
ejpam-6452	236	3	1	1	NUM
ejpam-6452	236	4	−1	−1	NOUN
ejpam-6452	236	5	(	(	PUNCT
ejpam-6452	236	6	qℓ	qℓ	NOUN
ejpam-6452	236	7	+	+	X
ejpam-6452	236	8	2	2	NUM
ejpam-6452	236	9	√	√	NUM
ejpam-6452	236	10	2pℓx)n−1dx	2pℓx)n−1dx	NUM
ejpam-6452	236	11	and	and	CCONJ
ejpam-6452	236	12	qℓn	qℓn	NOUN
ejpam-6452	236	13	=	=	SYM
ejpam-6452	236	14	1	1	NUM
ejpam-6452	236	15	2n	2n	NUM
ejpam-6452	236	16	∫	∫	PROPN
ejpam-6452	236	17	1	1	NUM
ejpam-6452	236	18	−1	−1	NOUN
ejpam-6452	236	19	(	(	PUNCT
ejpam-6452	236	20	qℓ	qℓ	NOUN
ejpam-6452	236	21	+	+	X
ejpam-6452	236	22	2	2	NUM
ejpam-6452	236	23	√	√	NUM
ejpam-6452	236	24	2(n	2(n	NUM
ejpam-6452	236	25	+	+	CCONJ
ejpam-6452	236	26	1)pℓx)(qℓ	1)pℓx)(qℓ	PRON
ejpam-6452	236	27	+	+	NUM
ejpam-6452	236	28	2	2	NUM
ejpam-6452	236	29	√	√	NUM
ejpam-6452	236	30	2pℓx)n−1dx	2pℓx)n−1dx	NUM
ejpam-6452	236	31	,	,	PUNCT
ejpam-6452	236	32	where	where	SCONJ
ejpam-6452	236	33	ℓ	ℓ	PROPN
ejpam-6452	236	34	and	and	CCONJ
ejpam-6452	236	35	n	n	PROPN
ejpam-6452	236	36	are	be	AUX
ejpam-6452	236	37	non	non	ADJ
ejpam-6452	236	38	-	-	ADJ
ejpam-6452	236	39	negative	negative	ADJ
ejpam-6452	236	40	integers	integer	NOUN
ejpam-6452	236	41	.	.	PUNCT
ejpam-6452	237	1	subsequently	subsequently	ADV
ejpam-6452	237	2	,	,	PUNCT
ejpam-6452	237	3	the	the	DET
ejpam-6452	237	4	authors	author	NOUN
ejpam-6452	237	5	[	[	X
ejpam-6452	237	6	11	11	NUM
ejpam-6452	237	7	,	,	PUNCT
ejpam-6452	237	8	18	18	NUM
ejpam-6452	237	9	,	,	PUNCT
ejpam-6452	237	10	19	19	NUM
ejpam-6452	237	11	]	]	PUNCT
ejpam-6452	237	12	give	give	VERB
ejpam-6452	237	13	new	new	ADJ
ejpam-6452	237	14	integral	integral	ADJ
ejpam-6452	237	15	representations	representation	NOUN
ejpam-6452	237	16	for	for	ADP
ejpam-6452	237	17	the	the	DET
ejpam-6452	237	18	general	general	NOUN
ejpam-6452	237	19	of	of	ADP
ejpam-6452	237	20	pell	pell	NOUN
ejpam-6452	237	21	and	and	CCONJ
ejpam-6452	237	22	pell	pell	NOUN
ejpam-6452	237	23	-	-	PUNCT
ejpam-6452	237	24	lucas	lucas	NOUN
ejpam-6452	237	25	numbers	number	NOUN
ejpam-6452	237	26	such	such	ADJ
ejpam-6452	237	27	as	as	ADP
ejpam-6452	237	28	the	the	DET
ejpam-6452	237	29	k	k	PROPN
ejpam-6452	237	30	-	-	NOUN
ejpam-6452	237	31	fibonacci	fibonacci	NOUN
ejpam-6452	237	32	,	,	PUNCT
ejpam-6452	237	33	k	k	NOUN
ejpam-6452	237	34	-	-	PUNCT
ejpam-6452	237	35	lucas	lucas	PROPN
ejpam-6452	237	36	,	,	PUNCT
ejpam-6452	237	37	k	k	NOUN
ejpam-6452	237	38	-	-	PUNCT
ejpam-6452	237	39	pell	pell	NOUN
ejpam-6452	237	40	,	,	PUNCT
ejpam-6452	237	41	k	k	NOUN
ejpam-6452	237	42	-	-	PUNCT
ejpam-6452	237	43	pell	pell	NOUN
ejpam-6452	237	44	-	-	PUNCT
ejpam-6452	237	45	lucas	lucas	NOUN
ejpam-6452	237	46	,	,	PUNCT
ejpam-6452	237	47	and	and	CCONJ
ejpam-6452	237	48	the	the	DET
ejpam-6452	237	49	k	k	NOUN
ejpam-6452	237	50	-	-	PUNCT
ejpam-6452	237	51	pell	pell	NOUN
ejpam-6452	237	52	-	-	PUNCT
ejpam-6452	237	53	lucas	lucas	NOUN
ejpam-6452	237	54	-	-	PUNCT
ejpam-6452	237	55	like	like	ADJ
ejpam-6452	237	56	numbers	number	NOUN
ejpam-6452	237	57	.	.	PUNCT
ejpam-6452	238	1	in	in	ADP
ejpam-6452	238	2	this	this	DET
ejpam-6452	238	3	section	section	NOUN
ejpam-6452	238	4	,	,	PUNCT
ejpam-6452	238	5	we	we	PRON
ejpam-6452	238	6	obtain	obtain	VERB
ejpam-6452	238	7	new	new	ADJ
ejpam-6452	238	8	integral	integral	ADJ
ejpam-6452	238	9	representations	representation	NOUN
ejpam-6452	238	10	for	for	ADP
ejpam-6452	238	11	the	the	DET
ejpam-6452	238	12	companion	companion	NOUN
ejpam-6452	238	13	generalized	generalize	VERB
ejpam-6452	238	14	pell	pell	NOUN
ejpam-6452	238	15	numbers	number	NOUN
ejpam-6452	238	16	.	.	PUNCT
ejpam-6452	239	1	we	we	PRON
ejpam-6452	239	2	start	start	VERB
ejpam-6452	239	3	with	with	ADP
ejpam-6452	239	4	the	the	DET
ejpam-6452	239	5	following	following	NOUN
ejpam-6452	239	6	theorem	theorem	NOUN
ejpam-6452	239	7	for	for	ADP
ejpam-6452	239	8	the	the	DET
ejpam-6452	239	9	generalized	generalize	VERB
ejpam-6452	239	10	pell	pell	NOUN
ejpam-6452	239	11	number	number	NOUN
ejpam-6452	239	12	pk,ℓn	pk,ℓn	NOUN
ejpam-6452	239	13	by	by	ADP
ejpam-6452	239	14	employing	employ	VERB
ejpam-6452	239	15	other	other	ADJ
ejpam-6452	239	16	known	know	VERB
ejpam-6452	239	17	relations	relation	NOUN
ejpam-6452	239	18	between	between	ADP
ejpam-6452	239	19	the	the	DET
ejpam-6452	239	20	two	two	NUM
ejpam-6452	239	21	numbers	number	NOUN
ejpam-6452	239	22	pk,ℓ	pk,ℓ	NOUN
ejpam-6452	239	23	and	and	CCONJ
ejpam-6452	239	24	lk,ℓ.	lk,ℓ.	ADJ
ejpam-6452	239	25	theorem	theorem	NOUN
ejpam-6452	239	26	8	8	NUM
ejpam-6452	239	27	.	.	PUNCT
ejpam-6452	240	1	let	let	VERB
ejpam-6452	240	2	k	k	NOUN
ejpam-6452	240	3	,	,	PUNCT
ejpam-6452	240	4	ℓ	ℓ	PROPN
ejpam-6452	240	5	and	and	CCONJ
ejpam-6452	240	6	n	n	CCONJ
ejpam-6452	240	7	be	be	VERB
ejpam-6452	240	8	non	non	ADJ
ejpam-6452	240	9	-	-	ADJ
ejpam-6452	240	10	negative	negative	ADJ
ejpam-6452	240	11	integers	integer	NOUN
ejpam-6452	240	12	with	with	ADP
ejpam-6452	240	13	k	k	PROPN
ejpam-6452	240	14	≥	≥	NUM
ejpam-6452	240	15	2	2	NUM
ejpam-6452	240	16	and	and	CCONJ
ejpam-6452	240	17	∆k	∆k	PROPN
ejpam-6452	240	18	=	=	SYM
ejpam-6452	240	19	√	√	PROPN
ejpam-6452	240	20	k2	k2	NOUN
ejpam-6452	241	1	+	+	CCONJ
ejpam-6452	241	2	4k	4k	NOUN
ejpam-6452	241	3	−	−	NOUN
ejpam-6452	242	1	4	4	X
ejpam-6452	242	2	.	.	PUNCT
ejpam-6452	242	3	then	then	ADV
ejpam-6452	242	4	pk,ℓn	pk,ℓn	PROPN
ejpam-6452	242	5	=	=	SYM
ejpam-6452	242	6	npk,ℓ	npk,ℓ	PROPN
ejpam-6452	242	7	2n∆k	2n∆k	NUM
ejpam-6452	242	8	∫	∫	PROPN
ejpam-6452	242	9	∆k	∆k	PROPN
ejpam-6452	242	10	−∆k	−∆k	PUNCT
ejpam-6452	243	1	(	(	PUNCT
ejpam-6452	243	2	lk,ℓ	lk,ℓ	X
ejpam-6452	243	3	+	+	CCONJ
ejpam-6452	243	4	pk,ℓ	pk,ℓ	NOUN
ejpam-6452	243	5	x)n−1	x)n−1	PROPN
ejpam-6452	243	6	dx	dx	PROPN
ejpam-6452	243	7	.	.	PUNCT
ejpam-6452	244	1	(	(	PUNCT
ejpam-6452	244	2	7	7	X
ejpam-6452	244	3	)	)	PUNCT
ejpam-6452	244	4	proof	proof	NOUN
ejpam-6452	244	5	.	.	PUNCT
ejpam-6452	245	1	for	for	ADP
ejpam-6452	245	2	n	n	NOUN
ejpam-6452	245	3	=	=	SYM
ejpam-6452	245	4	0	0	NUM
ejpam-6452	245	5	or	or	CCONJ
ejpam-6452	245	6	ℓ	ℓ	X
ejpam-6452	245	7	=	=	SYM
ejpam-6452	245	8	0	0	NUM
ejpam-6452	245	9	,	,	PUNCT
ejpam-6452	245	10	we	we	PRON
ejpam-6452	245	11	have	have	AUX
ejpam-6452	245	12	done	do	VERB
ejpam-6452	245	13	.	.	PUNCT
ejpam-6452	246	1	let	let	VERB
ejpam-6452	246	2	us	we	PRON
ejpam-6452	246	3	assume	assume	VERB
ejpam-6452	246	4	that	that	SCONJ
ejpam-6452	246	5	ℓ	ℓ	NOUN
ejpam-6452	246	6	,	,	PUNCT
ejpam-6452	246	7	n	n	X
ejpam-6452	246	8	>	>	X
ejpam-6452	246	9	0	0	X
ejpam-6452	246	10	.	.	PUNCT
ejpam-6452	246	11	using	use	VERB
ejpam-6452	246	12	integration	integration	NOUN
ejpam-6452	246	13	by	by	ADP
ejpam-6452	246	14	substitution	substitution	NOUN
ejpam-6452	246	15	,	,	PUNCT
ejpam-6452	246	16	we	we	PRON
ejpam-6452	246	17	get∫	get∫	VERB
ejpam-6452	246	18	∆k	∆k	X
ejpam-6452	246	19	−∆k	−∆k	PUNCT
ejpam-6452	247	1	(	(	PUNCT
ejpam-6452	247	2	lk,ℓ	lk,ℓ	X
ejpam-6452	247	3	+	+	CCONJ
ejpam-6452	247	4	pk,ℓx)n−1	pk,ℓx)n−1	PROPN
ejpam-6452	247	5	dx	dx	PROPN
ejpam-6452	248	1	=	=	SYM
ejpam-6452	248	2	1	1	NUM
ejpam-6452	248	3	npk,ℓ	npk,ℓ	NOUN
ejpam-6452	248	4	[	[	PUNCT
ejpam-6452	248	5	(	(	PUNCT
ejpam-6452	248	6	lk,ℓ	lk,ℓ	X
ejpam-6452	248	7	+	+	CCONJ
ejpam-6452	248	8	pk,ℓx)n	pk,ℓx)n	NOUN
ejpam-6452	248	9	]	]	PUNCT
ejpam-6452	248	10	∆k	∆k	X
ejpam-6452	248	11	−∆k	−∆k	PUNCT
ejpam-6452	249	1	=	=	SYM
ejpam-6452	249	2	1	1	NUM
ejpam-6452	249	3	npk,ℓ	npk,ℓ	PRON
ejpam-6452	249	4	[	[	X
ejpam-6452	249	5	(	(	PUNCT
ejpam-6452	249	6	lk,ℓ	lk,ℓ	X
ejpam-6452	249	7	+	+	X
ejpam-6452	249	8	∆kpk,ℓ	∆kpk,ℓ	NOUN
ejpam-6452	249	9	)	)	PUNCT
ejpam-6452	249	10	n	n	CCONJ
ejpam-6452	249	11	−	−	PROPN
ejpam-6452	249	12	(	(	PUNCT
ejpam-6452	249	13	lk,ℓ	lk,ℓ	X
ejpam-6452	249	14	−	−	PROPN
ejpam-6452	249	15	∆kpk,ℓ	∆kpk,ℓ	PROPN
ejpam-6452	249	16	)	)	PUNCT
ejpam-6452	249	17	n	n	CCONJ
ejpam-6452	249	18	]	]	PUNCT
ejpam-6452	249	19	.	.	PUNCT
ejpam-6452	250	1	it	it	PRON
ejpam-6452	250	2	follows	follow	VERB
ejpam-6452	250	3	from	from	ADP
ejpam-6452	250	4	(	(	PUNCT
ejpam-6452	250	5	i	i	NOUN
ejpam-6452	250	6	)	)	PUNCT
ejpam-6452	250	7	and	and	CCONJ
ejpam-6452	250	8	(	(	PUNCT
ejpam-6452	250	9	ii	ii	NOUN
ejpam-6452	250	10	)	)	PUNCT
ejpam-6452	250	11	of	of	ADP
ejpam-6452	250	12	theorem	theorem	NOUN
ejpam-6452	250	13	2	2	NUM
ejpam-6452	250	14	with	with	ADP
ejpam-6452	250	15	n	n	CCONJ
ejpam-6452	250	16	replaced	replace	VERB
ejpam-6452	250	17	with	with	ADP
ejpam-6452	250	18	ℓ	ℓ	PROPN
ejpam-6452	250	19	that∫	that∫	NOUN
ejpam-6452	250	20	∆k	∆k	PROPN
ejpam-6452	250	21	−∆k	−∆k	PUNCT
ejpam-6452	251	1	(	(	PUNCT
ejpam-6452	251	2	lk,ℓ	lk,ℓ	X
ejpam-6452	251	3	+	+	CCONJ
ejpam-6452	251	4	pk,ℓx)n−1	pk,ℓx)n−1	PROPN
ejpam-6452	251	5	dx	dx	PROPN
ejpam-6452	251	6	=	=	SYM
ejpam-6452	251	7	1	1	NUM
ejpam-6452	251	8	npk,ℓ	npk,ℓ	PRON
ejpam-6452	251	9	[	[	X
ejpam-6452	251	10	(	(	PUNCT
ejpam-6452	251	11	2σℓ	2σℓ	ADJ
ejpam-6452	251	12	k	k	PROPN
ejpam-6452	251	13	)	)	PUNCT
ejpam-6452	252	1	n	n	PROPN
ejpam-6452	252	2	−	−	PROPN
ejpam-6452	252	3	(	(	PUNCT
ejpam-6452	252	4	2	2	NUM
ejpam-6452	252	5	(	(	PUNCT
ejpam-6452	252	6	1	1	NUM
ejpam-6452	252	7	−	−	NOUN
ejpam-6452	252	8	k)ℓ	k)ℓ	PUNCT
ejpam-6452	252	9	σℓ	σℓ	X
ejpam-6452	252	10	k	k	PROPN
ejpam-6452	252	11	)	)	PUNCT
ejpam-6452	252	12	n	n	CCONJ
ejpam-6452	252	13	]	]	PUNCT
ejpam-6452	252	14	=	=	SYM
ejpam-6452	252	15	2n∆k	2n∆k	NUM
ejpam-6452	252	16	npk,ℓ	npk,ℓ	PRON
ejpam-6452	252	17	[	[	PUNCT
ejpam-6452	252	18	1	1	NUM
ejpam-6452	252	19	∆k	∆k	PROPN
ejpam-6452	252	20	(	(	PUNCT
ejpam-6452	252	21	σℓn	σℓn	PROPN
ejpam-6452	252	22	k	k	PROPN
ejpam-6452	252	23	−	−	PROPN
ejpam-6452	252	24	(	(	PUNCT
ejpam-6452	252	25	1	1	NUM
ejpam-6452	252	26	−	−	PROPN
ejpam-6452	253	1	k)ℓn	k)ℓn	PROPN
ejpam-6452	253	2	σℓn	σℓn	PROPN
ejpam-6452	253	3	k	k	PROPN
ejpam-6452	253	4	)	)	PUNCT
ejpam-6452	253	5	]	]	PUNCT
ejpam-6452	253	6	.	.	PUNCT
ejpam-6452	254	1	by	by	ADP
ejpam-6452	254	2	using	use	VERB
ejpam-6452	254	3	(	(	PUNCT
ejpam-6452	254	4	i	i	NOUN
ejpam-6452	254	5	)	)	PUNCT
ejpam-6452	254	6	of	of	ADP
ejpam-6452	254	7	corollary	corollary	ADJ
ejpam-6452	254	8	1	1	NUM
ejpam-6452	254	9	with	with	ADP
ejpam-6452	254	10	replace	replace	NOUN
ejpam-6452	254	11	n	n	ADV
ejpam-6452	254	12	by	by	ADP
ejpam-6452	254	13	ℓn	ℓn	ADV
ejpam-6452	254	14	,	,	PUNCT
ejpam-6452	254	15	we	we	PRON
ejpam-6452	254	16	have∫	have∫	VERB
ejpam-6452	254	17	∆k	∆k	PROPN
ejpam-6452	254	18	−∆k	−∆k	PUNCT
ejpam-6452	255	1	(	(	PUNCT
ejpam-6452	255	2	lk,ℓ	lk,ℓ	X
ejpam-6452	255	3	+	+	CCONJ
ejpam-6452	255	4	pk,ℓx)n−1	pk,ℓx)n−1	PROPN
ejpam-6452	255	5	dx	dx	PROPN
ejpam-6452	255	6	=	=	SYM
ejpam-6452	255	7	2n∆k	2n∆k	NUM
ejpam-6452	255	8	pk,ℓn	pk,ℓn	PROPN
ejpam-6452	255	9	npk,ℓ	npk,ℓ	PROPN
ejpam-6452	255	10	.	.	PUNCT
ejpam-6452	256	1	then	then	ADV
ejpam-6452	256	2	(	(	PUNCT
ejpam-6452	256	3	7	7	X
ejpam-6452	256	4	)	)	PUNCT
ejpam-6452	256	5	which	which	PRON
ejpam-6452	256	6	completes	complete	VERB
ejpam-6452	256	7	the	the	DET
ejpam-6452	256	8	proof	proof	NOUN
ejpam-6452	256	9	.	.	PUNCT
ejpam-6452	257	1	w.	w.	PROPN
ejpam-6452	257	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	257	3	,	,	PUNCT
ejpam-6452	257	4	a.	a.	NOUN
ejpam-6452	257	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	257	6	/	/	SYM
ejpam-6452	257	7	eur	eur	PROPN
ejpam-6452	257	8	.	.	PUNCT
ejpam-6452	258	1	j.	j.	PROPN
ejpam-6452	258	2	pure	pure	PROPN
ejpam-6452	258	3	appl	appl	PROPN
ejpam-6452	258	4	.	.	PROPN
ejpam-6452	258	5	math	math	PROPN
ejpam-6452	258	6	,	,	PUNCT
ejpam-6452	258	7	18	18	NUM
ejpam-6452	258	8	(	(	PUNCT
ejpam-6452	258	9	3	3	NUM
ejpam-6452	258	10	)	)	PUNCT
ejpam-6452	258	11	(	(	PUNCT
ejpam-6452	258	12	2025	2025	NUM
ejpam-6452	258	13	)	)	PUNCT
ejpam-6452	258	14	,	,	PUNCT
ejpam-6452	258	15	6452	6452	NUM
ejpam-6452	258	16	12	12	NUM
ejpam-6452	258	17	of	of	ADP
ejpam-6452	258	18	18	18	NUM
ejpam-6452	258	19	remark	remark	NOUN
ejpam-6452	258	20	2	2	NUM
ejpam-6452	258	21	.	.	PUNCT
ejpam-6452	259	1	as	as	ADP
ejpam-6452	259	2	in	in	ADP
ejpam-6452	259	3	theorem	theorem	ADJ
ejpam-6452	259	4	8	8	NUM
ejpam-6452	259	5	,	,	PUNCT
ejpam-6452	259	6	equation	equation	NOUN
ejpam-6452	259	7	(	(	PUNCT
ejpam-6452	259	8	7	7	X
ejpam-6452	259	9	)	)	PUNCT
ejpam-6452	259	10	is	be	AUX
ejpam-6452	259	11	equivalent	equivalent	ADJ
ejpam-6452	259	12	to	to	ADP
ejpam-6452	259	13	pk,ℓn	pk,ℓn	NOUN
ejpam-6452	259	14	=	=	SYM
ejpam-6452	259	15	npk,ℓ	npk,ℓ	PROPN
ejpam-6452	259	16	2n	2n	NUM
ejpam-6452	259	17	∫	∫	PROPN
ejpam-6452	259	18	1	1	NUM
ejpam-6452	259	19	−1	−1	NOUN
ejpam-6452	259	20	(	(	PUNCT
ejpam-6452	259	21	lk,ℓ	lk,ℓ	X
ejpam-6452	259	22	+	+	CCONJ
ejpam-6452	259	23	∆kpk,ℓ	∆kpk,ℓ	PROPN
ejpam-6452	259	24	t	t	PROPN
ejpam-6452	259	25	)	)	PUNCT
ejpam-6452	259	26	n−1	n−1	PROPN
ejpam-6452	259	27	dt	dt	PROPN
ejpam-6452	259	28	.	.	PUNCT
ejpam-6452	260	1	in	in	ADP
ejpam-6452	260	2	fact	fact	NOUN
ejpam-6452	260	3	,	,	PUNCT
ejpam-6452	260	4	substituting	substitute	VERB
ejpam-6452	260	5	t	t	NOUN
ejpam-6452	260	6	=	=	PUNCT
ejpam-6452	260	7	x	x	SYM
ejpam-6452	260	8	∆k	∆k	PROPN
ejpam-6452	260	9	produces	produce	VERB
ejpam-6452	260	10	dx	dx	PROPN
ejpam-6452	260	11	=	=	X
ejpam-6452	260	12	∆kdt	∆kdt	NOUN
ejpam-6452	260	13	and	and	CCONJ
ejpam-6452	260	14	the	the	DET
ejpam-6452	260	15	integration	integration	NOUN
ejpam-6452	260	16	limits	limit	NOUN
ejpam-6452	260	17	are	be	AUX
ejpam-6452	260	18	changed	change	VERB
ejpam-6452	260	19	to	to	ADP
ejpam-6452	260	20	−1	−1	NOUN
ejpam-6452	260	21	and	and	CCONJ
ejpam-6452	260	22	1	1	NUM
ejpam-6452	260	23	,	,	PUNCT
ejpam-6452	260	24	respectively	respectively	ADV
ejpam-6452	260	25	.	.	PUNCT
ejpam-6452	261	1	the	the	DET
ejpam-6452	261	2	generalized	generalize	VERB
ejpam-6452	261	3	pell	pell	NOUN
ejpam-6452	261	4	number	number	NOUN
ejpam-6452	261	5	pk,ℓn	pk,ℓn	NOUN
ejpam-6452	261	6	by	by	ADP
ejpam-6452	261	7	employing	employ	VERB
ejpam-6452	261	8	the	the	DET
ejpam-6452	261	9	two	two	NUM
ejpam-6452	261	10	numbers	number	NOUN
ejpam-6452	261	11	pk,ℓ	pk,ℓ	ADJ
ejpam-6452	261	12	and	and	CCONJ
ejpam-6452	261	13	qk,ℓ	qk,ℓ	X
ejpam-6452	261	14	is	be	AUX
ejpam-6452	261	15	presented	present	VERB
ejpam-6452	261	16	as	as	SCONJ
ejpam-6452	261	17	follows	follow	VERB
ejpam-6452	261	18	:	:	PUNCT
ejpam-6452	261	19	corollary	corollary	ADJ
ejpam-6452	261	20	13	13	NUM
ejpam-6452	261	21	(	(	PUNCT
ejpam-6452	261	22	[	[	X
ejpam-6452	261	23	11	11	NUM
ejpam-6452	261	24	,	,	PUNCT
ejpam-6452	261	25	theorem	theorem	VERB
ejpam-6452	261	26	2.3	2.3	NUM
ejpam-6452	261	27	]	]	PUNCT
ejpam-6452	261	28	)	)	PUNCT
ejpam-6452	261	29	.	.	PUNCT
ejpam-6452	262	1	let	let	VERB
ejpam-6452	262	2	k	k	NOUN
ejpam-6452	262	3	,	,	PUNCT
ejpam-6452	262	4	ℓ	ℓ	PROPN
ejpam-6452	262	5	and	and	CCONJ
ejpam-6452	262	6	n	n	CCONJ
ejpam-6452	262	7	be	be	VERB
ejpam-6452	262	8	non	non	ADJ
ejpam-6452	262	9	-	-	ADJ
ejpam-6452	262	10	negative	negative	ADJ
ejpam-6452	262	11	integers	integer	NOUN
ejpam-6452	262	12	with	with	ADP
ejpam-6452	262	13	k	k	PROPN
ejpam-6452	262	14	≥	≥	NUM
ejpam-6452	262	15	2	2	NUM
ejpam-6452	262	16	and	and	CCONJ
ejpam-6452	262	17	∆k	∆k	PROPN
ejpam-6452	262	18	=	=	SYM
ejpam-6452	262	19	√	√	PROPN
ejpam-6452	262	20	k2	k2	NOUN
ejpam-6452	262	21	+	+	CCONJ
ejpam-6452	262	22	4k	4k	NOUN
ejpam-6452	262	23	−	−	NOUN
ejpam-6452	263	1	4	4	X
ejpam-6452	263	2	.	.	PUNCT
ejpam-6452	263	3	then	then	ADV
ejpam-6452	263	4	pk,ℓn	pk,ℓn	PROPN
ejpam-6452	263	5	=	=	SYM
ejpam-6452	263	6	npk,ℓ	npk,ℓ	PROPN
ejpam-6452	263	7	2n∆k	2n∆k	NUM
ejpam-6452	263	8	∫	∫	PROPN
ejpam-6452	263	9	∆k	∆k	PROPN
ejpam-6452	263	10	−∆k	−∆k	X
ejpam-6452	263	11	(	(	PUNCT
ejpam-6452	263	12	qk,ℓ	qk,ℓ	X
ejpam-6452	263	13	+	+	CCONJ
ejpam-6452	263	14	(	(	PUNCT
ejpam-6452	263	15	k	k	NOUN
ejpam-6452	263	16	−	−	PROPN
ejpam-6452	263	17	2	2	NUM
ejpam-6452	263	18	+	+	CCONJ
ejpam-6452	263	19	x)pk,ℓ	x)pk,ℓ	NOUN
ejpam-6452	263	20	)	)	PUNCT
ejpam-6452	263	21	n−1	n−1	PROPN
ejpam-6452	263	22	dx	dx	PROPN
ejpam-6452	263	23	.	.	PUNCT
ejpam-6452	264	1	(	(	PUNCT
ejpam-6452	264	2	8)	8)	NUM
ejpam-6452	264	3	proof	proof	NOUN
ejpam-6452	264	4	.	.	PUNCT
ejpam-6452	265	1	from	from	ADP
ejpam-6452	265	2	(	(	PUNCT
ejpam-6452	265	3	i	i	NOUN
ejpam-6452	265	4	)	)	PUNCT
ejpam-6452	265	5	of	of	ADP
ejpam-6452	265	6	corollary	corollary	ADJ
ejpam-6452	265	7	3	3	NUM
ejpam-6452	265	8	,	,	PUNCT
ejpam-6452	265	9	we	we	PRON
ejpam-6452	265	10	have	have	VERB
ejpam-6452	265	11	lk,ℓ	lk,ℓ	X
ejpam-6452	265	12	=	=	SYM
ejpam-6452	265	13	qk,ℓ	qk,ℓ	X
ejpam-6452	265	14	+	+	CCONJ
ejpam-6452	266	1	(	(	PUNCT
ejpam-6452	266	2	k	k	ADJ
ejpam-6452	266	3	−	−	PROPN
ejpam-6452	266	4	2)pk,ℓ.	2)pk,ℓ.	NOUN
ejpam-6452	266	5	this	this	PRON
ejpam-6452	266	6	together	together	ADV
ejpam-6452	266	7	with	with	ADP
ejpam-6452	266	8	(	(	PUNCT
ejpam-6452	266	9	7	7	NUM
ejpam-6452	266	10	)	)	PUNCT
ejpam-6452	266	11	that	that	SCONJ
ejpam-6452	266	12	(	(	PUNCT
ejpam-6452	266	13	8)	8)	NUM
ejpam-6452	266	14	holds	hold	VERB
ejpam-6452	266	15	.	.	PUNCT
ejpam-6452	267	1	the	the	DET
ejpam-6452	267	2	integral	integral	ADJ
ejpam-6452	267	3	representations	representation	NOUN
ejpam-6452	267	4	of	of	ADP
ejpam-6452	267	5	the	the	DET
ejpam-6452	267	6	generalized	generalize	VERB
ejpam-6452	267	7	pell	pell	NOUN
ejpam-6452	267	8	number	number	NOUN
ejpam-6452	267	9	for	for	ADP
ejpam-6452	267	10	even	even	ADV
ejpam-6452	267	11	and	and	CCONJ
ejpam-6452	267	12	odd	odd	ADJ
ejpam-6452	267	13	orders	order	NOUN
ejpam-6452	267	14	are	be	AUX
ejpam-6452	267	15	shown	show	VERB
ejpam-6452	267	16	as	as	SCONJ
ejpam-6452	267	17	follows	follow	VERB
ejpam-6452	267	18	:	:	PUNCT
ejpam-6452	267	19	theorem	theorem	NOUN
ejpam-6452	267	20	9	9	NUM
ejpam-6452	267	21	.	.	PUNCT
ejpam-6452	268	1	let	let	VERB
ejpam-6452	268	2	k	k	NOUN
ejpam-6452	268	3	and	and	CCONJ
ejpam-6452	268	4	n	n	ADV
ejpam-6452	268	5	be	be	VERB
ejpam-6452	268	6	non	non	ADJ
ejpam-6452	268	7	-	-	ADJ
ejpam-6452	268	8	negative	negative	ADJ
ejpam-6452	268	9	integers	integer	NOUN
ejpam-6452	268	10	with	with	ADP
ejpam-6452	268	11	k	k	PROPN
ejpam-6452	268	12	≥	≥	NUM
ejpam-6452	268	13	2	2	NUM
ejpam-6452	268	14	and	and	CCONJ
ejpam-6452	268	15	∆k	∆k	PROPN
ejpam-6452	268	16	=	=	SYM
ejpam-6452	268	17	√	√	PROPN
ejpam-6452	268	18	k2	k2	NOUN
ejpam-6452	269	1	+	+	CCONJ
ejpam-6452	269	2	4k	4k	NOUN
ejpam-6452	269	3	−	−	NOUN
ejpam-6452	269	4	4	4	NUM
ejpam-6452	269	5	.	.	PUNCT
ejpam-6452	270	1	(	(	PUNCT
ejpam-6452	270	2	i	i	NOUN
ejpam-6452	270	3	)	)	PUNCT
ejpam-6452	270	4	the	the	DET
ejpam-6452	270	5	generalized	generalize	VERB
ejpam-6452	270	6	pell	pell	NOUN
ejpam-6452	270	7	number	number	NOUN
ejpam-6452	270	8	pk,2n	pk,2n	NOUN
ejpam-6452	270	9	can	can	AUX
ejpam-6452	270	10	be	be	AUX
ejpam-6452	270	11	represented	represent	VERB
ejpam-6452	270	12	by	by	ADP
ejpam-6452	270	13	pk,2n	pk,2n	PROPN
ejpam-6452	270	14	=	=	SYM
ejpam-6452	270	15	nk	nk	PROPN
ejpam-6452	270	16	2n∆k	2n∆k	NUM
ejpam-6452	270	17	∫	∫	PROPN
ejpam-6452	270	18	∆k	∆k	PROPN
ejpam-6452	270	19	−∆k	−∆k	PUNCT
ejpam-6452	271	1	(	(	PUNCT
ejpam-6452	271	2	k2	k2	X
ejpam-6452	271	3	+	+	CCONJ
ejpam-6452	271	4	2k	2k	NOUN
ejpam-6452	271	5	−	−	NUM
ejpam-6452	271	6	2	2	NUM
ejpam-6452	271	7	+	+	NUM
ejpam-6452	271	8	kx	kx	X
ejpam-6452	271	9	)	)	PUNCT
ejpam-6452	271	10	n−1	n−1	PROPN
ejpam-6452	271	11	dx	dx	PROPN
ejpam-6452	271	12	.	.	PUNCT
ejpam-6452	272	1	(	(	PUNCT
ejpam-6452	272	2	9	9	NUM
ejpam-6452	272	3	)	)	PUNCT
ejpam-6452	272	4	(	(	PUNCT
ejpam-6452	272	5	ii	ii	NOUN
ejpam-6452	272	6	)	)	PUNCT
ejpam-6452	272	7	the	the	DET
ejpam-6452	272	8	generalized	generalize	VERB
ejpam-6452	272	9	pell	pell	NOUN
ejpam-6452	272	10	number	number	NOUN
ejpam-6452	272	11	pk,2n+1	pk,2n+1	PROPN
ejpam-6452	272	12	can	can	AUX
ejpam-6452	272	13	be	be	AUX
ejpam-6452	272	14	represented	represent	VERB
ejpam-6452	272	15	by	by	ADP
ejpam-6452	272	16	pk,2n+1	pk,2n+1	PROPN
ejpam-6452	272	17	=	=	PROPN
ejpam-6452	272	18	1	1	NUM
ejpam-6452	272	19	2n+1∆k	2n+1∆k	NUM
ejpam-6452	272	20	∫	∫	NOUN
ejpam-6452	272	21	∆k	∆k	PROPN
ejpam-6452	272	22	−∆k	−∆k	PUNCT
ejpam-6452	273	1	(	(	PUNCT
ejpam-6452	273	2	2k	2k	NOUN
ejpam-6452	273	3	−	−	NUM
ejpam-6452	273	4	2	2	NUM
ejpam-6452	273	5	+	+	CCONJ
ejpam-6452	273	6	(	(	PUNCT
ejpam-6452	273	7	n	n	X
ejpam-6452	273	8	+	+	CCONJ
ejpam-6452	273	9	1)(k2	1)(k2	NUM
ejpam-6452	273	10	+	+	CCONJ
ejpam-6452	273	11	kx	kx	PROPN
ejpam-6452	273	12	)	)	PUNCT
ejpam-6452	273	13	)	)	PUNCT
ejpam-6452	274	1	(	(	PUNCT
ejpam-6452	274	2	k2	k2	X
ejpam-6452	274	3	+	+	CCONJ
ejpam-6452	274	4	2k	2k	NOUN
ejpam-6452	274	5	−	−	NUM
ejpam-6452	274	6	2	2	NUM
ejpam-6452	275	1	+	+	CCONJ
ejpam-6452	275	2	k	k	PROPN
ejpam-6452	275	3	x	x	X
ejpam-6452	275	4	)	)	PUNCT
ejpam-6452	275	5	n−1	n−1	PROPN
ejpam-6452	275	6	dx	dx	PROPN
ejpam-6452	275	7	.	.	PUNCT
ejpam-6452	275	8	proof	proof	NOUN
ejpam-6452	275	9	.	.	PUNCT
ejpam-6452	276	1	(	(	PUNCT
ejpam-6452	276	2	i	i	NOUN
ejpam-6452	276	3	)	)	PUNCT
ejpam-6452	276	4	notice	notice	VERB
ejpam-6452	276	5	that	that	SCONJ
ejpam-6452	276	6	pk,2	pk,2	PROPN
ejpam-6452	276	7	=	=	SYM
ejpam-6452	276	8	k	k	PROPN
ejpam-6452	276	9	and	and	CCONJ
ejpam-6452	276	10	lk,2	lk,2	PROPN
ejpam-6452	276	11	=	=	SYM
ejpam-6452	276	12	k2	k2	PROPN
ejpam-6452	276	13	+	+	CCONJ
ejpam-6452	276	14	2k	2k	NOUN
ejpam-6452	276	15	−	−	NOUN
ejpam-6452	276	16	2	2	X
ejpam-6452	276	17	.	.	PUNCT
ejpam-6452	276	18	setting	set	VERB
ejpam-6452	276	19	ℓ	ℓ	NOUN
ejpam-6452	276	20	=	=	SYM
ejpam-6452	276	21	2	2	NUM
ejpam-6452	276	22	in	in	ADP
ejpam-6452	276	23	(	(	PUNCT
ejpam-6452	276	24	7	7	NUM
ejpam-6452	276	25	)	)	PUNCT
ejpam-6452	276	26	,	,	PUNCT
ejpam-6452	276	27	we	we	PRON
ejpam-6452	276	28	have	have	VERB
ejpam-6452	276	29	pk,2n	pk,2n	NOUN
ejpam-6452	276	30	=	=	PUNCT
ejpam-6452	276	31	npk,2	npk,2	NOUN
ejpam-6452	276	32	2n∆k	2n∆k	NUM
ejpam-6452	276	33	∫	∫	PROPN
ejpam-6452	276	34	∆k	∆k	PROPN
ejpam-6452	276	35	−∆k	−∆k	PUNCT
ejpam-6452	277	1	(	(	PUNCT
ejpam-6452	277	2	lk,2	lk,2	NOUN
ejpam-6452	277	3	+	+	CCONJ
ejpam-6452	277	4	pk,2x)n−1	pk,2x)n−1	ADP
ejpam-6452	277	5	dx	dx	PROPN
ejpam-6452	277	6	=	=	SYM
ejpam-6452	277	7	nk	nk	PROPN
ejpam-6452	277	8	2n∆k	2n∆k	NUM
ejpam-6452	277	9	∫	∫	PROPN
ejpam-6452	277	10	∆k	∆k	PROPN
ejpam-6452	277	11	−∆k	−∆k	PUNCT
ejpam-6452	278	1	(	(	PUNCT
ejpam-6452	278	2	k2	k2	X
ejpam-6452	278	3	+	+	CCONJ
ejpam-6452	278	4	2k	2k	NOUN
ejpam-6452	278	5	−	−	NUM
ejpam-6452	278	6	2	2	NUM
ejpam-6452	278	7	+	+	NUM
ejpam-6452	278	8	kx	kx	X
ejpam-6452	278	9	)	)	PUNCT
ejpam-6452	278	10	n−1	n−1	PROPN
ejpam-6452	278	11	dx	dx	PROPN
ejpam-6452	278	12	.	.	PUNCT
ejpam-6452	278	13	(	(	PUNCT
ejpam-6452	278	14	ii	ii	NOUN
ejpam-6452	278	15	)	)	PUNCT
ejpam-6452	278	16	re	re	VERB
ejpam-6452	278	17	-	-	NOUN
ejpam-6452	278	18	indexing	indexing	ADJ
ejpam-6452	278	19	n	n	NOUN
ejpam-6452	278	20	by	by	ADP
ejpam-6452	278	21	n	n	PROPN
ejpam-6452	278	22	+	+	CCONJ
ejpam-6452	278	23	1	1	NUM
ejpam-6452	278	24	in	in	ADP
ejpam-6452	278	25	(	(	PUNCT
ejpam-6452	278	26	9	9	NUM
ejpam-6452	278	27	)	)	PUNCT
ejpam-6452	278	28	,	,	PUNCT
ejpam-6452	278	29	we	we	PRON
ejpam-6452	278	30	get	get	VERB
ejpam-6452	278	31	pk,2n+2	pk,2n+2	NOUN
ejpam-6452	278	32	=	=	PUNCT
ejpam-6452	278	33	(	(	PUNCT
ejpam-6452	278	34	n	n	X
ejpam-6452	278	35	+	+	CCONJ
ejpam-6452	278	36	1)k	1)k	NUM
ejpam-6452	278	37	2n+1∆k	2n+1∆k	NUM
ejpam-6452	278	38	∫	∫	NOUN
ejpam-6452	278	39	∆k	∆k	PROPN
ejpam-6452	278	40	−∆k	−∆k	PUNCT
ejpam-6452	279	1	(	(	PUNCT
ejpam-6452	279	2	k2	k2	X
ejpam-6452	279	3	+	+	CCONJ
ejpam-6452	279	4	2k	2k	NOUN
ejpam-6452	279	5	−	−	NUM
ejpam-6452	279	6	2	2	NUM
ejpam-6452	279	7	+	+	NUM
ejpam-6452	279	8	kx	kx	PROPN
ejpam-6452	279	9	)	)	PUNCT
ejpam-6452	279	10	n	n	PRON
ejpam-6452	279	11	dx	dx	PROPN
ejpam-6452	279	12	.	.	PUNCT
ejpam-6452	280	1	(	(	PUNCT
ejpam-6452	280	2	10	10	NUM
ejpam-6452	280	3	)	)	PUNCT
ejpam-6452	280	4	w.	w.	NOUN
ejpam-6452	280	5	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	280	6	,	,	PUNCT
ejpam-6452	280	7	a.	a.	NOUN
ejpam-6452	280	8	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	280	9	/	/	SYM
ejpam-6452	280	10	eur	eur	PROPN
ejpam-6452	280	11	.	.	PUNCT
ejpam-6452	281	1	j.	j.	PROPN
ejpam-6452	281	2	pure	pure	PROPN
ejpam-6452	281	3	appl	appl	PROPN
ejpam-6452	281	4	.	.	PROPN
ejpam-6452	281	5	math	math	PROPN
ejpam-6452	281	6	,	,	PUNCT
ejpam-6452	281	7	18	18	NUM
ejpam-6452	281	8	(	(	PUNCT
ejpam-6452	281	9	3	3	NUM
ejpam-6452	281	10	)	)	PUNCT
ejpam-6452	281	11	(	(	PUNCT
ejpam-6452	281	12	2025	2025	NUM
ejpam-6452	281	13	)	)	PUNCT
ejpam-6452	281	14	,	,	PUNCT
ejpam-6452	281	15	6452	6452	NUM
ejpam-6452	281	16	13	13	NUM
ejpam-6452	281	17	of	of	ADP
ejpam-6452	281	18	18	18	NUM
ejpam-6452	281	19	using	use	VERB
ejpam-6452	281	20	pk,2n+2	pk,2n+2	PROPN
ejpam-6452	281	21	=	=	PUNCT
ejpam-6452	281	22	kpk,2n+1	kpk,2n+1	PROPN
ejpam-6452	282	1	+	+	SYM
ejpam-6452	282	2	(	(	PUNCT
ejpam-6452	282	3	k	k	PROPN
ejpam-6452	282	4	−	−	PROPN
ejpam-6452	282	5	1)pk,2n	1)pk,2n	PROPN
ejpam-6452	282	6	with	with	ADP
ejpam-6452	282	7	(	(	PUNCT
ejpam-6452	282	8	9	9	NUM
ejpam-6452	282	9	)	)	PUNCT
ejpam-6452	282	10	and	and	CCONJ
ejpam-6452	282	11	(	(	PUNCT
ejpam-6452	282	12	10	10	NUM
ejpam-6452	282	13	)	)	PUNCT
ejpam-6452	282	14	,	,	PUNCT
ejpam-6452	282	15	we	we	PRON
ejpam-6452	282	16	obtain	obtain	VERB
ejpam-6452	282	17	pk,2n+1	pk,2n+1	NOUN
ejpam-6452	282	18	=	=	SYM
ejpam-6452	282	19	1	1	NUM
ejpam-6452	282	20	k	k	X
ejpam-6452	282	21	pk,2n+2	pk,2n+2	PROPN
ejpam-6452	282	22	−	−	PROPN
ejpam-6452	283	1	k	k	NOUN
ejpam-6452	284	1	−	−	PROPN
ejpam-6452	284	2	1	1	NUM
ejpam-6452	284	3	k	k	NOUN
ejpam-6452	284	4	pk,2n	pk,2n	NOUN
ejpam-6452	284	5	=	=	SYM
ejpam-6452	284	6	(	(	PUNCT
ejpam-6452	284	7	n	n	PROPN
ejpam-6452	284	8	+	+	CCONJ
ejpam-6452	284	9	1	1	NUM
ejpam-6452	284	10	)	)	PUNCT
ejpam-6452	284	11	2n+1∆k	2n+1∆k	NUM
ejpam-6452	284	12	∫	∫	NOUN
ejpam-6452	284	13	∆k	∆k	PROPN
ejpam-6452	284	14	−∆k	−∆k	PUNCT
ejpam-6452	285	1	(	(	PUNCT
ejpam-6452	285	2	k2	k2	X
ejpam-6452	285	3	+	+	CCONJ
ejpam-6452	285	4	2k	2k	NOUN
ejpam-6452	285	5	−	−	NUM
ejpam-6452	285	6	2	2	NUM
ejpam-6452	285	7	+	+	CCONJ
ejpam-6452	285	8	kx	kx	X
ejpam-6452	285	9	)	)	PUNCT
ejpam-6452	285	10	n	n	CCONJ
ejpam-6452	285	11	dx−	dx−	PROPN
ejpam-6452	285	12	n(k	n(k	PROPN
ejpam-6452	285	13	−	−	PROPN
ejpam-6452	285	14	1	1	NUM
ejpam-6452	285	15	)	)	PUNCT
ejpam-6452	285	16	2n∆k	2n∆k	NUM
ejpam-6452	285	17	∫	∫	PROPN
ejpam-6452	285	18	∆k	∆k	PROPN
ejpam-6452	285	19	−∆k	−∆k	PUNCT
ejpam-6452	286	1	(	(	PUNCT
ejpam-6452	286	2	k2	k2	X
ejpam-6452	286	3	+	+	CCONJ
ejpam-6452	286	4	2k	2k	NOUN
ejpam-6452	286	5	−	−	NUM
ejpam-6452	286	6	2	2	NUM
ejpam-6452	286	7	+	+	NUM
ejpam-6452	286	8	kx	kx	X
ejpam-6452	286	9	)	)	PUNCT
ejpam-6452	287	1	n−1	n−1	PROPN
ejpam-6452	287	2	dx	dx	NOUN
ejpam-6452	287	3	=	=	SYM
ejpam-6452	287	4	1	1	NUM
ejpam-6452	287	5	2n+1∆k	2n+1∆k	NUM
ejpam-6452	287	6	∫	∫	NOUN
ejpam-6452	287	7	∆k	∆k	PROPN
ejpam-6452	287	8	−∆k	−∆k	PUNCT
ejpam-6452	288	1	(	(	PUNCT
ejpam-6452	288	2	2k	2k	NOUN
ejpam-6452	288	3	−	−	NUM
ejpam-6452	288	4	2	2	NUM
ejpam-6452	288	5	+	+	CCONJ
ejpam-6452	288	6	(	(	PUNCT
ejpam-6452	288	7	n	n	X
ejpam-6452	288	8	+	+	CCONJ
ejpam-6452	288	9	1)(k2	1)(k2	NUM
ejpam-6452	288	10	+	+	CCONJ
ejpam-6452	288	11	kx	kx	PROPN
ejpam-6452	288	12	)	)	PUNCT
ejpam-6452	288	13	)	)	PUNCT
ejpam-6452	289	1	(	(	PUNCT
ejpam-6452	289	2	k2	k2	X
ejpam-6452	289	3	+	+	CCONJ
ejpam-6452	289	4	2k	2k	NOUN
ejpam-6452	289	5	−	−	NUM
ejpam-6452	289	6	2	2	NUM
ejpam-6452	290	1	+	+	CCONJ
ejpam-6452	290	2	k	k	PROPN
ejpam-6452	290	3	x	x	X
ejpam-6452	290	4	)	)	PUNCT
ejpam-6452	290	5	n−1	n−1	PROPN
ejpam-6452	290	6	dx	dx	PROPN
ejpam-6452	290	7	.	.	PUNCT
ejpam-6452	291	1	this	this	PRON
ejpam-6452	291	2	completes	complete	VERB
ejpam-6452	291	3	the	the	DET
ejpam-6452	291	4	proof	proof	NOUN
ejpam-6452	291	5	.	.	PUNCT
ejpam-6452	292	1	setting	set	VERB
ejpam-6452	292	2	k	k	X
ejpam-6452	292	3	=	=	SYM
ejpam-6452	292	4	2	2	NUM
ejpam-6452	292	5	in	in	ADP
ejpam-6452	292	6	theorem	theorem	ADJ
ejpam-6452	292	7	8	8	NUM
ejpam-6452	292	8	and	and	CCONJ
ejpam-6452	292	9	remark	remark	NOUN
ejpam-6452	292	10	2	2	NUM
ejpam-6452	292	11	,	,	PUNCT
ejpam-6452	292	12	we	we	PRON
ejpam-6452	292	13	have	have	VERB
ejpam-6452	292	14	the	the	DET
ejpam-6452	292	15	following	follow	VERB
ejpam-6452	292	16	corollary	corollary	NOUN
ejpam-6452	292	17	.	.	PUNCT
ejpam-6452	293	1	corollary	corollary	ADJ
ejpam-6452	293	2	14	14	NUM
ejpam-6452	293	3	(	(	PUNCT
ejpam-6452	293	4	[	[	X
ejpam-6452	293	5	17	17	NUM
ejpam-6452	293	6	,	,	PUNCT
ejpam-6452	293	7	theorem	theorem	VERB
ejpam-6452	293	8	3.1	3.1	NUM
ejpam-6452	293	9	]	]	PUNCT
ejpam-6452	293	10	)	)	PUNCT
ejpam-6452	293	11	.	.	PUNCT
ejpam-6452	294	1	let	let	VERB
ejpam-6452	294	2	ℓ	ℓ	NOUN
ejpam-6452	294	3	and	and	CCONJ
ejpam-6452	294	4	n	n	CCONJ
ejpam-6452	294	5	be	be	VERB
ejpam-6452	294	6	non	non	ADJ
ejpam-6452	294	7	-	-	ADJ
ejpam-6452	294	8	negative	negative	ADJ
ejpam-6452	294	9	integers	integer	NOUN
ejpam-6452	294	10	.	.	PUNCT
ejpam-6452	295	1	then	then	ADV
ejpam-6452	295	2	pℓn	pℓn	INTJ
ejpam-6452	295	3	=	=	PUNCT
ejpam-6452	295	4	npℓ	npℓ	ADJ
ejpam-6452	295	5	2n+1	2n+1	NOUN
ejpam-6452	295	6	√	√	NUM
ejpam-6452	295	7	2	2	NUM
ejpam-6452	295	8	∫	∫	NOUN
ejpam-6452	295	9	2	2	NUM
ejpam-6452	295	10	√	√	NUM
ejpam-6452	295	11	2	2	NUM
ejpam-6452	295	12	−2	−2	NOUN
ejpam-6452	295	13	√	√	NOUN
ejpam-6452	295	14	2	2	NUM
ejpam-6452	295	15	(	(	PUNCT
ejpam-6452	295	16	qℓ	qℓ	NOUN
ejpam-6452	295	17	+	+	X
ejpam-6452	295	18	pℓ	pℓ	ADV
ejpam-6452	295	19	x)n−1dx	x)n−1dx	PROPN
ejpam-6452	296	1	=	=	PUNCT
ejpam-6452	296	2	npℓ	npℓ	PROPN
ejpam-6452	297	1	2n	2n	NUM
ejpam-6452	297	2	∫	∫	PROPN
ejpam-6452	297	3	1	1	NUM
ejpam-6452	297	4	−1	−1	NOUN
ejpam-6452	297	5	(	(	PUNCT
ejpam-6452	297	6	qℓ	qℓ	NOUN
ejpam-6452	297	7	+	+	X
ejpam-6452	297	8	2	2	NUM
ejpam-6452	297	9	√	√	NUM
ejpam-6452	297	10	2pℓ	2pℓ	NOUN
ejpam-6452	298	1	x)n−1dx	x)n−1dx	PROPN
ejpam-6452	298	2	.	.	PUNCT
ejpam-6452	299	1	next	next	ADV
ejpam-6452	299	2	,	,	PUNCT
ejpam-6452	299	3	we	we	PRON
ejpam-6452	299	4	provide	provide	VERB
ejpam-6452	299	5	integral	integral	ADJ
ejpam-6452	299	6	representations	representation	NOUN
ejpam-6452	299	7	for	for	ADP
ejpam-6452	299	8	the	the	DET
ejpam-6452	299	9	generalized	generalize	VERB
ejpam-6452	299	10	pell	pell	NOUN
ejpam-6452	299	11	-	-	PUNCT
ejpam-6452	299	12	lucas	lucas	NOUN
ejpam-6452	299	13	-	-	PUNCT
ejpam-6452	299	14	like	like	ADJ
ejpam-6452	299	15	number	number	NOUN
ejpam-6452	299	16	lk,ℓn	lk,ℓn	PROPN
ejpam-6452	299	17	based	base	VERB
ejpam-6452	299	18	on	on	ADP
ejpam-6452	299	19	the	the	DET
ejpam-6452	299	20	two	two	NUM
ejpam-6452	299	21	numbers	number	NOUN
ejpam-6452	299	22	pk,ℓ	pk,ℓ	NOUN
ejpam-6452	299	23	and	and	CCONJ
ejpam-6452	299	24	lk,ℓ.	lk,ℓ.	ADJ
ejpam-6452	299	25	theorem	theorem	ADJ
ejpam-6452	299	26	10	10	NUM
ejpam-6452	299	27	.	.	PUNCT
ejpam-6452	300	1	let	let	VERB
ejpam-6452	300	2	k	k	NOUN
ejpam-6452	300	3	,	,	PUNCT
ejpam-6452	300	4	ℓ	ℓ	PROPN
ejpam-6452	300	5	and	and	CCONJ
ejpam-6452	300	6	n	n	CCONJ
ejpam-6452	300	7	be	be	VERB
ejpam-6452	300	8	non	non	ADJ
ejpam-6452	300	9	-	-	ADJ
ejpam-6452	300	10	negative	negative	ADJ
ejpam-6452	300	11	integers	integer	NOUN
ejpam-6452	300	12	with	with	ADP
ejpam-6452	300	13	k	k	PROPN
ejpam-6452	300	14	≥	≥	NUM
ejpam-6452	300	15	2	2	NUM
ejpam-6452	300	16	and	and	CCONJ
ejpam-6452	300	17	∆k	∆k	PROPN
ejpam-6452	300	18	=	=	SYM
ejpam-6452	300	19	√	√	PROPN
ejpam-6452	300	20	k2	k2	NOUN
ejpam-6452	301	1	+	+	CCONJ
ejpam-6452	301	2	4k	4k	NOUN
ejpam-6452	301	3	−	−	NOUN
ejpam-6452	302	1	4	4	X
ejpam-6452	302	2	.	.	PUNCT
ejpam-6452	302	3	then	then	ADV
ejpam-6452	302	4	lk,ℓn	lk,ℓn	PROPN
ejpam-6452	302	5	=	=	SYM
ejpam-6452	302	6	1	1	NUM
ejpam-6452	302	7	2n∆k	2n∆k	NUM
ejpam-6452	302	8	∫	∫	NOUN
ejpam-6452	302	9	∆k	∆k	PROPN
ejpam-6452	302	10	−∆k	−∆k	PUNCT
ejpam-6452	303	1	(	(	PUNCT
ejpam-6452	303	2	lk,ℓ	lk,ℓ	X
ejpam-6452	303	3	+	+	CCONJ
ejpam-6452	303	4	(	(	PUNCT
ejpam-6452	303	5	n	n	X
ejpam-6452	303	6	+	+	NUM
ejpam-6452	303	7	1)pk,ℓx	1)pk,ℓx	NUM
ejpam-6452	303	8	)	)	PUNCT
ejpam-6452	303	9	(	(	PUNCT
ejpam-6452	303	10	lk,ℓ	lk,ℓ	X
ejpam-6452	303	11	+	+	CCONJ
ejpam-6452	303	12	pk,ℓx)n−1	pk,ℓx)n−1	PROPN
ejpam-6452	303	13	dx	dx	PROPN
ejpam-6452	303	14	.	.	PUNCT
ejpam-6452	304	1	(	(	PUNCT
ejpam-6452	304	2	11	11	NUM
ejpam-6452	304	3	)	)	PUNCT
ejpam-6452	304	4	proof	proof	NOUN
ejpam-6452	304	5	.	.	PUNCT
ejpam-6452	305	1	for	for	ADP
ejpam-6452	305	2	n	n	NOUN
ejpam-6452	305	3	=	=	SYM
ejpam-6452	305	4	0	0	NUM
ejpam-6452	305	5	or	or	CCONJ
ejpam-6452	305	6	ℓ	ℓ	X
ejpam-6452	305	7	=	=	SYM
ejpam-6452	305	8	0	0	NUM
ejpam-6452	305	9	,	,	PUNCT
ejpam-6452	305	10	it	it	PRON
ejpam-6452	305	11	is	be	AUX
ejpam-6452	305	12	easy	easy	ADJ
ejpam-6452	305	13	to	to	PART
ejpam-6452	305	14	see	see	VERB
ejpam-6452	305	15	that	that	PRON
ejpam-6452	305	16	(	(	PUNCT
ejpam-6452	305	17	11	11	NUM
ejpam-6452	305	18	)	)	PUNCT
ejpam-6452	305	19	holds	hold	VERB
ejpam-6452	305	20	.	.	PUNCT
ejpam-6452	306	1	we	we	PRON
ejpam-6452	306	2	assume	assume	VERB
ejpam-6452	306	3	now	now	ADV
ejpam-6452	306	4	that	that	SCONJ
ejpam-6452	306	5	ℓ	ℓ	NOUN
ejpam-6452	306	6	,	,	PUNCT
ejpam-6452	306	7	n	n	X
ejpam-6452	306	8	>	>	X
ejpam-6452	306	9	0	0	X
ejpam-6452	306	10	.	.	PUNCT
ejpam-6452	307	1	we	we	PRON
ejpam-6452	307	2	will	will	AUX
ejpam-6452	307	3	solve	solve	VERB
ejpam-6452	307	4	(	(	PUNCT
ejpam-6452	307	5	11	11	NUM
ejpam-6452	307	6	)	)	PUNCT
ejpam-6452	307	7	using	use	VERB
ejpam-6452	307	8	integration	integration	NOUN
ejpam-6452	307	9	by	by	ADP
ejpam-6452	307	10	parts	part	NOUN
ejpam-6452	307	11	.	.	PUNCT
ejpam-6452	308	1	let	let	VERB
ejpam-6452	308	2	u	u	PRON
ejpam-6452	308	3	and	and	CCONJ
ejpam-6452	308	4	v	v	NOUN
ejpam-6452	308	5	be	be	AUX
ejpam-6452	308	6	such	such	ADJ
ejpam-6452	308	7	that	that	SCONJ
ejpam-6452	308	8	u(x	u(x	NOUN
ejpam-6452	308	9	)	)	PUNCT
ejpam-6452	308	10	=	=	PUNCT
ejpam-6452	308	11	lk,ℓ	lk,ℓ	X
ejpam-6452	308	12	+	+	CCONJ
ejpam-6452	308	13	(	(	PUNCT
ejpam-6452	308	14	n	n	X
ejpam-6452	308	15	+	+	CCONJ
ejpam-6452	308	16	1)pk,ℓx	1)pk,ℓx	NUM
ejpam-6452	308	17	and	and	CCONJ
ejpam-6452	308	18	dv	dv	PROPN
ejpam-6452	309	1	=	=	PUNCT
ejpam-6452	309	2	(	(	PUNCT
ejpam-6452	309	3	lk,ℓ	lk,ℓ	X
ejpam-6452	309	4	+	+	CCONJ
ejpam-6452	309	5	pk,ℓx)n−1	pk,ℓx)n−1	PROPN
ejpam-6452	309	6	dx	dx	PROPN
ejpam-6452	309	7	.	.	PUNCT
ejpam-6452	310	1	then	then	ADV
ejpam-6452	310	2	i	i	PRON
ejpam-6452	310	3	=	=	NOUN
ejpam-6452	310	4	1	1	NUM
ejpam-6452	310	5	2n∆k	2n∆k	NUM
ejpam-6452	310	6	∫	∫	NOUN
ejpam-6452	310	7	∆k	∆k	PROPN
ejpam-6452	310	8	−∆k	−∆k	PUNCT
ejpam-6452	311	1	(	(	PUNCT
ejpam-6452	311	2	lk,ℓ	lk,ℓ	X
ejpam-6452	311	3	+	+	CCONJ
ejpam-6452	311	4	(	(	PUNCT
ejpam-6452	311	5	n	n	X
ejpam-6452	311	6	+	+	NUM
ejpam-6452	311	7	1)pk,ℓx	1)pk,ℓx	NUM
ejpam-6452	311	8	)	)	PUNCT
ejpam-6452	311	9	(	(	PUNCT
ejpam-6452	311	10	lk,ℓ	lk,ℓ	X
ejpam-6452	311	11	+	+	CCONJ
ejpam-6452	311	12	pk,ℓx)n−1	pk,ℓx)n−1	PROPN
ejpam-6452	311	13	dx	dx	PROPN
ejpam-6452	312	1	=	=	SYM
ejpam-6452	312	2	1	1	NUM
ejpam-6452	312	3	n2n∆k	n2n∆k	X
ejpam-6452	312	4	pk,ℓ	pk,ℓ	PUNCT
ejpam-6452	312	5	[	[	PUNCT
ejpam-6452	312	6	(	(	PUNCT
ejpam-6452	312	7	lk,ℓ	lk,ℓ	X
ejpam-6452	312	8	+	+	CCONJ
ejpam-6452	312	9	(	(	PUNCT
ejpam-6452	312	10	n	n	X
ejpam-6452	312	11	+	+	NUM
ejpam-6452	312	12	1)pk,ℓx	1)pk,ℓx	NUM
ejpam-6452	312	13	)	)	PUNCT
ejpam-6452	312	14	(	(	PUNCT
ejpam-6452	312	15	lk,ℓ	lk,ℓ	X
ejpam-6452	312	16	+	+	CCONJ
ejpam-6452	312	17	pk,ℓx)n	pk,ℓx)n	NOUN
ejpam-6452	312	18	]	]	PUNCT
ejpam-6452	312	19	∆k	∆k	PROPN
ejpam-6452	312	20	−∆k	−∆k	NOUN
ejpam-6452	313	1	−	−	PROPN
ejpam-6452	313	2	(	(	PUNCT
ejpam-6452	313	3	n	n	NOUN
ejpam-6452	313	4	+	+	CCONJ
ejpam-6452	313	5	1	1	NUM
ejpam-6452	313	6	)	)	PUNCT
ejpam-6452	313	7	n2n∆k	n2n∆k	VERB
ejpam-6452	313	8	∫	∫	PROPN
ejpam-6452	313	9	∆k	∆k	PROPN
ejpam-6452	313	10	−∆k	−∆k	PUNCT
ejpam-6452	314	1	(	(	PUNCT
ejpam-6452	314	2	lk,ℓ	lk,ℓ	X
ejpam-6452	314	3	+	+	CCONJ
ejpam-6452	314	4	pk,ℓx)n	pk,ℓx)n	ADJ
ejpam-6452	314	5	dx	dx	PROPN
ejpam-6452	314	6	.	.	PUNCT
ejpam-6452	315	1	(	(	PUNCT
ejpam-6452	315	2	12	12	NUM
ejpam-6452	315	3	)	)	PUNCT
ejpam-6452	315	4	replacing	replace	VERB
ejpam-6452	315	5	n	n	PRON
ejpam-6452	315	6	by	by	ADP
ejpam-6452	315	7	n	n	PROPN
ejpam-6452	315	8	+	+	CCONJ
ejpam-6452	315	9	1	1	NUM
ejpam-6452	315	10	in	in	ADP
ejpam-6452	315	11	(	(	PUNCT
ejpam-6452	315	12	7	7	NUM
ejpam-6452	315	13	)	)	PUNCT
ejpam-6452	315	14	becomes	become	VERB
ejpam-6452	315	15	pk,ℓn+ℓ	pk,ℓn+ℓ	PROPN
ejpam-6452	315	16	=	=	PUNCT
ejpam-6452	315	17	(	(	PUNCT
ejpam-6452	315	18	n	n	X
ejpam-6452	315	19	+	+	CCONJ
ejpam-6452	315	20	1)pk,ℓ	1)pk,ℓ	NUM
ejpam-6452	315	21	2n+1∆k	2n+1∆k	NUM
ejpam-6452	315	22	∫	∫	NOUN
ejpam-6452	315	23	∆k	∆k	PROPN
ejpam-6452	315	24	−∆k	−∆k	PUNCT
ejpam-6452	316	1	(	(	PUNCT
ejpam-6452	316	2	lk,ℓ	lk,ℓ	X
ejpam-6452	316	3	+	+	CCONJ
ejpam-6452	316	4	pk,ℓx)n	pk,ℓx)n	ADJ
ejpam-6452	316	5	dx	dx	PROPN
ejpam-6452	316	6	and	and	CCONJ
ejpam-6452	316	7	so	so	ADV
ejpam-6452	316	8	2pk,ℓn+ℓ	2pk,ℓn+ℓ	NUM
ejpam-6452	316	9	npk,ℓ	npk,ℓ	NOUN
ejpam-6452	316	10	=	=	SYM
ejpam-6452	316	11	(	(	PUNCT
ejpam-6452	316	12	n	n	PROPN
ejpam-6452	316	13	+	+	CCONJ
ejpam-6452	316	14	1	1	NUM
ejpam-6452	316	15	)	)	PUNCT
ejpam-6452	316	16	n2n∆k	n2n∆k	VERB
ejpam-6452	316	17	∫	∫	PROPN
ejpam-6452	316	18	∆k	∆k	PROPN
ejpam-6452	316	19	−∆k	−∆k	PUNCT
ejpam-6452	316	20	(	(	PUNCT
ejpam-6452	316	21	lk,ℓ	lk,ℓ	X
ejpam-6452	316	22	+	+	CCONJ
ejpam-6452	316	23	pk,ℓx)n	pk,ℓx)n	ADJ
ejpam-6452	316	24	dx	dx	PROPN
ejpam-6452	316	25	.	.	PUNCT
ejpam-6452	317	1	w.	w.	PROPN
ejpam-6452	317	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	317	3	,	,	PUNCT
ejpam-6452	317	4	a.	a.	NOUN
ejpam-6452	317	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	317	6	/	/	SYM
ejpam-6452	317	7	eur	eur	PROPN
ejpam-6452	317	8	.	.	PUNCT
ejpam-6452	318	1	j.	j.	PROPN
ejpam-6452	318	2	pure	pure	PROPN
ejpam-6452	318	3	appl	appl	PROPN
ejpam-6452	318	4	.	.	PROPN
ejpam-6452	318	5	math	math	PROPN
ejpam-6452	318	6	,	,	PUNCT
ejpam-6452	318	7	18	18	NUM
ejpam-6452	318	8	(	(	PUNCT
ejpam-6452	318	9	3	3	NUM
ejpam-6452	318	10	)	)	PUNCT
ejpam-6452	318	11	(	(	PUNCT
ejpam-6452	318	12	2025	2025	NUM
ejpam-6452	318	13	)	)	PUNCT
ejpam-6452	318	14	,	,	PUNCT
ejpam-6452	318	15	6452	6452	NUM
ejpam-6452	318	16	14	14	NUM
ejpam-6452	318	17	of	of	ADP
ejpam-6452	318	18	18	18	NUM
ejpam-6452	318	19	this	this	PRON
ejpam-6452	318	20	together	together	ADV
ejpam-6452	318	21	with	with	ADP
ejpam-6452	318	22	(	(	PUNCT
ejpam-6452	318	23	12	12	NUM
ejpam-6452	318	24	)	)	PUNCT
ejpam-6452	318	25	gives	give	VERB
ejpam-6452	318	26	i	i	PRON
ejpam-6452	318	27	=	=	SYM
ejpam-6452	318	28	1	1	NUM
ejpam-6452	318	29	n2n∆k	n2n∆k	X
ejpam-6452	318	30	pk,ℓ	pk,ℓ	PUNCT
ejpam-6452	318	31	[	[	PUNCT
ejpam-6452	318	32	(	(	PUNCT
ejpam-6452	318	33	lk,ℓ	lk,ℓ	X
ejpam-6452	318	34	+	+	CCONJ
ejpam-6452	318	35	(	(	PUNCT
ejpam-6452	318	36	n	n	X
ejpam-6452	318	37	+	+	NOUN
ejpam-6452	318	38	1)∆kpk,ℓ	1)∆kpk,ℓ	NUM
ejpam-6452	318	39	)	)	PUNCT
ejpam-6452	318	40	(	(	PUNCT
ejpam-6452	318	41	lk,ℓ	lk,ℓ	X
ejpam-6452	318	42	+	+	CCONJ
ejpam-6452	318	43	∆kpk,ℓ	∆kpk,ℓ	NUM
ejpam-6452	318	44	)	)	PUNCT
ejpam-6452	318	45	n	n	NOUN
ejpam-6452	318	46	]	]	PUNCT
ejpam-6452	318	47	−	−	PROPN
ejpam-6452	318	48	1	1	NUM
ejpam-6452	318	49	n2n∆k	n2n∆k	X
ejpam-6452	318	50	pk,ℓ	pk,ℓ	PUNCT
ejpam-6452	318	51	[	[	PUNCT
ejpam-6452	318	52	(	(	PUNCT
ejpam-6452	318	53	lk,ℓ	lk,ℓ	X
ejpam-6452	318	54	−	−	PROPN
ejpam-6452	318	55	(	(	PUNCT
ejpam-6452	318	56	n	n	NOUN
ejpam-6452	318	57	+	+	NOUN
ejpam-6452	318	58	1)∆kpk,ℓ	1)∆kpk,ℓ	NUM
ejpam-6452	318	59	)	)	PUNCT
ejpam-6452	318	60	(	(	PUNCT
ejpam-6452	318	61	lk,ℓ	lk,ℓ	X
ejpam-6452	318	62	−	−	PROPN
ejpam-6452	318	63	∆kpk,ℓ	∆kpk,ℓ	PROPN
ejpam-6452	318	64	)	)	PUNCT
ejpam-6452	318	65	n	n	NOUN
ejpam-6452	318	66	]	]	PUNCT
ejpam-6452	318	67	−	−	PROPN
ejpam-6452	318	68	2pk,ℓn+ℓ	2pk,ℓn+ℓ	NUM
ejpam-6452	318	69	npk,ℓ	npk,ℓ	NUM
ejpam-6452	318	70	.	.	PUNCT
ejpam-6452	319	1	(	(	PUNCT
ejpam-6452	319	2	13	13	X
ejpam-6452	319	3	)	)	PUNCT
ejpam-6452	319	4	applying	apply	VERB
ejpam-6452	319	5	(	(	PUNCT
ejpam-6452	319	6	i	i	NOUN
ejpam-6452	319	7	)	)	PUNCT
ejpam-6452	319	8	and	and	CCONJ
ejpam-6452	319	9	(	(	PUNCT
ejpam-6452	319	10	ii	ii	NOUN
ejpam-6452	319	11	)	)	PUNCT
ejpam-6452	319	12	of	of	ADP
ejpam-6452	319	13	theorem	theorem	NOUN
ejpam-6452	319	14	2	2	NUM
ejpam-6452	319	15	to	to	ADP
ejpam-6452	319	16	(	(	PUNCT
ejpam-6452	319	17	13	13	NUM
ejpam-6452	319	18	)	)	PUNCT
ejpam-6452	319	19	gives	give	VERB
ejpam-6452	319	20	i	i	PRON
ejpam-6452	319	21	=	=	SYM
ejpam-6452	319	22	1	1	NUM
ejpam-6452	319	23	n2n∆k	n2n∆k	X
ejpam-6452	319	24	pk,ℓ	pk,ℓ	PUNCT
ejpam-6452	319	25	[	[	PUNCT
ejpam-6452	319	26	2nσℓn	2nσℓn	NUM
ejpam-6452	319	27	k	k	X
ejpam-6452	319	28	(	(	PUNCT
ejpam-6452	319	29	lk,ℓ	lk,ℓ	X
ejpam-6452	319	30	+	+	CCONJ
ejpam-6452	319	31	(	(	PUNCT
ejpam-6452	319	32	n	n	X
ejpam-6452	319	33	+	+	NOUN
ejpam-6452	319	34	1)∆kpk,ℓ	1)∆kpk,ℓ	NUM
ejpam-6452	319	35	)	)	PUNCT
ejpam-6452	319	36	]	]	PUNCT
ejpam-6452	320	1	−	−	PROPN
ejpam-6452	320	2	1	1	NUM
ejpam-6452	320	3	n2n∆k	n2n∆k	X
ejpam-6452	320	4	pk,ℓ	pk,ℓ	PUNCT
ejpam-6452	320	5	[	[	PUNCT
ejpam-6452	320	6	2n	2n	NUM
ejpam-6452	320	7	(	(	PUNCT
ejpam-6452	320	8	1	1	NUM
ejpam-6452	320	9	−	−	PROPN
ejpam-6452	321	1	k)ℓn	k)ℓn	PROPN
ejpam-6452	321	2	σℓn	σℓn	PROPN
ejpam-6452	321	3	k	k	PROPN
ejpam-6452	321	4	(	(	PUNCT
ejpam-6452	321	5	lk,ℓ	lk,ℓ	X
ejpam-6452	321	6	−	−	PROPN
ejpam-6452	321	7	(	(	PUNCT
ejpam-6452	321	8	n	n	NOUN
ejpam-6452	321	9	+	+	NOUN
ejpam-6452	321	10	1)∆kpk,ℓ	1)∆kpk,ℓ	NUM
ejpam-6452	321	11	)	)	PUNCT
ejpam-6452	321	12	]	]	PUNCT
ejpam-6452	322	1	−	−	PROPN
ejpam-6452	322	2	2pk,ℓn+ℓ	2pk,ℓn+ℓ	NUM
ejpam-6452	322	3	npk,ℓ	npk,ℓ	NOUN
ejpam-6452	322	4	=	=	SYM
ejpam-6452	322	5	1	1	NUM
ejpam-6452	322	6	npk,ℓ	npk,ℓ	NOUN
ejpam-6452	322	7	[	[	PUNCT
ejpam-6452	322	8	1	1	NUM
ejpam-6452	322	9	∆k	∆k	PROPN
ejpam-6452	322	10	(	(	PUNCT
ejpam-6452	322	11	σℓn	σℓn	PROPN
ejpam-6452	322	12	k	k	PROPN
ejpam-6452	323	1	−	−	PROPN
ejpam-6452	323	2	(	(	PUNCT
ejpam-6452	323	3	1	1	NUM
ejpam-6452	323	4	−	−	PROPN
ejpam-6452	323	5	k)ℓn	k)ℓn	PROPN
ejpam-6452	323	6	σℓn	σℓn	PROPN
ejpam-6452	323	7	k	k	PROPN
ejpam-6452	323	8	)	)	PUNCT
ejpam-6452	323	9	lk,ℓ	lk,ℓ	X
ejpam-6452	324	1	+	+	CCONJ
ejpam-6452	324	2	(	(	PUNCT
ejpam-6452	324	3	σℓn	σℓn	PROPN
ejpam-6452	324	4	k	k	PROPN
ejpam-6452	325	1	+	+	CCONJ
ejpam-6452	325	2	(	(	PUNCT
ejpam-6452	325	3	1	1	NUM
ejpam-6452	325	4	−	−	PROPN
ejpam-6452	325	5	k)ℓn	k)ℓn	PROPN
ejpam-6452	325	6	σℓn	σℓn	PROPN
ejpam-6452	325	7	k	k	PROPN
ejpam-6452	325	8	)	)	PUNCT
ejpam-6452	325	9	(	(	PUNCT
ejpam-6452	325	10	n	n	X
ejpam-6452	325	11	+	+	CCONJ
ejpam-6452	325	12	1)pk,ℓ	1)pk,ℓ	NUM
ejpam-6452	325	13	−	−	PROPN
ejpam-6452	325	14	2pk,ℓn+ℓ	2pk,ℓn+ℓ	NUM
ejpam-6452	325	15	]	]	PUNCT
ejpam-6452	325	16	.	.	PUNCT
ejpam-6452	326	1	using	use	VERB
ejpam-6452	326	2	(	(	PUNCT
ejpam-6452	326	3	i	i	NOUN
ejpam-6452	326	4	)	)	PUNCT
ejpam-6452	326	5	and	and	CCONJ
ejpam-6452	326	6	(	(	PUNCT
ejpam-6452	326	7	ii	ii	NOUN
ejpam-6452	326	8	)	)	PUNCT
ejpam-6452	326	9	of	of	ADP
ejpam-6452	326	10	corollary	corollary	ADJ
ejpam-6452	326	11	1	1	NUM
ejpam-6452	326	12	,	,	PUNCT
ejpam-6452	326	13	and	and	CCONJ
ejpam-6452	326	14	(	(	PUNCT
ejpam-6452	326	15	i	i	NOUN
ejpam-6452	326	16	)	)	PUNCT
ejpam-6452	326	17	of	of	ADP
ejpam-6452	326	18	theorem	theorem	NOUN
ejpam-6452	326	19	4	4	NUM
ejpam-6452	326	20	,	,	PUNCT
ejpam-6452	326	21	it	it	PRON
ejpam-6452	326	22	follows	follow	VERB
ejpam-6452	326	23	that	that	SCONJ
ejpam-6452	327	1	i	i	PRON
ejpam-6452	327	2	=	=	NOUN
ejpam-6452	327	3	1	1	NUM
ejpam-6452	327	4	npk,ℓ	npk,ℓ	NOUN
ejpam-6452	327	5	[	[	PUNCT
ejpam-6452	327	6	pk,ℓnlk,ℓ	pk,ℓnlk,ℓ	PROPN
ejpam-6452	327	7	+	+	CCONJ
ejpam-6452	327	8	(	(	PUNCT
ejpam-6452	327	9	n	n	PROPN
ejpam-6452	327	10	+	+	CCONJ
ejpam-6452	327	11	1)pk,ℓlk,ℓn	1)pk,ℓlk,ℓn	NUM
ejpam-6452	327	12	−	−	NOUN
ejpam-6452	327	13	2pk,ℓn+ℓ	2pk,ℓn+ℓ	NUM
ejpam-6452	327	14	]	]	X
ejpam-6452	327	15	=	=	SYM
ejpam-6452	327	16	1	1	NUM
ejpam-6452	327	17	npk,ℓ	npk,ℓ	NOUN
ejpam-6452	327	18	[	[	PUNCT
ejpam-6452	327	19	pk,ℓnlk,ℓ	pk,ℓnlk,ℓ	PROPN
ejpam-6452	327	20	+	+	CCONJ
ejpam-6452	327	21	pk,ℓlk,ℓn	pk,ℓlk,ℓn	PROPN
ejpam-6452	327	22	−	−	NOUN
ejpam-6452	327	23	2pk,ℓn+ℓ	2pk,ℓn+ℓ	NUM
ejpam-6452	327	24	]	]	X
ejpam-6452	328	1	+	+	CCONJ
ejpam-6452	328	2	lk,ℓn	lk,ℓn	X
ejpam-6452	328	3	=	=	SYM
ejpam-6452	328	4	lk,ℓn	lk,ℓn	PROPN
ejpam-6452	328	5	.	.	PUNCT
ejpam-6452	329	1	this	this	PRON
ejpam-6452	329	2	completes	complete	VERB
ejpam-6452	329	3	the	the	DET
ejpam-6452	329	4	proof	proof	NOUN
ejpam-6452	329	5	.	.	PUNCT
ejpam-6452	330	1	now	now	ADV
ejpam-6452	330	2	,	,	PUNCT
ejpam-6452	330	3	new	new	ADJ
ejpam-6452	330	4	integral	integral	ADJ
ejpam-6452	330	5	representations	representation	NOUN
ejpam-6452	330	6	for	for	ADP
ejpam-6452	330	7	the	the	DET
ejpam-6452	330	8	companion	companion	NOUN
ejpam-6452	330	9	generalized	generalize	VERB
ejpam-6452	330	10	pell	pell	NOUN
ejpam-6452	330	11	numbers	number	NOUN
ejpam-6452	330	12	associated	associate	VERB
ejpam-6452	330	13	with	with	ADP
ejpam-6452	330	14	the	the	DET
ejpam-6452	330	15	generalized	generalized	ADJ
ejpam-6452	330	16	pell	pell	NOUN
ejpam-6452	330	17	and	and	CCONJ
ejpam-6452	330	18	generalized	generalize	VERB
ejpam-6452	330	19	pell	pell	NOUN
ejpam-6452	330	20	-	-	PUNCT
ejpam-6452	330	21	lucas	lucas	NOUN
ejpam-6452	330	22	-	-	PUNCT
ejpam-6452	330	23	like	like	ADJ
ejpam-6452	330	24	numbers	number	NOUN
ejpam-6452	330	25	are	be	AUX
ejpam-6452	330	26	presented	present	VERB
ejpam-6452	330	27	as	as	SCONJ
ejpam-6452	330	28	follows	follow	VERB
ejpam-6452	330	29	:	:	PUNCT
ejpam-6452	330	30	theorem	theorem	NOUN
ejpam-6452	330	31	11	11	NUM
ejpam-6452	330	32	.	.	PUNCT
ejpam-6452	331	1	let	let	VERB
ejpam-6452	331	2	k	k	NOUN
ejpam-6452	331	3	,	,	PUNCT
ejpam-6452	331	4	ℓ	ℓ	PROPN
ejpam-6452	331	5	and	and	CCONJ
ejpam-6452	331	6	n	n	CCONJ
ejpam-6452	331	7	be	be	VERB
ejpam-6452	331	8	non	non	ADJ
ejpam-6452	331	9	-	-	ADJ
ejpam-6452	331	10	negative	negative	ADJ
ejpam-6452	331	11	integers	integer	NOUN
ejpam-6452	331	12	with	with	ADP
ejpam-6452	331	13	k	k	PROPN
ejpam-6452	331	14	≥	≥	NUM
ejpam-6452	331	15	2	2	NUM
ejpam-6452	331	16	and	and	CCONJ
ejpam-6452	331	17	∆k	∆k	PROPN
ejpam-6452	331	18	=	=	SYM
ejpam-6452	331	19	√	√	PROPN
ejpam-6452	331	20	k2	k2	NOUN
ejpam-6452	332	1	+	+	CCONJ
ejpam-6452	332	2	4k	4k	NOUN
ejpam-6452	332	3	−	−	NOUN
ejpam-6452	332	4	4	4	NUM
ejpam-6452	332	5	.	.	PUNCT
ejpam-6452	333	1	the	the	DET
ejpam-6452	333	2	companion	companion	NOUN
ejpam-6452	333	3	generalized	generalize	VERB
ejpam-6452	333	4	pell	pell	NOUN
ejpam-6452	333	5	numbers	number	NOUN
ejpam-6452	333	6	gpk,ℓn	gpk,ℓn	X
ejpam-6452	333	7	are	be	AUX
ejpam-6452	333	8	represented	represent	VERB
ejpam-6452	333	9	by	by	ADP
ejpam-6452	333	10	gpk,ℓn	gpk,ℓn	PROPN
ejpam-6452	333	11	=	=	SYM
ejpam-6452	333	12	1	1	NUM
ejpam-6452	333	13	2n+1∆k	2n+1∆k	NUM
ejpam-6452	333	14	∫	∫	NOUN
ejpam-6452	333	15	∆k	∆k	PROPN
ejpam-6452	333	16	−∆k	−∆k	PUNCT
ejpam-6452	334	1	(	(	PUNCT
ejpam-6452	334	2	alk,ℓ	alk,ℓ	PROPN
ejpam-6452	334	3	+	+	CCONJ
ejpam-6452	334	4	(	(	PUNCT
ejpam-6452	334	5	2b−	2b−	NUM
ejpam-6452	334	6	ak)npk,ℓ	ak)npk,ℓ	NOUN
ejpam-6452	334	7	+	+	CCONJ
ejpam-6452	334	8	a(n	a(n	PROPN
ejpam-6452	334	9	+	+	NUM
ejpam-6452	334	10	1)pk,ℓx	1)pk,ℓx	NUM
ejpam-6452	334	11	)	)	PUNCT
ejpam-6452	334	12	(	(	PUNCT
ejpam-6452	334	13	lk,ℓ	lk,ℓ	X
ejpam-6452	334	14	+	+	CCONJ
ejpam-6452	334	15	pk,ℓx)n−1	pk,ℓx)n−1	PROPN
ejpam-6452	334	16	dx	dx	PROPN
ejpam-6452	334	17	.	.	PUNCT
ejpam-6452	335	1	proof	proof	NOUN
ejpam-6452	335	2	.	.	PUNCT
ejpam-6452	336	1	from	from	ADP
ejpam-6452	336	2	theorem	theorem	ADJ
ejpam-6452	336	3	3	3	NUM
ejpam-6452	336	4	,	,	PUNCT
ejpam-6452	336	5	we	we	PRON
ejpam-6452	336	6	obtain	obtain	VERB
ejpam-6452	336	7	gpk,ℓn	gpk,ℓn	X
ejpam-6452	336	8	=	=	PUNCT
ejpam-6452	336	9	a	a	DET
ejpam-6452	336	10	2	2	NUM
ejpam-6452	336	11	lk,ℓn	lk,ℓn	NOUN
ejpam-6452	336	12	+	+	CCONJ
ejpam-6452	336	13	2b−	2b−	NUM
ejpam-6452	336	14	ak	ak	PROPN
ejpam-6452	336	15	2	2	NUM
ejpam-6452	336	16	pk,ℓn	pk,ℓn	NOUN
ejpam-6452	336	17	.	.	PUNCT
ejpam-6452	337	1	(	(	PUNCT
ejpam-6452	337	2	14	14	NUM
ejpam-6452	337	3	)	)	PUNCT
ejpam-6452	337	4	applying	apply	VERB
ejpam-6452	337	5	the	the	DET
ejpam-6452	337	6	integral	integral	ADJ
ejpam-6452	337	7	representations	representation	NOUN
ejpam-6452	337	8	of	of	ADP
ejpam-6452	337	9	pk,ℓn	pk,ℓn	NOUN
ejpam-6452	337	10	and	and	CCONJ
ejpam-6452	337	11	lk,ℓn	lk,ℓn	NOUN
ejpam-6452	337	12	from	from	ADP
ejpam-6452	337	13	theorems	theorem	NOUN
ejpam-6452	337	14	8	8	NUM
ejpam-6452	337	15	and	and	CCONJ
ejpam-6452	337	16	10	10	NUM
ejpam-6452	337	17	to	to	PART
ejpam-6452	337	18	(	(	PUNCT
ejpam-6452	337	19	14	14	NUM
ejpam-6452	337	20	)	)	PUNCT
ejpam-6452	337	21	,	,	PUNCT
ejpam-6452	337	22	this	this	PRON
ejpam-6452	337	23	completes	complete	VERB
ejpam-6452	337	24	the	the	DET
ejpam-6452	337	25	proof	proof	NOUN
ejpam-6452	337	26	.	.	PUNCT
ejpam-6452	338	1	remark	remark	VERB
ejpam-6452	338	2	3	3	NUM
ejpam-6452	338	3	.	.	PUNCT
ejpam-6452	339	1	as	as	ADP
ejpam-6452	339	2	in	in	ADP
ejpam-6452	339	3	theorems	theorem	NOUN
ejpam-6452	339	4	3	3	NUM
ejpam-6452	339	5	and	and	CCONJ
ejpam-6452	339	6	11	11	NUM
ejpam-6452	339	7	,	,	PUNCT
ejpam-6452	339	8	we	we	PRON
ejpam-6452	339	9	have	have	VERB
ejpam-6452	339	10	the	the	DET
ejpam-6452	339	11	following	follow	VERB
ejpam-6452	339	12	results	result	NOUN
ejpam-6452	339	13	.	.	PUNCT
ejpam-6452	340	1	w.	w.	NOUN
ejpam-6452	340	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	340	3	,	,	PUNCT
ejpam-6452	340	4	a.	a.	NOUN
ejpam-6452	340	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	340	6	/	/	SYM
ejpam-6452	340	7	eur	eur	PROPN
ejpam-6452	340	8	.	.	PUNCT
ejpam-6452	341	1	j.	j.	PROPN
ejpam-6452	341	2	pure	pure	PROPN
ejpam-6452	341	3	appl	appl	PROPN
ejpam-6452	341	4	.	.	PROPN
ejpam-6452	341	5	math	math	PROPN
ejpam-6452	341	6	,	,	PUNCT
ejpam-6452	341	7	18	18	NUM
ejpam-6452	341	8	(	(	PUNCT
ejpam-6452	341	9	3	3	NUM
ejpam-6452	341	10	)	)	PUNCT
ejpam-6452	341	11	(	(	PUNCT
ejpam-6452	341	12	2025	2025	NUM
ejpam-6452	341	13	)	)	PUNCT
ejpam-6452	341	14	,	,	PUNCT
ejpam-6452	341	15	6452	6452	NUM
ejpam-6452	341	16	15	15	NUM
ejpam-6452	341	17	of	of	ADP
ejpam-6452	341	18	18	18	NUM
ejpam-6452	341	19	(	(	PUNCT
ejpam-6452	341	20	i	i	NOUN
ejpam-6452	341	21	)	)	PUNCT
ejpam-6452	341	22	if	if	SCONJ
ejpam-6452	341	23	a	a	PRON
ejpam-6452	341	24	=	=	SYM
ejpam-6452	341	25	0	0	NUM
ejpam-6452	341	26	,	,	PUNCT
ejpam-6452	341	27	then	then	ADV
ejpam-6452	341	28	gpk	gpk	PROPN
ejpam-6452	341	29	,	,	PUNCT
ejpam-6452	341	30	n	n	PROPN
ejpam-6452	341	31	=	=	SYM
ejpam-6452	341	32	bpk	bpk	PROPN
ejpam-6452	341	33	,	,	PUNCT
ejpam-6452	341	34	n	n	NOUN
ejpam-6452	341	35	and	and	CCONJ
ejpam-6452	341	36	gpk,ℓn	gpk,ℓn	PROPN
ejpam-6452	341	37	=	=	SYM
ejpam-6452	341	38	bnpk,ℓ	bnpk,ℓ	NUM
ejpam-6452	341	39	2n	2n	NUM
ejpam-6452	341	40	∫	∫	PROPN
ejpam-6452	341	41	∆k	∆k	PROPN
ejpam-6452	341	42	−∆k	−∆k	PUNCT
ejpam-6452	341	43	(	(	PUNCT
ejpam-6452	341	44	lk,ℓ	lk,ℓ	X
ejpam-6452	341	45	+	+	X
ejpam-6452	341	46	pk,ℓx)n−1dx	pk,ℓx)n−1dx	X
ejpam-6452	341	47	.	.	PUNCT
ejpam-6452	342	1	(	(	PUNCT
ejpam-6452	342	2	ii	ii	NOUN
ejpam-6452	342	3	)	)	PUNCT
ejpam-6452	342	4	if	if	SCONJ
ejpam-6452	342	5	ak	ak	PROPN
ejpam-6452	342	6	=	=	PROPN
ejpam-6452	342	7	2b	2b	NOUN
ejpam-6452	342	8	,	,	PUNCT
ejpam-6452	342	9	then	then	ADV
ejpam-6452	342	10	gpk	gpk	PROPN
ejpam-6452	342	11	,	,	PUNCT
ejpam-6452	342	12	n	n	NOUN
ejpam-6452	342	13	=	=	PUNCT
ejpam-6452	342	14	a	a	DET
ejpam-6452	342	15	2lk	2lk	ADJ
ejpam-6452	342	16	,	,	PUNCT
ejpam-6452	342	17	n	n	NOUN
ejpam-6452	342	18	and	and	CCONJ
ejpam-6452	342	19	gpk,ℓn	gpk,ℓn	PROPN
ejpam-6452	342	20	=	=	PUNCT
ejpam-6452	342	21	a	a	DET
ejpam-6452	342	22	2n+1	2n+1	PROPN
ejpam-6452	342	23	∫	∫	PROPN
ejpam-6452	342	24	∆k	∆k	PROPN
ejpam-6452	342	25	−∆k	−∆k	PUNCT
ejpam-6452	342	26	(	(	PUNCT
ejpam-6452	342	27	lk,ℓ	lk,ℓ	X
ejpam-6452	342	28	+	+	CCONJ
ejpam-6452	342	29	(	(	PUNCT
ejpam-6452	342	30	n	n	CCONJ
ejpam-6452	342	31	+	+	CCONJ
ejpam-6452	342	32	1)pk,ℓx)(lk,ℓ	1)pk,ℓx)(lk,ℓ	NUM
ejpam-6452	342	33	+	+	CCONJ
ejpam-6452	342	34	(	(	PUNCT
ejpam-6452	342	35	k	k	X
ejpam-6452	342	36	+	+	PROPN
ejpam-6452	342	37	1)pk,ℓx)n−1dx	1)pk,ℓx)n−1dx	NUM
ejpam-6452	342	38	.	.	PUNCT
ejpam-6452	343	1	setting	set	VERB
ejpam-6452	343	2	(	(	PUNCT
ejpam-6452	343	3	a	a	DET
ejpam-6452	343	4	,	,	PUNCT
ejpam-6452	343	5	b	b	NOUN
ejpam-6452	343	6	)	)	PUNCT
ejpam-6452	343	7	=	=	SYM
ejpam-6452	343	8	(	(	PUNCT
ejpam-6452	343	9	2	2	NUM
ejpam-6452	343	10	,	,	PUNCT
ejpam-6452	343	11	k	k	NOUN
ejpam-6452	343	12	)	)	PUNCT
ejpam-6452	343	13	in	in	ADP
ejpam-6452	343	14	theorem	theorem	ADJ
ejpam-6452	343	15	11	11	NUM
ejpam-6452	343	16	and	and	CCONJ
ejpam-6452	343	17	using	use	VERB
ejpam-6452	343	18	(	(	PUNCT
ejpam-6452	343	19	i	i	NOUN
ejpam-6452	343	20	)	)	PUNCT
ejpam-6452	343	21	of	of	ADP
ejpam-6452	343	22	corollary	corollary	ADJ
ejpam-6452	343	23	3	3	NUM
ejpam-6452	343	24	,	,	PUNCT
ejpam-6452	343	25	we	we	PRON
ejpam-6452	343	26	have	have	VERB
ejpam-6452	343	27	the	the	DET
ejpam-6452	343	28	following	follow	VERB
ejpam-6452	343	29	corollary	corollary	NOUN
ejpam-6452	343	30	.	.	PUNCT
ejpam-6452	344	1	corollary	corollary	ADJ
ejpam-6452	344	2	15	15	NUM
ejpam-6452	344	3	(	(	PUNCT
ejpam-6452	344	4	[	[	X
ejpam-6452	344	5	11	11	NUM
ejpam-6452	344	6	,	,	PUNCT
ejpam-6452	344	7	theorem	theorem	VERB
ejpam-6452	344	8	2.6	2.6	NUM
ejpam-6452	344	9	]	]	PUNCT
ejpam-6452	344	10	)	)	PUNCT
ejpam-6452	344	11	.	.	PUNCT
ejpam-6452	345	1	let	let	VERB
ejpam-6452	345	2	k	k	NOUN
ejpam-6452	345	3	,	,	PUNCT
ejpam-6452	345	4	ℓ	ℓ	PROPN
ejpam-6452	345	5	and	and	CCONJ
ejpam-6452	345	6	n	n	CCONJ
ejpam-6452	345	7	be	be	VERB
ejpam-6452	345	8	non	non	ADJ
ejpam-6452	345	9	-	-	ADJ
ejpam-6452	345	10	negative	negative	ADJ
ejpam-6452	345	11	integers	integer	NOUN
ejpam-6452	345	12	with	with	ADP
ejpam-6452	345	13	k	k	PROPN
ejpam-6452	345	14	≥	≥	NUM
ejpam-6452	345	15	2	2	NUM
ejpam-6452	345	16	and	and	CCONJ
ejpam-6452	345	17	∆k	∆k	PROPN
ejpam-6452	345	18	=	=	SYM
ejpam-6452	345	19	√	√	PROPN
ejpam-6452	345	20	k2	k2	NOUN
ejpam-6452	345	21	+	+	CCONJ
ejpam-6452	345	22	4k	4k	NOUN
ejpam-6452	345	23	−	−	NOUN
ejpam-6452	346	1	4	4	X
ejpam-6452	346	2	.	.	PUNCT
ejpam-6452	346	3	then	then	ADV
ejpam-6452	346	4	qk,ℓn	qk,ℓn	NOUN
ejpam-6452	346	5	=	=	PUNCT
ejpam-6452	346	6	1	1	NUM
ejpam-6452	346	7	2n∆k	2n∆k	NUM
ejpam-6452	346	8	∫	∫	NOUN
ejpam-6452	346	9	∆k	∆k	PROPN
ejpam-6452	346	10	−∆k	−∆k	X
ejpam-6452	346	11	(	(	PUNCT
ejpam-6452	346	12	qk,ℓ	qk,ℓ	X
ejpam-6452	346	13	+	+	CCONJ
ejpam-6452	346	14	(	(	PUNCT
ejpam-6452	346	15	k	k	NOUN
ejpam-6452	346	16	−	−	PROPN
ejpam-6452	346	17	2	2	NUM
ejpam-6452	346	18	+	+	NUM
ejpam-6452	346	19	x−	x−	PROPN
ejpam-6452	346	20	n(k	n(k	PROPN
ejpam-6452	346	21	−	−	PROPN
ejpam-6452	346	22	2x))pk,ℓ)(qk,ℓ	2x))pk,ℓ)(qk,ℓ	NUM
ejpam-6452	346	23	+	+	CCONJ
ejpam-6452	346	24	(	(	PUNCT
ejpam-6452	346	25	k	k	NOUN
ejpam-6452	346	26	−	−	PROPN
ejpam-6452	346	27	2	2	NUM
ejpam-6452	346	28	+	+	CCONJ
ejpam-6452	346	29	x)pk,ℓ	x)pk,ℓ	NOUN
ejpam-6452	346	30	)	)	PUNCT
ejpam-6452	346	31	n−1dx	n−1dx	NOUN
ejpam-6452	346	32	.	.	PUNCT
ejpam-6452	347	1	setting	set	VERB
ejpam-6452	347	2	k	k	X
ejpam-6452	347	3	=	=	SYM
ejpam-6452	347	4	2	2	NUM
ejpam-6452	347	5	in	in	ADP
ejpam-6452	347	6	theorem	theorem	NOUN
ejpam-6452	347	7	11	11	NUM
ejpam-6452	347	8	,	,	PUNCT
ejpam-6452	347	9	we	we	PRON
ejpam-6452	347	10	have	have	VERB
ejpam-6452	347	11	the	the	DET
ejpam-6452	347	12	following	follow	VERB
ejpam-6452	347	13	corollary	corollary	NOUN
ejpam-6452	347	14	.	.	PUNCT
ejpam-6452	348	1	corollary	corollary	ADJ
ejpam-6452	348	2	16	16	NUM
ejpam-6452	348	3	.	.	PUNCT
ejpam-6452	349	1	let	let	VERB
ejpam-6452	349	2	ℓ	ℓ	NOUN
ejpam-6452	349	3	and	and	CCONJ
ejpam-6452	349	4	n	n	CCONJ
ejpam-6452	349	5	be	be	VERB
ejpam-6452	349	6	non	non	ADJ
ejpam-6452	349	7	-	-	ADJ
ejpam-6452	349	8	negative	negative	ADJ
ejpam-6452	349	9	integers	integer	NOUN
ejpam-6452	349	10	.	.	PUNCT
ejpam-6452	350	1	then	then	ADV
ejpam-6452	350	2	ha	ha	INTJ
ejpam-6452	350	3	,	,	PUNCT
ejpam-6452	350	4	b	b	NOUN
ejpam-6452	350	5	ℓn	ℓn	ADV
ejpam-6452	350	6	=	=	SYM
ejpam-6452	350	7	1	1	NUM
ejpam-6452	350	8	2n+2	2n+2	NUM
ejpam-6452	350	9	√	√	ADV
ejpam-6452	350	10	2	2	NUM
ejpam-6452	350	11	∫	∫	NOUN
ejpam-6452	350	12	2	2	NUM
ejpam-6452	350	13	√	√	NUM
ejpam-6452	350	14	2	2	NUM
ejpam-6452	350	15	−2	−2	NOUN
ejpam-6452	350	16	√	√	NOUN
ejpam-6452	350	17	2	2	NUM
ejpam-6452	350	18	(	(	PUNCT
ejpam-6452	350	19	aqℓ	aqℓ	X
ejpam-6452	350	20	+	+	CCONJ
ejpam-6452	350	21	2(b−	2(b−	NUM
ejpam-6452	350	22	a)npℓ	a)npℓ	NOUN
ejpam-6452	350	23	+	+	CCONJ
ejpam-6452	350	24	a(n	a(n	ADP
ejpam-6452	350	25	+	+	NOUN
ejpam-6452	350	26	1)pℓx	1)pℓx	NUM
ejpam-6452	350	27	)	)	PUNCT
ejpam-6452	350	28	(	(	PUNCT
ejpam-6452	350	29	qℓ	qℓ	X
ejpam-6452	350	30	+	+	X
ejpam-6452	350	31	pℓx)n−1	pℓx)n−1	PROPN
ejpam-6452	350	32	dx	dx	PROPN
ejpam-6452	350	33	.	.	PUNCT
ejpam-6452	351	1	setting	set	VERB
ejpam-6452	351	2	k	k	PROPN
ejpam-6452	351	3	=	=	SYM
ejpam-6452	351	4	2	2	NUM
ejpam-6452	351	5	in	in	ADP
ejpam-6452	351	6	theorem	theorem	ADJ
ejpam-6452	351	7	10	10	NUM
ejpam-6452	351	8	or	or	CCONJ
ejpam-6452	351	9	(	(	PUNCT
ejpam-6452	351	10	a	a	PRON
ejpam-6452	351	11	,	,	PUNCT
ejpam-6452	351	12	b	b	NOUN
ejpam-6452	351	13	)	)	PUNCT
ejpam-6452	351	14	=	=	SYM
ejpam-6452	351	15	(	(	PUNCT
ejpam-6452	351	16	2	2	NUM
ejpam-6452	351	17	,	,	PUNCT
ejpam-6452	351	18	2	2	NUM
ejpam-6452	351	19	)	)	PUNCT
ejpam-6452	351	20	in	in	ADP
ejpam-6452	351	21	corollary	corollary	ADJ
ejpam-6452	351	22	16	16	NUM
ejpam-6452	351	23	,	,	PUNCT
ejpam-6452	351	24	we	we	PRON
ejpam-6452	351	25	have	have	VERB
ejpam-6452	351	26	the	the	DET
ejpam-6452	351	27	following	follow	VERB
ejpam-6452	351	28	corollary	corollary	NOUN
ejpam-6452	351	29	.	.	PUNCT
ejpam-6452	352	1	corollary	corollary	ADJ
ejpam-6452	352	2	17	17	NUM
ejpam-6452	352	3	(	(	PUNCT
ejpam-6452	352	4	[	[	X
ejpam-6452	352	5	17	17	NUM
ejpam-6452	352	6	,	,	PUNCT
ejpam-6452	352	7	theorem	theorem	VERB
ejpam-6452	352	8	3.4	3.4	NUM
ejpam-6452	352	9	]	]	PUNCT
ejpam-6452	352	10	)	)	PUNCT
ejpam-6452	352	11	.	.	PUNCT
ejpam-6452	353	1	let	let	VERB
ejpam-6452	353	2	ℓ	ℓ	NOUN
ejpam-6452	353	3	and	and	CCONJ
ejpam-6452	353	4	n	n	CCONJ
ejpam-6452	353	5	be	be	VERB
ejpam-6452	353	6	non	non	ADJ
ejpam-6452	353	7	-	-	ADJ
ejpam-6452	353	8	negative	negative	ADJ
ejpam-6452	353	9	integers	integer	NOUN
ejpam-6452	353	10	.	.	PUNCT
ejpam-6452	354	1	then	then	ADV
ejpam-6452	354	2	qℓn	qℓn	NOUN
ejpam-6452	354	3	=	=	SYM
ejpam-6452	354	4	1	1	NUM
ejpam-6452	354	5	2n+1	2n+1	NOUN
ejpam-6452	354	6	√	√	NUM
ejpam-6452	354	7	2	2	NUM
ejpam-6452	354	8	∫	∫	NOUN
ejpam-6452	354	9	2	2	NUM
ejpam-6452	354	10	√	√	NUM
ejpam-6452	354	11	2	2	NUM
ejpam-6452	354	12	−2	−2	NOUN
ejpam-6452	354	13	√	√	NOUN
ejpam-6452	354	14	2	2	NUM
ejpam-6452	354	15	(	(	PUNCT
ejpam-6452	354	16	qℓ	qℓ	PROPN
ejpam-6452	354	17	+	+	X
ejpam-6452	354	18	(	(	PUNCT
ejpam-6452	354	19	n	n	X
ejpam-6452	354	20	+	+	NOUN
ejpam-6452	354	21	1)pℓx	1)pℓx	NUM
ejpam-6452	354	22	)	)	PUNCT
ejpam-6452	354	23	(	(	PUNCT
ejpam-6452	354	24	qℓ	qℓ	X
ejpam-6452	354	25	+	+	X
ejpam-6452	354	26	pℓx)n−1	pℓx)n−1	PROPN
ejpam-6452	354	27	dx	dx	PROPN
ejpam-6452	354	28	.	.	PUNCT
ejpam-6452	355	1	finally	finally	ADV
ejpam-6452	355	2	,	,	PUNCT
ejpam-6452	355	3	both	both	CCONJ
ejpam-6452	355	4	pk,ℓn	pk,ℓn	NOUN
ejpam-6452	355	5	and	and	CCONJ
ejpam-6452	355	6	lk,ℓn	lk,ℓn	PROPN
ejpam-6452	355	7	are	be	AUX
ejpam-6452	355	8	then	then	ADV
ejpam-6452	355	9	used	use	VERB
ejpam-6452	355	10	to	to	PART
ejpam-6452	355	11	establish	establish	VERB
ejpam-6452	355	12	integral	integral	ADJ
ejpam-6452	355	13	representations	representation	NOUN
ejpam-6452	355	14	for	for	ADP
ejpam-6452	355	15	pk,ℓn+r	pk,ℓn+r	NOUN
ejpam-6452	355	16	and	and	CCONJ
ejpam-6452	355	17	lk,ℓn+r	lk,ℓn+r	NOUN
ejpam-6452	355	18	as	as	ADP
ejpam-6452	355	19	the	the	DET
ejpam-6452	355	20	following	follow	VERB
ejpam-6452	355	21	theorems	theorem	NOUN
ejpam-6452	355	22	.	.	PUNCT
ejpam-6452	356	1	theorem	theorem	NOUN
ejpam-6452	356	2	12	12	NUM
ejpam-6452	356	3	.	.	PUNCT
ejpam-6452	357	1	let	let	VERB
ejpam-6452	357	2	k	k	NOUN
ejpam-6452	357	3	,	,	PUNCT
ejpam-6452	357	4	ℓ	ℓ	PROPN
ejpam-6452	357	5	,	,	PUNCT
ejpam-6452	357	6	n	n	NOUN
ejpam-6452	357	7	and	and	CCONJ
ejpam-6452	357	8	r	r	AUX
ejpam-6452	357	9	be	be	VERB
ejpam-6452	357	10	non	non	ADJ
ejpam-6452	357	11	-	-	ADJ
ejpam-6452	357	12	negative	negative	ADJ
ejpam-6452	357	13	integers	integer	NOUN
ejpam-6452	357	14	with	with	ADP
ejpam-6452	357	15	k	k	PROPN
ejpam-6452	357	16	≥	≥	NUM
ejpam-6452	357	17	2	2	NUM
ejpam-6452	357	18	and	and	CCONJ
ejpam-6452	357	19	∆k	∆k	PROPN
ejpam-6452	357	20	=	=	SYM
ejpam-6452	357	21	√	√	PROPN
ejpam-6452	357	22	k2	k2	NOUN
ejpam-6452	358	1	+	+	CCONJ
ejpam-6452	358	2	4k	4k	NOUN
ejpam-6452	358	3	−	−	NOUN
ejpam-6452	359	1	4	4	X
ejpam-6452	359	2	.	.	PUNCT
ejpam-6452	360	1	then	then	ADV
ejpam-6452	360	2	pk,ℓn+r	pk,ℓn+r	NOUN
ejpam-6452	360	3	=	=	SYM
ejpam-6452	360	4	1	1	NUM
ejpam-6452	360	5	2n+1∆k	2n+1∆k	NUM
ejpam-6452	360	6	∫	∫	NOUN
ejpam-6452	360	7	∆k	∆k	PROPN
ejpam-6452	360	8	−∆k	−∆k	X
ejpam-6452	360	9	(	(	PUNCT
ejpam-6452	360	10	npk,ℓlk	npk,ℓlk	PROPN
ejpam-6452	360	11	,	,	PUNCT
ejpam-6452	360	12	r	r	NOUN
ejpam-6452	360	13	+	+	SYM
ejpam-6452	360	14	pk	pk	NOUN
ejpam-6452	360	15	,	,	PUNCT
ejpam-6452	360	16	rlk,ℓ	rlk,ℓ	NOUN
ejpam-6452	360	17	+	+	CCONJ
ejpam-6452	360	18	(	(	PUNCT
ejpam-6452	360	19	n	n	X
ejpam-6452	360	20	+	+	CCONJ
ejpam-6452	360	21	1)pk,ℓpk	1)pk,ℓpk	NUM
ejpam-6452	360	22	,	,	PUNCT
ejpam-6452	360	23	rx	rx	ADJ
ejpam-6452	360	24	)	)	PUNCT
ejpam-6452	360	25	(	(	PUNCT
ejpam-6452	360	26	lk,ℓ	lk,ℓ	X
ejpam-6452	360	27	+	+	X
ejpam-6452	360	28	pk,ℓx)n−1dx	pk,ℓx)n−1dx	X
ejpam-6452	360	29	.	.	PUNCT
ejpam-6452	361	1	proof	proof	NOUN
ejpam-6452	361	2	.	.	PUNCT
ejpam-6452	362	1	using	use	VERB
ejpam-6452	362	2	(	(	PUNCT
ejpam-6452	362	3	i	i	NOUN
ejpam-6452	362	4	)	)	PUNCT
ejpam-6452	362	5	of	of	ADP
ejpam-6452	362	6	theorem	theorem	NOUN
ejpam-6452	362	7	4	4	NUM
ejpam-6452	362	8	with	with	ADP
ejpam-6452	362	9	m	m	PROPN
ejpam-6452	362	10	and	and	CCONJ
ejpam-6452	362	11	n	n	PRON
ejpam-6452	362	12	replaced	replace	VERB
ejpam-6452	362	13	by	by	ADP
ejpam-6452	362	14	ℓn	ℓn	ADV
ejpam-6452	362	15	and	and	CCONJ
ejpam-6452	362	16	r	r	VERB
ejpam-6452	362	17	respectively	respectively	ADV
ejpam-6452	362	18	,	,	PUNCT
ejpam-6452	362	19	we	we	PRON
ejpam-6452	362	20	get	get	VERB
ejpam-6452	362	21	pk,ℓn+r	pk,ℓn+r	NOUN
ejpam-6452	362	22	=	=	SYM
ejpam-6452	362	23	1	1	NUM
ejpam-6452	362	24	2	2	NUM
ejpam-6452	362	25	pk,ℓnlk	pk,ℓnlk	NOUN
ejpam-6452	362	26	,	,	PUNCT
ejpam-6452	362	27	r	r	NOUN
ejpam-6452	362	28	+	+	CCONJ
ejpam-6452	362	29	1	1	NUM
ejpam-6452	362	30	2	2	NUM
ejpam-6452	362	31	pk	pk	NOUN
ejpam-6452	362	32	,	,	PUNCT
ejpam-6452	362	33	rlk,ℓn	rlk,ℓn	PROPN
ejpam-6452	362	34	.	.	PUNCT
ejpam-6452	363	1	w.	w.	PROPN
ejpam-6452	363	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	363	3	,	,	PUNCT
ejpam-6452	363	4	a.	a.	NOUN
ejpam-6452	363	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	363	6	/	/	SYM
ejpam-6452	363	7	eur	eur	PROPN
ejpam-6452	363	8	.	.	PUNCT
ejpam-6452	364	1	j.	j.	PROPN
ejpam-6452	364	2	pure	pure	PROPN
ejpam-6452	364	3	appl	appl	PROPN
ejpam-6452	364	4	.	.	PROPN
ejpam-6452	364	5	math	math	PROPN
ejpam-6452	364	6	,	,	PUNCT
ejpam-6452	364	7	18	18	NUM
ejpam-6452	364	8	(	(	PUNCT
ejpam-6452	364	9	3	3	NUM
ejpam-6452	364	10	)	)	PUNCT
ejpam-6452	364	11	(	(	PUNCT
ejpam-6452	364	12	2025	2025	NUM
ejpam-6452	364	13	)	)	PUNCT
ejpam-6452	364	14	,	,	PUNCT
ejpam-6452	364	15	6452	6452	NUM
ejpam-6452	364	16	16	16	NUM
ejpam-6452	364	17	of	of	ADP
ejpam-6452	364	18	18	18	NUM
ejpam-6452	364	19	applying	apply	VERB
ejpam-6452	364	20	the	the	DET
ejpam-6452	364	21	integral	integral	ADJ
ejpam-6452	364	22	representations	representation	NOUN
ejpam-6452	364	23	of	of	ADP
ejpam-6452	364	24	pk,ℓn	pk,ℓn	NOUN
ejpam-6452	364	25	and	and	CCONJ
ejpam-6452	364	26	lk,ℓn	lk,ℓn	NOUN
ejpam-6452	364	27	from	from	ADP
ejpam-6452	364	28	theorems	theorem	NOUN
ejpam-6452	364	29	8	8	NUM
ejpam-6452	364	30	and	and	CCONJ
ejpam-6452	364	31	10	10	NUM
ejpam-6452	364	32	,	,	PUNCT
ejpam-6452	364	33	we	we	PRON
ejpam-6452	364	34	obtain	obtain	VERB
ejpam-6452	364	35	pk,ℓn+r	pk,ℓn+r	NOUN
ejpam-6452	364	36	=	=	SYM
ejpam-6452	364	37	1	1	NUM
ejpam-6452	364	38	2	2	NUM
ejpam-6452	364	39	(	(	PUNCT
ejpam-6452	364	40	npk,ℓ	npk,ℓ	PROPN
ejpam-6452	364	41	2n∆k	2n∆k	NUM
ejpam-6452	364	42	∫	∫	PROPN
ejpam-6452	364	43	∆k	∆k	PROPN
ejpam-6452	364	44	−∆k	−∆k	PUNCT
ejpam-6452	365	1	(	(	PUNCT
ejpam-6452	365	2	lk,ℓ	lk,ℓ	X
ejpam-6452	365	3	+	+	X
ejpam-6452	365	4	pk,ℓx)n−1dx	pk,ℓx)n−1dx	NOUN
ejpam-6452	365	5	)	)	PUNCT
ejpam-6452	365	6	lk	lk	PROPN
ejpam-6452	365	7	,	,	PUNCT
ejpam-6452	365	8	r	r	NOUN
ejpam-6452	365	9	+	+	CCONJ
ejpam-6452	365	10	1	1	NUM
ejpam-6452	365	11	2	2	NUM
ejpam-6452	365	12	pk	pk	NOUN
ejpam-6452	365	13	,	,	PUNCT
ejpam-6452	365	14	r	r	NOUN
ejpam-6452	365	15	(	(	PUNCT
ejpam-6452	365	16	1	1	NUM
ejpam-6452	365	17	2n∆k	2n∆k	NUM
ejpam-6452	365	18	∫	∫	NOUN
ejpam-6452	365	19	∆k	∆k	PROPN
ejpam-6452	365	20	−∆k	−∆k	PUNCT
ejpam-6452	365	21	(	(	PUNCT
ejpam-6452	365	22	lk,ℓ	lk,ℓ	X
ejpam-6452	365	23	+	+	CCONJ
ejpam-6452	365	24	(	(	PUNCT
ejpam-6452	365	25	n	n	CCONJ
ejpam-6452	365	26	+	+	CCONJ
ejpam-6452	365	27	1)pk,ℓx)(lk,ℓ	1)pk,ℓx)(lk,ℓ	NUM
ejpam-6452	365	28	+	+	CCONJ
ejpam-6452	365	29	pk,ℓx)n−1dx	pk,ℓx)n−1dx	NOUN
ejpam-6452	365	30	)	)	PUNCT
ejpam-6452	365	31	=	=	SYM
ejpam-6452	366	1	1	1	NUM
ejpam-6452	366	2	2n+1∆k	2n+1∆k	NUM
ejpam-6452	366	3	∫	∫	NOUN
ejpam-6452	366	4	∆k	∆k	PROPN
ejpam-6452	366	5	−∆k	−∆k	X
ejpam-6452	366	6	(	(	PUNCT
ejpam-6452	366	7	npk,ℓlk	npk,ℓlk	PROPN
ejpam-6452	366	8	,	,	PUNCT
ejpam-6452	366	9	r	r	NOUN
ejpam-6452	366	10	+	+	SYM
ejpam-6452	366	11	pk	pk	NOUN
ejpam-6452	366	12	,	,	PUNCT
ejpam-6452	366	13	rlk,ℓ	rlk,ℓ	NOUN
ejpam-6452	366	14	+	+	CCONJ
ejpam-6452	366	15	(	(	PUNCT
ejpam-6452	366	16	n	n	X
ejpam-6452	366	17	+	+	CCONJ
ejpam-6452	366	18	1)pk,ℓpk	1)pk,ℓpk	NUM
ejpam-6452	366	19	,	,	PUNCT
ejpam-6452	366	20	rx	rx	ADJ
ejpam-6452	366	21	)	)	PUNCT
ejpam-6452	366	22	(	(	PUNCT
ejpam-6452	366	23	lk,ℓ	lk,ℓ	X
ejpam-6452	366	24	+	+	X
ejpam-6452	366	25	pk,ℓx)n−1dx	pk,ℓx)n−1dx	X
ejpam-6452	366	26	.	.	PUNCT
ejpam-6452	367	1	this	this	PRON
ejpam-6452	367	2	completes	complete	VERB
ejpam-6452	367	3	the	the	DET
ejpam-6452	367	4	proof	proof	NOUN
ejpam-6452	367	5	.	.	PUNCT
ejpam-6452	368	1	setting	set	VERB
ejpam-6452	368	2	k	k	X
ejpam-6452	368	3	=	=	SYM
ejpam-6452	368	4	2	2	NUM
ejpam-6452	368	5	in	in	ADP
ejpam-6452	368	6	theorem	theorem	NOUN
ejpam-6452	368	7	12	12	NUM
ejpam-6452	368	8	,	,	PUNCT
ejpam-6452	368	9	we	we	PRON
ejpam-6452	368	10	have	have	VERB
ejpam-6452	368	11	the	the	DET
ejpam-6452	368	12	following	follow	VERB
ejpam-6452	368	13	corollary	corollary	NOUN
ejpam-6452	368	14	.	.	PUNCT
ejpam-6452	369	1	corollary	corollary	ADJ
ejpam-6452	369	2	18	18	NUM
ejpam-6452	369	3	(	(	PUNCT
ejpam-6452	369	4	[	[	X
ejpam-6452	369	5	17	17	NUM
ejpam-6452	369	6	]	]	PUNCT
ejpam-6452	369	7	,	,	PUNCT
ejpam-6452	369	8	theorem	theorem	VERB
ejpam-6452	369	9	3.5	3.5	NUM
ejpam-6452	369	10	)	)	PUNCT
ejpam-6452	369	11	.	.	PUNCT
ejpam-6452	370	1	let	let	VERB
ejpam-6452	370	2	ℓ	ℓ	NOUN
ejpam-6452	370	3	,	,	PUNCT
ejpam-6452	370	4	n	n	PROPN
ejpam-6452	370	5	and	and	CCONJ
ejpam-6452	370	6	r	r	AUX
ejpam-6452	370	7	be	be	VERB
ejpam-6452	370	8	non	non	ADJ
ejpam-6452	370	9	-	-	ADJ
ejpam-6452	370	10	negative	negative	ADJ
ejpam-6452	370	11	integers	integer	NOUN
ejpam-6452	370	12	.	.	PUNCT
ejpam-6452	371	1	then	then	ADV
ejpam-6452	371	2	pℓn+r	pℓn+r	PROPN
ejpam-6452	371	3	=	=	SYM
ejpam-6452	371	4	1	1	NUM
ejpam-6452	371	5	2n+2	2n+2	NUM
ejpam-6452	371	6	√	√	ADV
ejpam-6452	371	7	2	2	NUM
ejpam-6452	371	8	∫	∫	NOUN
ejpam-6452	371	9	2	2	NUM
ejpam-6452	371	10	√	√	NUM
ejpam-6452	371	11	2	2	NUM
ejpam-6452	371	12	−2	−2	NOUN
ejpam-6452	371	13	√	√	NOUN
ejpam-6452	371	14	2	2	NUM
ejpam-6452	371	15	(	(	PUNCT
ejpam-6452	371	16	npℓqr	npℓqr	NOUN
ejpam-6452	371	17	+	+	NUM
ejpam-6452	371	18	prqℓ	prqℓ	NOUN
ejpam-6452	371	19	+	+	CCONJ
ejpam-6452	371	20	(	(	PUNCT
ejpam-6452	371	21	n	n	X
ejpam-6452	371	22	+	+	CCONJ
ejpam-6452	371	23	1)pℓprx	1)pℓprx	NUM
ejpam-6452	371	24	)	)	PUNCT
ejpam-6452	371	25	(	(	PUNCT
ejpam-6452	371	26	qℓ	qℓ	PROPN
ejpam-6452	371	27	+	+	CCONJ
ejpam-6452	371	28	pℓx)n−1dx	pℓx)n−1dx	PROPN
ejpam-6452	371	29	.	.	PUNCT
ejpam-6452	371	30	theorem	theorem	VERB
ejpam-6452	371	31	13	13	NUM
ejpam-6452	371	32	.	.	PUNCT
ejpam-6452	372	1	let	let	VERB
ejpam-6452	372	2	k	k	NOUN
ejpam-6452	372	3	,	,	PUNCT
ejpam-6452	372	4	ℓ	ℓ	PROPN
ejpam-6452	372	5	,	,	PUNCT
ejpam-6452	372	6	n	n	NOUN
ejpam-6452	372	7	and	and	CCONJ
ejpam-6452	372	8	r	r	AUX
ejpam-6452	372	9	be	be	VERB
ejpam-6452	372	10	non	non	ADJ
ejpam-6452	372	11	-	-	ADJ
ejpam-6452	372	12	negative	negative	ADJ
ejpam-6452	372	13	integers	integer	NOUN
ejpam-6452	372	14	with	with	ADP
ejpam-6452	372	15	k	k	PROPN
ejpam-6452	372	16	≥	≥	NUM
ejpam-6452	372	17	2	2	NUM
ejpam-6452	372	18	and	and	CCONJ
ejpam-6452	372	19	∆k	∆k	PROPN
ejpam-6452	372	20	=	=	SYM
ejpam-6452	372	21	√	√	PROPN
ejpam-6452	372	22	k2	k2	NOUN
ejpam-6452	373	1	+	+	CCONJ
ejpam-6452	373	2	4k	4k	NOUN
ejpam-6452	373	3	−	−	NOUN
ejpam-6452	374	1	4	4	X
ejpam-6452	374	2	.	.	PUNCT
ejpam-6452	374	3	then	then	ADV
ejpam-6452	374	4	lk,ℓn+r	lk,ℓn+r	PROPN
ejpam-6452	374	5	=	=	SYM
ejpam-6452	374	6	1	1	NUM
ejpam-6452	374	7	2n+1∆k	2n+1∆k	NUM
ejpam-6452	374	8	∫	∫	NOUN
ejpam-6452	374	9	∆k	∆k	PROPN
ejpam-6452	374	10	−∆k	−∆k	PUNCT
ejpam-6452	375	1	(	(	PUNCT
ejpam-6452	375	2	n∆2	n∆2	NOUN
ejpam-6452	375	3	kpk,ℓpk	kpk,ℓpk	NOUN
ejpam-6452	375	4	,	,	PUNCT
ejpam-6452	375	5	r	r	NOUN
ejpam-6452	375	6	+	+	PUNCT
ejpam-6452	375	7	lk,ℓlk	lk,ℓlk	ADJ
ejpam-6452	375	8	,	,	PUNCT
ejpam-6452	375	9	r	r	NOUN
ejpam-6452	375	10	+	+	CCONJ
ejpam-6452	375	11	(	(	PUNCT
ejpam-6452	375	12	n	n	X
ejpam-6452	375	13	+	+	NUM
ejpam-6452	375	14	1)pk,ℓlk	1)pk,ℓlk	NUM
ejpam-6452	375	15	,	,	PUNCT
ejpam-6452	375	16	rx	rx	ADJ
ejpam-6452	375	17	)	)	PUNCT
ejpam-6452	375	18	(	(	PUNCT
ejpam-6452	375	19	lk,ℓ	lk,ℓ	X
ejpam-6452	375	20	+	+	X
ejpam-6452	375	21	pk,ℓx)n−1dx	pk,ℓx)n−1dx	X
ejpam-6452	375	22	.	.	PUNCT
ejpam-6452	376	1	proof	proof	NOUN
ejpam-6452	376	2	.	.	PUNCT
ejpam-6452	377	1	using	use	VERB
ejpam-6452	377	2	(	(	PUNCT
ejpam-6452	377	3	ii	ii	NOUN
ejpam-6452	377	4	)	)	PUNCT
ejpam-6452	377	5	of	of	ADP
ejpam-6452	377	6	theorem	theorem	NOUN
ejpam-6452	377	7	4	4	NUM
ejpam-6452	377	8	with	with	ADP
ejpam-6452	377	9	m	m	PROPN
ejpam-6452	377	10	and	and	CCONJ
ejpam-6452	377	11	n	n	PRON
ejpam-6452	377	12	replaced	replace	VERB
ejpam-6452	377	13	by	by	ADP
ejpam-6452	377	14	ℓn	ℓn	ADV
ejpam-6452	377	15	and	and	CCONJ
ejpam-6452	377	16	r	r	VERB
ejpam-6452	377	17	respectively	respectively	ADV
ejpam-6452	377	18	,	,	PUNCT
ejpam-6452	377	19	we	we	PRON
ejpam-6452	377	20	get	get	VERB
ejpam-6452	377	21	lk,ℓn+r	lk,ℓn+r	X
ejpam-6452	377	22	=	=	SYM
ejpam-6452	377	23	1	1	NUM
ejpam-6452	377	24	2	2	NUM
ejpam-6452	377	25	lk,ℓnlk	lk,ℓnlk	PROPN
ejpam-6452	377	26	,	,	PUNCT
ejpam-6452	377	27	r	r	NOUN
ejpam-6452	377	28	+	+	PROPN
ejpam-6452	377	29	∆2	∆2	PROPN
ejpam-6452	377	30	k	k	PROPN
ejpam-6452	377	31	2	2	NUM
ejpam-6452	377	32	pk,ℓnpk	pk,ℓnpk	PROPN
ejpam-6452	377	33	,	,	PUNCT
ejpam-6452	377	34	r.	r.	VERB
ejpam-6452	377	35	this	this	PRON
ejpam-6452	377	36	together	together	ADV
ejpam-6452	377	37	with	with	ADP
ejpam-6452	377	38	theorems	theorem	NOUN
ejpam-6452	377	39	8	8	NUM
ejpam-6452	377	40	and	and	CCONJ
ejpam-6452	377	41	10	10	NUM
ejpam-6452	377	42	gives	give	VERB
ejpam-6452	377	43	that	that	SCONJ
ejpam-6452	377	44	the	the	DET
ejpam-6452	377	45	proof	proof	NOUN
ejpam-6452	377	46	is	be	AUX
ejpam-6452	377	47	finish	finish	ADJ
ejpam-6452	377	48	.	.	PUNCT
ejpam-6452	378	1	setting	set	VERB
ejpam-6452	378	2	k	k	X
ejpam-6452	378	3	=	=	SYM
ejpam-6452	378	4	2	2	NUM
ejpam-6452	378	5	in	in	ADP
ejpam-6452	378	6	theorem	theorem	NOUN
ejpam-6452	378	7	13	13	NUM
ejpam-6452	378	8	,	,	PUNCT
ejpam-6452	378	9	we	we	PRON
ejpam-6452	378	10	have	have	VERB
ejpam-6452	378	11	the	the	DET
ejpam-6452	378	12	following	follow	VERB
ejpam-6452	378	13	corollary	corollary	NOUN
ejpam-6452	378	14	.	.	PUNCT
ejpam-6452	379	1	corollary	corollary	ADJ
ejpam-6452	379	2	19	19	NUM
ejpam-6452	379	3	(	(	PUNCT
ejpam-6452	379	4	[	[	X
ejpam-6452	379	5	17	17	NUM
ejpam-6452	379	6	]	]	PUNCT
ejpam-6452	379	7	,	,	PUNCT
ejpam-6452	379	8	theorem	theorem	VERB
ejpam-6452	379	9	3.6	3.6	NUM
ejpam-6452	379	10	)	)	PUNCT
ejpam-6452	379	11	.	.	PUNCT
ejpam-6452	380	1	let	let	VERB
ejpam-6452	380	2	ℓ	ℓ	NOUN
ejpam-6452	380	3	,	,	PUNCT
ejpam-6452	380	4	n	n	PROPN
ejpam-6452	380	5	and	and	CCONJ
ejpam-6452	380	6	r	r	AUX
ejpam-6452	380	7	be	be	VERB
ejpam-6452	380	8	non	non	ADJ
ejpam-6452	380	9	-	-	ADJ
ejpam-6452	380	10	negative	negative	ADJ
ejpam-6452	380	11	integers	integer	NOUN
ejpam-6452	380	12	.	.	PUNCT
ejpam-6452	381	1	then	then	ADV
ejpam-6452	381	2	qℓn+r	qℓn+r	PROPN
ejpam-6452	381	3	=	=	SYM
ejpam-6452	381	4	1	1	NUM
ejpam-6452	381	5	2n+2	2n+2	NUM
ejpam-6452	381	6	√	√	ADV
ejpam-6452	381	7	2	2	NUM
ejpam-6452	381	8	∫	∫	NOUN
ejpam-6452	381	9	2	2	NUM
ejpam-6452	381	10	√	√	NUM
ejpam-6452	381	11	2	2	NUM
ejpam-6452	381	12	−2	−2	NOUN
ejpam-6452	381	13	√	√	ADV
ejpam-6452	381	14	2	2	NUM
ejpam-6452	381	15	(	(	PUNCT
ejpam-6452	381	16	8npℓpr	8npℓpr	NUM
ejpam-6452	381	17	+	+	NUM
ejpam-6452	381	18	qℓqr	qℓqr	ADV
ejpam-6452	381	19	+	+	CCONJ
ejpam-6452	381	20	(	(	PUNCT
ejpam-6452	381	21	n	n	X
ejpam-6452	381	22	+	+	CCONJ
ejpam-6452	381	23	1)pℓqrx	1)pℓqrx	NUM
ejpam-6452	381	24	)	)	PUNCT
ejpam-6452	381	25	(	(	PUNCT
ejpam-6452	381	26	qℓ	qℓ	PROPN
ejpam-6452	381	27	+	+	CCONJ
ejpam-6452	381	28	pℓx)n−1dx	pℓx)n−1dx	PROPN
ejpam-6452	381	29	.	.	PUNCT
ejpam-6452	381	30	remark	remark	PROPN
ejpam-6452	381	31	4	4	NUM
ejpam-6452	381	32	.	.	PUNCT
ejpam-6452	382	1	the	the	DET
ejpam-6452	382	2	integral	integral	ADJ
ejpam-6452	382	3	representations	representation	NOUN
ejpam-6452	382	4	for	for	ADP
ejpam-6452	382	5	the	the	DET
ejpam-6452	382	6	companion	companion	NOUN
ejpam-6452	382	7	generalized	generalize	VERB
ejpam-6452	382	8	pell	pell	NOUN
ejpam-6452	382	9	numbers	number	NOUN
ejpam-6452	382	10	gpk,ℓn+r	gpk,ℓn+r	PROPN
ejpam-6452	382	11	are	be	AUX
ejpam-6452	382	12	established	establish	VERB
ejpam-6452	382	13	by	by	ADP
ejpam-6452	382	14	applying	apply	VERB
ejpam-6452	382	15	theorems	theorem	NOUN
ejpam-6452	382	16	3	3	NUM
ejpam-6452	382	17	,	,	PUNCT
ejpam-6452	382	18	12	12	NUM
ejpam-6452	382	19	and	and	CCONJ
ejpam-6452	382	20	13	13	NUM
ejpam-6452	382	21	.	.	PUNCT
ejpam-6452	383	1	w.	w.	PROPN
ejpam-6452	383	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	383	3	,	,	PUNCT
ejpam-6452	383	4	a.	a.	NOUN
ejpam-6452	383	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	383	6	/	/	SYM
ejpam-6452	383	7	eur	eur	PROPN
ejpam-6452	383	8	.	.	PUNCT
ejpam-6452	384	1	j.	j.	PROPN
ejpam-6452	384	2	pure	pure	PROPN
ejpam-6452	384	3	appl	appl	PROPN
ejpam-6452	384	4	.	.	PROPN
ejpam-6452	384	5	math	math	PROPN
ejpam-6452	384	6	,	,	PUNCT
ejpam-6452	384	7	18	18	NUM
ejpam-6452	384	8	(	(	PUNCT
ejpam-6452	384	9	3	3	NUM
ejpam-6452	384	10	)	)	PUNCT
ejpam-6452	384	11	(	(	PUNCT
ejpam-6452	384	12	2025	2025	NUM
ejpam-6452	384	13	)	)	PUNCT
ejpam-6452	384	14	,	,	PUNCT
ejpam-6452	384	15	6452	6452	NUM
ejpam-6452	384	16	17	17	NUM
ejpam-6452	384	17	of	of	ADP
ejpam-6452	384	18	18	18	NUM
ejpam-6452	384	19	4	4	NUM
ejpam-6452	384	20	.	.	PUNCT
ejpam-6452	385	1	conclusions	conclusion	NOUN
ejpam-6452	385	2	this	this	DET
ejpam-6452	385	3	paper	paper	NOUN
ejpam-6452	385	4	presents	present	VERB
ejpam-6452	385	5	a	a	DET
ejpam-6452	385	6	comprehensive	comprehensive	ADJ
ejpam-6452	385	7	study	study	NOUN
ejpam-6452	385	8	on	on	ADP
ejpam-6452	385	9	one	one	NUM
ejpam-6452	385	10	-	-	PUNCT
ejpam-6452	385	11	parameter	parameter	NOUN
ejpam-6452	385	12	generalizations	generalization	NOUN
ejpam-6452	385	13	of	of	ADP
ejpam-6452	385	14	pell	pell	NOUN
ejpam-6452	385	15	numbers	number	NOUN
ejpam-6452	385	16	and	and	CCONJ
ejpam-6452	385	17	their	their	PRON
ejpam-6452	385	18	associated	associated	ADJ
ejpam-6452	385	19	sequences	sequence	NOUN
ejpam-6452	385	20	,	,	PUNCT
ejpam-6452	385	21	introducing	introduce	VERB
ejpam-6452	385	22	generalized	generalize	VERB
ejpam-6452	385	23	pell	pell	NOUN
ejpam-6452	385	24	-	-	PUNCT
ejpam-6452	385	25	lucas	lucas	NOUN
ejpam-6452	385	26	-	-	PUNCT
ejpam-6452	385	27	like	like	ADJ
ejpam-6452	385	28	numbers	number	NOUN
ejpam-6452	385	29	and	and	CCONJ
ejpam-6452	385	30	their	their	PRON
ejpam-6452	385	31	integral	integral	ADJ
ejpam-6452	385	32	representations	representation	NOUN
ejpam-6452	385	33	.	.	PUNCT
ejpam-6452	386	1	the	the	DET
ejpam-6452	386	2	paper	paper	NOUN
ejpam-6452	386	3	further	far	ADV
ejpam-6452	386	4	extends	extend	VERB
ejpam-6452	386	5	known	known	ADJ
ejpam-6452	386	6	identities	identity	NOUN
ejpam-6452	386	7	,	,	PUNCT
ejpam-6452	386	8	derives	derive	VERB
ejpam-6452	386	9	binet	binet	NOUN
ejpam-6452	386	10	-	-	PUNCT
ejpam-6452	386	11	type	type	NOUN
ejpam-6452	386	12	formulas	formula	NOUN
ejpam-6452	386	13	,	,	PUNCT
ejpam-6452	386	14	and	and	CCONJ
ejpam-6452	386	15	proposes	propose	VERB
ejpam-6452	386	16	several	several	ADJ
ejpam-6452	386	17	new	new	ADJ
ejpam-6452	386	18	integral	integral	ADJ
ejpam-6452	386	19	formulations	formulation	NOUN
ejpam-6452	386	20	that	that	PRON
ejpam-6452	386	21	encompass	encompass	VERB
ejpam-6452	386	22	and	and	CCONJ
ejpam-6452	386	23	generalize	generalize	VERB
ejpam-6452	386	24	classical	classical	ADJ
ejpam-6452	386	25	results	result	NOUN
ejpam-6452	386	26	.	.	PUNCT
ejpam-6452	387	1	our	our	PRON
ejpam-6452	387	2	results	result	NOUN
ejpam-6452	387	3	not	not	PART
ejpam-6452	387	4	only	only	ADV
ejpam-6452	387	5	generalize	generalize	VERB
ejpam-6452	387	6	the	the	DET
ejpam-6452	387	7	integral	integral	ADJ
ejpam-6452	387	8	representations	representation	NOUN
ejpam-6452	387	9	of	of	ADP
ejpam-6452	387	10	the	the	DET
ejpam-6452	387	11	pell	pell	NOUN
ejpam-6452	387	12	and	and	CCONJ
ejpam-6452	387	13	pell	pell	NOUN
ejpam-6452	387	14	-	-	PUNCT
ejpam-6452	387	15	lucas	lucas	NOUN
ejpam-6452	387	16	numbers	number	NOUN
ejpam-6452	387	17	but	but	CCONJ
ejpam-6452	387	18	also	also	ADV
ejpam-6452	387	19	apply	apply	VERB
ejpam-6452	387	20	to	to	ADP
ejpam-6452	387	21	all	all	DET
ejpam-6452	387	22	the	the	DET
ejpam-6452	387	23	companion	companion	NOUN
ejpam-6452	387	24	numbers	number	NOUN
ejpam-6452	387	25	of	of	ADP
ejpam-6452	387	26	generalized	generalized	ADJ
ejpam-6452	387	27	pell	pell	NOUN
ejpam-6452	387	28	numbers	number	NOUN
ejpam-6452	387	29	.	.	PUNCT
ejpam-6452	388	1	acknowledgements	acknowledgement	NOUN
ejpam-6452	388	2	we	we	PRON
ejpam-6452	388	3	would	would	AUX
ejpam-6452	388	4	like	like	VERB
ejpam-6452	388	5	to	to	PART
ejpam-6452	388	6	thank	thank	VERB
ejpam-6452	388	7	the	the	DET
ejpam-6452	388	8	referees	referee	NOUN
ejpam-6452	388	9	for	for	ADP
ejpam-6452	388	10	their	their	PRON
ejpam-6452	388	11	comments	comment	NOUN
ejpam-6452	388	12	and	and	CCONJ
ejpam-6452	388	13	suggestions	suggestion	NOUN
ejpam-6452	388	14	on	on	ADP
ejpam-6452	388	15	the	the	DET
ejpam-6452	388	16	manuscript	manuscript	NOUN
ejpam-6452	388	17	.	.	PUNCT
ejpam-6452	389	1	the	the	DET
ejpam-6452	389	2	first	first	ADJ
ejpam-6452	389	3	author	author	NOUN
ejpam-6452	389	4	is	be	AUX
ejpam-6452	389	5	supported	support	VERB
ejpam-6452	389	6	by	by	ADP
ejpam-6452	389	7	the	the	DET
ejpam-6452	389	8	department	department	NOUN
ejpam-6452	389	9	of	of	ADP
ejpam-6452	389	10	mathematics	mathematic	NOUN
ejpam-6452	389	11	,	,	PUNCT
ejpam-6452	389	12	statistics	statistic	NOUN
ejpam-6452	389	13	and	and	CCONJ
ejpam-6452	389	14	computer	computer	NOUN
ejpam-6452	389	15	,	,	PUNCT
ejpam-6452	389	16	faculty	faculty	NOUN
ejpam-6452	389	17	of	of	ADP
ejpam-6452	389	18	science	science	NOUN
ejpam-6452	389	19	,	,	PUNCT
ejpam-6452	389	20	ubon	ubon	PROPN
ejpam-6452	389	21	ratchatani	ratchatani	PROPN
ejpam-6452	389	22	university	university	PROPN
ejpam-6452	389	23	.	.	PUNCT
ejpam-6452	390	1	the	the	DET
ejpam-6452	390	2	second	second	ADJ
ejpam-6452	390	3	author	author	NOUN
ejpam-6452	390	4	is	be	AUX
ejpam-6452	390	5	supported	support	VERB
ejpam-6452	390	6	by	by	ADP
ejpam-6452	390	7	the	the	DET
ejpam-6452	390	8	department	department	NOUN
ejpam-6452	390	9	of	of	ADP
ejpam-6452	390	10	mathematics	mathematic	NOUN
ejpam-6452	390	11	,	,	PUNCT
ejpam-6452	390	12	faculty	faculty	NOUN
ejpam-6452	390	13	of	of	ADP
ejpam-6452	390	14	science	science	NOUN
ejpam-6452	390	15	,	,	PUNCT
ejpam-6452	390	16	ubon	ubon	PROPN
ejpam-6452	390	17	ratchatani	ratchatani	PROPN
ejpam-6452	390	18	rajabhat	rajabhat	PROPN
ejpam-6452	390	19	university	university	PROPN
ejpam-6452	390	20	.	.	PUNCT
ejpam-6452	391	1	references	reference	NOUN
ejpam-6452	391	2	[	[	X
ejpam-6452	391	3	1	1	NUM
ejpam-6452	391	4	]	]	PUNCT
ejpam-6452	391	5	sergio	sergio	PROPN
ejpam-6452	391	6	falcón	falcón	PROPN
ejpam-6452	391	7	and	and	CCONJ
ejpam-6452	391	8	ángel	ángel	PROPN
ejpam-6452	391	9	plaza	plaza	PROPN
ejpam-6452	391	10	.	.	PUNCT
ejpam-6452	392	1	on	on	ADP
ejpam-6452	392	2	the	the	DET
ejpam-6452	392	3	fibonacci	fibonacci	NOUN
ejpam-6452	392	4	k	k	NOUN
ejpam-6452	392	5	-	-	PUNCT
ejpam-6452	392	6	numbers	number	NOUN
ejpam-6452	392	7	.	.	PUNCT
ejpam-6452	393	1	chaos	chaos	NOUN
ejpam-6452	393	2	solitons	soliton	NOUN
ejpam-6452	393	3	&	&	CCONJ
ejpam-6452	393	4	fractals	fractal	NOUN
ejpam-6452	393	5	,	,	PUNCT
ejpam-6452	393	6	32:1615–1624	32:1615–1624	NUM
ejpam-6452	393	7	,	,	PUNCT
ejpam-6452	393	8	2007	2007	NUM
ejpam-6452	393	9	.	.	PUNCT
ejpam-6452	394	1	[	[	X
ejpam-6452	394	2	2	2	NUM
ejpam-6452	394	3	]	]	PUNCT
ejpam-6452	394	4	sergio	sergio	PROPN
ejpam-6452	394	5	falcón	falcón	PROPN
ejpam-6452	394	6	.	.	PUNCT
ejpam-6452	395	1	on	on	ADP
ejpam-6452	395	2	the	the	DET
ejpam-6452	395	3	k	k	PROPN
ejpam-6452	395	4	-	-	PUNCT
ejpam-6452	395	5	lucas	lucas	ADJ
ejpam-6452	395	6	numbers	number	NOUN
ejpam-6452	395	7	.	.	PUNCT
ejpam-6452	396	1	international	international	ADJ
ejpam-6452	396	2	journal	journal	PROPN
ejpam-6452	396	3	of	of	ADP
ejpam-6452	396	4	contemporary	contemporary	PROPN
ejpam-6452	396	5	mathematical	mathematical	PROPN
ejpam-6452	396	6	sciences	sciences	PROPN
ejpam-6452	396	7	,	,	PUNCT
ejpam-6452	396	8	6:1039–1050	6:1039–1050	NUM
ejpam-6452	396	9	,	,	PUNCT
ejpam-6452	396	10	2011	2011	NUM
ejpam-6452	396	11	.	.	PUNCT
ejpam-6452	397	1	[	[	X
ejpam-6452	397	2	3	3	X
ejpam-6452	397	3	]	]	X
ejpam-6452	397	4	paula	paula	PROPN
ejpam-6452	397	5	catarino	catarino	PROPN
ejpam-6452	397	6	.	.	PUNCT
ejpam-6452	398	1	on	on	ADP
ejpam-6452	398	2	some	some	DET
ejpam-6452	398	3	identities	identity	NOUN
ejpam-6452	398	4	and	and	CCONJ
ejpam-6452	398	5	generating	generating	NOUN
ejpam-6452	398	6	functions	function	NOUN
ejpam-6452	398	7	for	for	ADP
ejpam-6452	398	8	k	k	ADJ
ejpam-6452	398	9	-	-	PUNCT
ejpam-6452	398	10	pell	pell	ADJ
ejpam-6452	398	11	numbers	number	NOUN
ejpam-6452	398	12	.	.	PUNCT
ejpam-6452	399	1	international	international	ADJ
ejpam-6452	399	2	journal	journal	PROPN
ejpam-6452	399	3	of	of	ADP
ejpam-6452	399	4	mathematical	mathematical	ADJ
ejpam-6452	399	5	analysis	analysis	NOUN
ejpam-6452	399	6	,	,	PUNCT
ejpam-6452	399	7	7(37	7(37	NUM
ejpam-6452	399	8	-	-	SYM
ejpam-6452	399	9	40):1877–1884	40):1877–1884	PROPN
ejpam-6452	399	10	,	,	PUNCT
ejpam-6452	399	11	2013	2013	NUM
ejpam-6452	399	12	.	.	PUNCT
ejpam-6452	400	1	[	[	X
ejpam-6452	400	2	4	4	X
ejpam-6452	400	3	]	]	PUNCT
ejpam-6452	400	4	paula	paula	PROPN
ejpam-6452	400	5	catarino	catarino	PROPN
ejpam-6452	400	6	and	and	CCONJ
ejpam-6452	400	7	paulo	paulo	PROPN
ejpam-6452	400	8	vasco	vasco	PROPN
ejpam-6452	400	9	.	.	PUNCT
ejpam-6452	401	1	on	on	ADP
ejpam-6452	401	2	some	some	DET
ejpam-6452	401	3	identities	identity	NOUN
ejpam-6452	401	4	and	and	CCONJ
ejpam-6452	401	5	generating	generating	NOUN
ejpam-6452	401	6	functions	function	NOUN
ejpam-6452	401	7	for	for	ADP
ejpam-6452	401	8	k	k	NOUN
ejpam-6452	401	9	-	-	PUNCT
ejpam-6452	401	10	pell	pell	ADJ
ejpam-6452	401	11	-	-	PUNCT
ejpam-6452	401	12	lucas	lucas	NOUN
ejpam-6452	401	13	sequence	sequence	NOUN
ejpam-6452	401	14	.	.	PUNCT
ejpam-6452	402	1	applied	apply	VERB
ejpam-6452	402	2	mathematical	mathematical	ADJ
ejpam-6452	402	3	sciences	science	NOUN
ejpam-6452	402	4	,	,	PUNCT
ejpam-6452	402	5	7(97	7(97	NUM
ejpam-6452	402	6	-	-	SYM
ejpam-6452	402	7	100):4867–4873	100):4867–4873	NUM
ejpam-6452	402	8	,	,	PUNCT
ejpam-6452	402	9	2013	2013	NUM
ejpam-6452	402	10	.	.	PUNCT
ejpam-6452	403	1	[	[	X
ejpam-6452	403	2	5	5	NUM
ejpam-6452	403	3	]	]	PUNCT
ejpam-6452	403	4	lucyna	lucyna	PROPN
ejpam-6452	403	5	trojnar	trojnar	NOUN
ejpam-6452	403	6	-	-	PUNCT
ejpam-6452	403	7	spelina	spelina	NOUN
ejpam-6452	403	8	and	and	CCONJ
ejpam-6452	403	9	iwona	iwona	NOUN
ejpam-6452	403	10	w	w	PROPN
ejpam-6452	403	11	l	l	NOUN
ejpam-6452	403	12	och	och	PROPN
ejpam-6452	403	13	.	.	PUNCT
ejpam-6452	404	1	on	on	ADP
ejpam-6452	404	2	generalized	generalized	ADJ
ejpam-6452	404	3	pell	pell	NOUN
ejpam-6452	404	4	and	and	CCONJ
ejpam-6452	404	5	pell	pell	NOUN
ejpam-6452	404	6	-	-	PUNCT
ejpam-6452	404	7	lucas	lucas	NOUN
ejpam-6452	404	8	numbers	number	NOUN
ejpam-6452	404	9	.	.	PUNCT
ejpam-6452	405	1	iranian	iranian	ADJ
ejpam-6452	405	2	journal	journal	PROPN
ejpam-6452	405	3	of	of	ADP
ejpam-6452	405	4	science	science	NOUN
ejpam-6452	405	5	and	and	CCONJ
ejpam-6452	405	6	technology	technology	NOUN
ejpam-6452	405	7	.	.	PUNCT
ejpam-6452	406	1	transaction	transaction	NOUN
ejpam-6452	406	2	a.	a.	NOUN
ejpam-6452	406	3	science	science	PROPN
ejpam-6452	406	4	,	,	PUNCT
ejpam-6452	406	5	43(6):2871	43(6):2871	NUM
ejpam-6452	406	6	–	–	PUNCT
ejpam-6452	406	7	2877	2877	NUM
ejpam-6452	406	8	,	,	PUNCT
ejpam-6452	406	9	2019	2019	NUM
ejpam-6452	406	10	.	.	PUNCT
ejpam-6452	407	1	[	[	X
ejpam-6452	407	2	6	6	NUM
ejpam-6452	407	3	]	]	PUNCT
ejpam-6452	407	4	gül	gül	PROPN
ejpam-6452	407	5	özkan	özkan	PROPN
ejpam-6452	407	6	kızılırmak	kızılırmak	PROPN
ejpam-6452	407	7	.	.	PUNCT
ejpam-6452	408	1	on	on	ADP
ejpam-6452	408	2	some	some	DET
ejpam-6452	408	3	identities	identity	NOUN
ejpam-6452	408	4	and	and	CCONJ
ejpam-6452	408	5	hankel	hankel	NOUN
ejpam-6452	408	6	matrices	matrix	NOUN
ejpam-6452	408	7	norms	norm	NOUN
ejpam-6452	408	8	involving	involve	VERB
ejpam-6452	408	9	new	new	ADJ
ejpam-6452	408	10	defined	define	VERB
ejpam-6452	408	11	generalized	generalize	VERB
ejpam-6452	408	12	modified	modify	VERB
ejpam-6452	408	13	pell	pell	NOUN
ejpam-6452	408	14	numbers	number	NOUN
ejpam-6452	408	15	.	.	PUNCT
ejpam-6452	409	1	journal	journal	NOUN
ejpam-6452	409	2	of	of	ADP
ejpam-6452	409	3	new	new	ADJ
ejpam-6452	409	4	results	result	NOUN
ejpam-6452	409	5	in	in	ADP
ejpam-6452	409	6	science	science	NOUN
ejpam-6452	409	7	,	,	PUNCT
ejpam-6452	409	8	10(3):60–66	10(3):60–66	NUM
ejpam-6452	409	9	,	,	PUNCT
ejpam-6452	409	10	2021	2021	NUM
ejpam-6452	409	11	.	.	PUNCT
ejpam-6452	410	1	[	[	X
ejpam-6452	410	2	7	7	X
ejpam-6452	410	3	]	]	X
ejpam-6452	410	4	edouard	edouard	PROPN
ejpam-6452	410	5	lucas	lucas	PROPN
ejpam-6452	410	6	.	.	PUNCT
ejpam-6452	411	1	theorie	theorie	PROPN
ejpam-6452	411	2	des	des	PROPN
ejpam-6452	411	3	fonctions	fonctions	PROPN
ejpam-6452	411	4	numeriques	numerique	NOUN
ejpam-6452	411	5	simplement	simplement	PROPN
ejpam-6452	411	6	periodiques	periodiques	PROPN
ejpam-6452	411	7	.	.	PUNCT
ejpam-6452	412	1	american	american	PROPN
ejpam-6452	412	2	journal	journal	PROPN
ejpam-6452	412	3	of	of	ADP
ejpam-6452	412	4	mathematics	mathematic	NOUN
ejpam-6452	412	5	,	,	PUNCT
ejpam-6452	412	6	1(4):289–321	1(4):289–321	NUM
ejpam-6452	412	7	,	,	PUNCT
ejpam-6452	412	8	1878	1878	NUM
ejpam-6452	412	9	.	.	PUNCT
ejpam-6452	413	1	[	[	X
ejpam-6452	413	2	8	8	NUM
ejpam-6452	413	3	]	]	X
ejpam-6452	413	4	neil	neil	PROPN
ejpam-6452	413	5	j.	j.	PROPN
ejpam-6452	413	6	a.	a.	PROPN
ejpam-6452	413	7	sloane	sloane	PROPN
ejpam-6452	413	8	.	.	PUNCT
ejpam-6452	414	1	the	the	DET
ejpam-6452	414	2	on	on	ADP
ejpam-6452	414	3	-	-	PUNCT
ejpam-6452	414	4	line	line	NOUN
ejpam-6452	414	5	encyclopedia	encyclopedia	NOUN
ejpam-6452	414	6	of	of	ADP
ejpam-6452	414	7	integer	integer	NOUN
ejpam-6452	414	8	sequences	sequence	NOUN
ejpam-6452	414	9	.	.	PUNCT
ejpam-6452	415	1	the	the	DET
ejpam-6452	415	2	oeis	oeis	PROPN
ejpam-6452	415	3	foundation	foundation	PROPN
ejpam-6452	415	4	inc	inc	PROPN
ejpam-6452	415	5	.	.	PROPN
ejpam-6452	415	6	,	,	PUNCT
ejpam-6452	415	7	2024	2024	NUM
ejpam-6452	415	8	.	.	PUNCT
ejpam-6452	416	1	[	[	X
ejpam-6452	416	2	9	9	NUM
ejpam-6452	416	3	]	]	PUNCT
ejpam-6452	416	4	a.	a.	NOUN
ejpam-6452	416	5	f.	f.	PROPN
ejpam-6452	416	6	horadam	horadam	PROPN
ejpam-6452	416	7	.	.	PUNCT
ejpam-6452	417	1	basic	basic	ADJ
ejpam-6452	417	2	properties	property	NOUN
ejpam-6452	417	3	of	of	ADP
ejpam-6452	417	4	a	a	DET
ejpam-6452	417	5	certain	certain	ADJ
ejpam-6452	417	6	generalized	generalized	ADJ
ejpam-6452	417	7	sequence	sequence	NOUN
ejpam-6452	417	8	of	of	ADP
ejpam-6452	417	9	numbers	number	NOUN
ejpam-6452	417	10	.	.	PUNCT
ejpam-6452	418	1	the	the	DET
ejpam-6452	418	2	fibonacci	fibonacci	NOUN
ejpam-6452	418	3	quarterly	quarterly	ADV
ejpam-6452	418	4	,	,	PUNCT
ejpam-6452	418	5	3:161–176	3:161–176	PROPN
ejpam-6452	418	6	,	,	PUNCT
ejpam-6452	418	7	1965	1965	NUM
ejpam-6452	418	8	.	.	PUNCT
ejpam-6452	419	1	[	[	X
ejpam-6452	419	2	10	10	NUM
ejpam-6452	419	3	]	]	X
ejpam-6452	419	4	w.	w.	PROPN
ejpam-6452	419	5	m.	m.	PROPN
ejpam-6452	419	6	abd	abd	PROPN
ejpam-6452	419	7	-	-	PUNCT
ejpam-6452	419	8	elhameed	elhameed	PROPN
ejpam-6452	419	9	and	and	CCONJ
ejpam-6452	419	10	n.	n.	NOUN
ejpam-6452	419	11	a.	a.	PROPN
ejpam-6452	419	12	zeyada	zeyada	PROPN
ejpam-6452	419	13	.	.	PUNCT
ejpam-6452	420	1	a	a	DET
ejpam-6452	420	2	generalization	generalization	NOUN
ejpam-6452	420	3	of	of	ADP
ejpam-6452	420	4	generalized	generalized	ADJ
ejpam-6452	420	5	fibonacci	fibonacci	NOUN
ejpam-6452	420	6	and	and	CCONJ
ejpam-6452	420	7	generalized	generalized	ADJ
ejpam-6452	420	8	pell	pell	NOUN
ejpam-6452	420	9	numbers	number	NOUN
ejpam-6452	420	10	.	.	PUNCT
ejpam-6452	421	1	international	international	ADJ
ejpam-6452	421	2	journal	journal	PROPN
ejpam-6452	421	3	of	of	ADP
ejpam-6452	421	4	mathematical	mathematical	ADJ
ejpam-6452	421	5	education	education	NOUN
ejpam-6452	421	6	in	in	ADP
ejpam-6452	421	7	science	science	NOUN
ejpam-6452	421	8	and	and	CCONJ
ejpam-6452	421	9	technology	technology	NOUN
ejpam-6452	421	10	,	,	PUNCT
ejpam-6452	421	11	48(1):102–107	48(1):102–107	PROPN
ejpam-6452	421	12	,	,	PUNCT
ejpam-6452	421	13	2017	2017	NUM
ejpam-6452	421	14	.	.	PUNCT
ejpam-6452	422	1	w.	w.	PROPN
ejpam-6452	422	2	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	422	3	,	,	PUNCT
ejpam-6452	422	4	a.	a.	NOUN
ejpam-6452	422	5	nilsrakoo	nilsrakoo	PROPN
ejpam-6452	422	6	/	/	SYM
ejpam-6452	422	7	eur	eur	PROPN
ejpam-6452	422	8	.	.	PUNCT
ejpam-6452	423	1	j.	j.	PROPN
ejpam-6452	423	2	pure	pure	PROPN
ejpam-6452	423	3	appl	appl	PROPN
ejpam-6452	423	4	.	.	PROPN
ejpam-6452	423	5	math	math	PROPN
ejpam-6452	423	6	,	,	PUNCT
ejpam-6452	423	7	18	18	NUM
ejpam-6452	423	8	(	(	PUNCT
ejpam-6452	423	9	3	3	NUM
ejpam-6452	423	10	)	)	PUNCT
ejpam-6452	423	11	(	(	PUNCT
ejpam-6452	423	12	2025	2025	NUM
ejpam-6452	423	13	)	)	PUNCT
ejpam-6452	423	14	,	,	PUNCT
ejpam-6452	423	15	6452	6452	NUM
ejpam-6452	423	16	18	18	NUM
ejpam-6452	423	17	of	of	ADP
ejpam-6452	423	18	18	18	NUM
ejpam-6452	423	19	[	[	SYM
ejpam-6452	423	20	11	11	NUM
ejpam-6452	423	21	]	]	PUNCT
ejpam-6452	423	22	achariya	achariya	NOUN
ejpam-6452	423	23	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	423	24	and	and	CCONJ
ejpam-6452	423	25	weerayuth	weerayuth	NOUN
ejpam-6452	423	26	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	423	27	.	.	PUNCT
ejpam-6452	424	1	integral	integral	ADJ
ejpam-6452	424	2	aspects	aspect	NOUN
ejpam-6452	424	3	of	of	ADP
ejpam-6452	424	4	the	the	DET
ejpam-6452	424	5	generalized	generalized	ADJ
ejpam-6452	424	6	pell	pell	NOUN
ejpam-6452	424	7	and	and	CCONJ
ejpam-6452	424	8	pell	pell	NOUN
ejpam-6452	424	9	-	-	PUNCT
ejpam-6452	424	10	lucas	lucas	NOUN
ejpam-6452	424	11	numbers	number	NOUN
ejpam-6452	424	12	.	.	PUNCT
ejpam-6452	425	1	international	international	ADJ
ejpam-6452	425	2	journal	journal	NOUN
ejpam-6452	425	3	of	of	ADP
ejpam-6452	425	4	mathematics	mathematic	NOUN
ejpam-6452	425	5	and	and	CCONJ
ejpam-6452	425	6	computer	computer	NOUN
ejpam-6452	425	7	science	science	NOUN
ejpam-6452	425	8	,	,	PUNCT
ejpam-6452	425	9	20(2):469–473	20(2):469–473	PROPN
ejpam-6452	425	10	,	,	PUNCT
ejpam-6452	425	11	2025	2025	NUM
ejpam-6452	425	12	.	.	PUNCT
ejpam-6452	426	1	[	[	X
ejpam-6452	426	2	12	12	NUM
ejpam-6452	426	3	]	]	PUNCT
ejpam-6452	426	4	dorin	dorin	PROPN
ejpam-6452	426	5	andrica	andrica	PROPN
ejpam-6452	426	6	and	and	CCONJ
ejpam-6452	426	7	ovidiu	ovidiu	PROPN
ejpam-6452	426	8	bagdasar	bagdasar	PROPN
ejpam-6452	426	9	.	.	PUNCT
ejpam-6452	427	1	recurrent	recurrent	ADJ
ejpam-6452	427	2	sequences	sequence	NOUN
ejpam-6452	427	3	—	—	PUNCT
ejpam-6452	427	4	key	key	ADJ
ejpam-6452	427	5	results	result	NOUN
ejpam-6452	427	6	,	,	PUNCT
ejpam-6452	427	7	applications	application	NOUN
ejpam-6452	427	8	,	,	PUNCT
ejpam-6452	427	9	and	and	CCONJ
ejpam-6452	427	10	problems	problem	NOUN
ejpam-6452	427	11	.	.	PUNCT
ejpam-6452	428	1	problem	problem	NOUN
ejpam-6452	428	2	books	book	NOUN
ejpam-6452	428	3	in	in	ADP
ejpam-6452	428	4	mathematics	mathematic	NOUN
ejpam-6452	428	5	.	.	PUNCT
ejpam-6452	429	1	springer	springer	PROPN
ejpam-6452	429	2	,	,	PUNCT
ejpam-6452	429	3	cham	cham	PROPN
ejpam-6452	429	4	,	,	PUNCT
ejpam-6452	429	5	2020	2020	NUM
ejpam-6452	429	6	.	.	PUNCT
ejpam-6452	430	1	[	[	X
ejpam-6452	430	2	13	13	NUM
ejpam-6452	430	3	]	]	PUNCT
ejpam-6452	430	4	thierry	thierry	PROPN
ejpam-6452	430	5	dana	dana	PROPN
ejpam-6452	430	6	-	-	PUNCT
ejpam-6452	430	7	picard	picard	PROPN
ejpam-6452	430	8	.	.	PUNCT
ejpam-6452	431	1	integral	integral	ADJ
ejpam-6452	431	2	presentations	presentation	NOUN
ejpam-6452	431	3	of	of	ADP
ejpam-6452	431	4	catalan	catalan	NOUN
ejpam-6452	431	5	numbers	number	NOUN
ejpam-6452	431	6	.	.	PUNCT
ejpam-6452	432	1	internat	internat	PROPN
ejpam-6452	432	2	.	.	PUNCT
ejpam-6452	433	1	j.	j.	PROPN
ejpam-6452	433	2	math	math	PROPN
ejpam-6452	433	3	.	.	PUNCT
ejpam-6452	434	1	ed	ed	NOUN
ejpam-6452	434	2	.	.	PUNCT
ejpam-6452	435	1	sci	sci	PROPN
ejpam-6452	435	2	.	.	PUNCT
ejpam-6452	435	3	tech	tech	PROPN
ejpam-6452	435	4	.	.	PUNCT
ejpam-6452	435	5	,	,	PUNCT
ejpam-6452	435	6	41(1):63–69	41(1):63–69	NUM
ejpam-6452	435	7	,	,	PUNCT
ejpam-6452	435	8	2010	2010	NUM
ejpam-6452	435	9	.	.	PUNCT
ejpam-6452	436	1	[	[	X
ejpam-6452	436	2	14	14	NUM
ejpam-6452	436	3	]	]	X
ejpam-6452	436	4	karl	karl	PROPN
ejpam-6452	436	5	dilcher	dilcher	PROPN
ejpam-6452	436	6	.	.	PUNCT
ejpam-6452	437	1	hypergeometric	hypergeometric	ADJ
ejpam-6452	437	2	functions	function	NOUN
ejpam-6452	437	3	and	and	CCONJ
ejpam-6452	437	4	fibonacci	fibonacci	NOUN
ejpam-6452	437	5	numbers	number	NOUN
ejpam-6452	437	6	.	.	PUNCT
ejpam-6452	438	1	the	the	DET
ejpam-6452	438	2	fibonacci	fibonacci	NOUN
ejpam-6452	438	3	quarterly	quarterly	PROPN
ejpam-6452	438	4	,	,	PUNCT
ejpam-6452	438	5	38(4):342–363	38(4):342–363	PROPN
ejpam-6452	438	6	,	,	PUNCT
ejpam-6452	438	7	2000	2000	NUM
ejpam-6452	438	8	.	.	PUNCT
ejpam-6452	439	1	[	[	X
ejpam-6452	439	2	15	15	NUM
ejpam-6452	439	3	]	]	X
ejpam-6452	439	4	m.	m.	NOUN
ejpam-6452	439	5	lawrence	lawrence	PROPN
ejpam-6452	439	6	glasser	glasser	PROPN
ejpam-6452	439	7	and	and	CCONJ
ejpam-6452	439	8	yajun	yajun	PROPN
ejpam-6452	439	9	zhou	zhou	PROPN
ejpam-6452	439	10	.	.	PUNCT
ejpam-6452	440	1	an	an	DET
ejpam-6452	440	2	integral	integral	ADJ
ejpam-6452	440	3	representation	representation	NOUN
ejpam-6452	440	4	for	for	ADP
ejpam-6452	440	5	the	the	DET
ejpam-6452	440	6	fibonacci	fibonacci	NOUN
ejpam-6452	440	7	numbers	number	NOUN
ejpam-6452	440	8	and	and	CCONJ
ejpam-6452	440	9	their	their	PRON
ejpam-6452	440	10	generalization	generalization	NOUN
ejpam-6452	440	11	.	.	PUNCT
ejpam-6452	441	1	the	the	DET
ejpam-6452	441	2	fibonacci	fibonacci	NOUN
ejpam-6452	441	3	quarterly	quarterly	PROPN
ejpam-6452	441	4	,	,	PUNCT
ejpam-6452	441	5	53(4):313–318	53(4):313–318	PROPN
ejpam-6452	441	6	,	,	PUNCT
ejpam-6452	441	7	2015	2015	NUM
ejpam-6452	441	8	.	.	PUNCT
ejpam-6452	442	1	[	[	X
ejpam-6452	442	2	16	16	NUM
ejpam-6452	442	3	]	]	X
ejpam-6452	442	4	wen	wen	PROPN
ejpam-6452	442	5	-	-	PROPN
ejpam-6452	442	6	hui	hui	PROPN
ejpam-6452	442	7	li	li	PROPN
ejpam-6452	442	8	,	,	PUNCT
ejpam-6452	442	9	omran	omran	PROPN
ejpam-6452	442	10	kouba	kouba	PROPN
ejpam-6452	442	11	,	,	PUNCT
ejpam-6452	442	12	issam	issam	PROPN
ejpam-6452	442	13	kaddoura	kaddoura	PROPN
ejpam-6452	442	14	,	,	PUNCT
ejpam-6452	442	15	and	and	CCONJ
ejpam-6452	442	16	feng	feng	PROPN
ejpam-6452	442	17	qi	qi	PROPN
ejpam-6452	442	18	.	.	PUNCT
ejpam-6452	443	1	a	a	DET
ejpam-6452	443	2	further	further	ADJ
ejpam-6452	443	3	generalization	generalization	NOUN
ejpam-6452	443	4	of	of	ADP
ejpam-6452	443	5	the	the	DET
ejpam-6452	443	6	catalan	catalan	NOUN
ejpam-6452	443	7	numbers	number	NOUN
ejpam-6452	443	8	and	and	CCONJ
ejpam-6452	443	9	its	its	PRON
ejpam-6452	443	10	explicit	explicit	ADJ
ejpam-6452	443	11	formula	formula	NOUN
ejpam-6452	443	12	and	and	CCONJ
ejpam-6452	443	13	integral	integral	ADJ
ejpam-6452	443	14	representation	representation	NOUN
ejpam-6452	443	15	.	.	PUNCT
ejpam-6452	444	1	filomat	filomat	NOUN
ejpam-6452	444	2	,	,	PUNCT
ejpam-6452	444	3	37(19):6505–6524	37(19):6505–6524	NUM
ejpam-6452	444	4	,	,	PUNCT
ejpam-6452	444	5	2023	2023	NUM
ejpam-6452	444	6	.	.	PUNCT
ejpam-6452	445	1	[	[	X
ejpam-6452	445	2	17	17	NUM
ejpam-6452	445	3	]	]	PUNCT
ejpam-6452	445	4	achariya	achariya	NOUN
ejpam-6452	445	5	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	445	6	.	.	PUNCT
ejpam-6452	446	1	integral	integral	ADJ
ejpam-6452	446	2	representations	representation	NOUN
ejpam-6452	446	3	of	of	ADP
ejpam-6452	446	4	the	the	DET
ejpam-6452	446	5	pell	pell	NOUN
ejpam-6452	446	6	and	and	CCONJ
ejpam-6452	446	7	pell	pell	NOUN
ejpam-6452	446	8	-	-	PUNCT
ejpam-6452	446	9	lucas	lucas	NOUN
ejpam-6452	446	10	numbers	number	NOUN
ejpam-6452	446	11	.	.	PUNCT
ejpam-6452	447	1	journal	journal	PROPN
ejpam-6452	447	2	of	of	ADP
ejpam-6452	447	3	science	science	NOUN
ejpam-6452	447	4	and	and	CCONJ
ejpam-6452	447	5	science	science	NOUN
ejpam-6452	447	6	education	education	NOUN
ejpam-6452	447	7	,	,	PUNCT
ejpam-6452	447	8	7(2):272–281	7(2):272–281	NUM
ejpam-6452	447	9	,	,	PUNCT
ejpam-6452	447	10	2024	2024	NUM
ejpam-6452	447	11	.	.	PUNCT
ejpam-6452	448	1	[	[	X
ejpam-6452	448	2	18	18	NUM
ejpam-6452	448	3	]	]	PUNCT
ejpam-6452	448	4	achariya	achariya	NOUN
ejpam-6452	448	5	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	448	6	and	and	CCONJ
ejpam-6452	448	7	weerayuth	weerayuth	NOUN
ejpam-6452	448	8	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	448	9	.	.	PUNCT
ejpam-6452	449	1	on	on	ADP
ejpam-6452	449	2	the	the	DET
ejpam-6452	449	3	integral	integral	ADJ
ejpam-6452	449	4	representations	representation	NOUN
ejpam-6452	449	5	of	of	ADP
ejpam-6452	449	6	the	the	DET
ejpam-6452	449	7	k	k	NOUN
ejpam-6452	449	8	-	-	PUNCT
ejpam-6452	449	9	pell	pell	NOUN
ejpam-6452	449	10	and	and	CCONJ
ejpam-6452	449	11	k	k	NOUN
ejpam-6452	449	12	-	-	PUNCT
ejpam-6452	449	13	pell	pell	ADJ
ejpam-6452	449	14	-	-	PUNCT
ejpam-6452	449	15	lucas	lucas	NOUN
ejpam-6452	449	16	numbers	number	NOUN
ejpam-6452	449	17	.	.	PUNCT
ejpam-6452	450	1	thai	thai	PROPN
ejpam-6452	450	2	journal	journal	PROPN
ejpam-6452	450	3	of	of	ADP
ejpam-6452	450	4	mathematics	mathematic	NOUN
ejpam-6452	450	5	,	,	PUNCT
ejpam-6452	450	6	23(1):9–19	23(1):9–19	NUM
ejpam-6452	450	7	,	,	PUNCT
ejpam-6452	450	8	2025	2025	NUM
ejpam-6452	450	9	.	.	PUNCT
ejpam-6452	451	1	[	[	X
ejpam-6452	451	2	19	19	NUM
ejpam-6452	451	3	]	]	X
ejpam-6452	451	4	weerayuth	weerayuth	NOUN
ejpam-6452	451	5	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	451	6	and	and	CCONJ
ejpam-6452	451	7	achariya	achariya	NOUN
ejpam-6452	451	8	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	451	9	.	.	PUNCT
ejpam-6452	452	1	on	on	ADP
ejpam-6452	452	2	the	the	DET
ejpam-6452	452	3	integral	integral	ADJ
ejpam-6452	452	4	representations	representation	NOUN
ejpam-6452	452	5	of	of	ADP
ejpam-6452	452	6	the	the	DET
ejpam-6452	452	7	k	k	NOUN
ejpam-6452	452	8	-	-	NOUN
ejpam-6452	452	9	fibonacci	fibonacci	PROPN
ejpam-6452	452	10	and	and	CCONJ
ejpam-6452	452	11	k	k	ADJ
ejpam-6452	452	12	-	-	PUNCT
ejpam-6452	452	13	lucas	lucas	NOUN
ejpam-6452	452	14	numbers	number	NOUN
ejpam-6452	452	15	.	.	PUNCT
ejpam-6452	453	1	wseas	wseas	NOUN
ejpam-6452	453	2	transactions	transaction	NOUN
ejpam-6452	453	3	on	on	ADP
ejpam-6452	453	4	mathematics	mathematic	NOUN
ejpam-6452	453	5	,	,	PUNCT
ejpam-6452	453	6	23:791	23:791	NUM
ejpam-6452	453	7	–	–	PUNCT
ejpam-6452	453	8	801	801	NUM
ejpam-6452	453	9	,	,	PUNCT
ejpam-6452	453	10	2024	2024	NUM
ejpam-6452	453	11	.	.	PUNCT
ejpam-6452	454	1	[	[	X
ejpam-6452	454	2	20	20	NUM
ejpam-6452	454	3	]	]	PUNCT
ejpam-6452	454	4	weerayuth	weerayuth	NOUN
ejpam-6452	454	5	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	454	6	and	and	CCONJ
ejpam-6452	454	7	achariya	achariya	NOUN
ejpam-6452	454	8	nilsrakoo	nilsrakoo	NOUN
ejpam-6452	454	9	.	.	PUNCT
ejpam-6452	455	1	on	on	ADP
ejpam-6452	455	2	one	one	NUM
ejpam-6452	455	3	-	-	PUNCT
ejpam-6452	455	4	parameter	parameter	NOUN
ejpam-6452	455	5	generalization	generalization	NOUN
ejpam-6452	455	6	of	of	ADP
ejpam-6452	455	7	jacobsthal	jacobsthal	ADJ
ejpam-6452	455	8	numbers	number	NOUN
ejpam-6452	455	9	.	.	PUNCT
ejpam-6452	456	1	wseas	wseas	NOUN
ejpam-6452	456	2	transactions	transaction	NOUN
ejpam-6452	456	3	on	on	ADP
ejpam-6452	456	4	mathematics	mathematic	NOUN
ejpam-6452	456	5	,	,	PUNCT
ejpam-6452	456	6	24:51–61	24:51–61	NUM
ejpam-6452	456	7	,	,	PUNCT
ejpam-6452	456	8	2025	2025	NUM
ejpam-6452	456	9	.	.	PUNCT
ejpam-6452	457	1	[	[	X
ejpam-6452	457	2	21	21	NUM
ejpam-6452	457	3	]	]	PUNCT
ejpam-6452	457	4	seán	seán	PROPN
ejpam-6452	457	5	m.	m.	PROPN
ejpam-6452	457	6	stewart	stewart	PROPN
ejpam-6452	457	7	.	.	PUNCT
ejpam-6452	458	1	simple	simple	ADJ
ejpam-6452	458	2	integral	integral	ADJ
ejpam-6452	458	3	representations	representation	NOUN
ejpam-6452	458	4	for	for	ADP
ejpam-6452	458	5	the	the	DET
ejpam-6452	458	6	fibonacci	fibonacci	NOUN
ejpam-6452	458	7	and	and	CCONJ
ejpam-6452	458	8	lucas	lucas	PROPN
ejpam-6452	458	9	numbers	number	NOUN
ejpam-6452	458	10	.	.	PUNCT
ejpam-6452	459	1	australian	australian	ADJ
ejpam-6452	459	2	journal	journal	NOUN
ejpam-6452	459	3	of	of	ADP
ejpam-6452	459	4	mathematical	mathematical	ADJ
ejpam-6452	459	5	analysis	analysis	NOUN
ejpam-6452	459	6	and	and	CCONJ
ejpam-6452	459	7	applications	application	NOUN
ejpam-6452	459	8	,	,	PUNCT
ejpam-6452	459	9	19(2):art	19(2):art	NUM
ejpam-6452	459	10	.	.	PUNCT
ejpam-6452	460	1	2	2	NUM
ejpam-6452	460	2	,	,	PUNCT
ejpam-6452	460	3	5	5	NUM
ejpam-6452	460	4	pp	pp	NOUN
ejpam-6452	460	5	.	.	PUNCT
ejpam-6452	460	6	,	,	PUNCT
ejpam-6452	460	7	2022	2022	NUM
ejpam-6452	460	8	.	.	PUNCT
ejpam-6452	461	1	[	[	X
ejpam-6452	461	2	22	22	NUM
ejpam-6452	461	3	]	]	PUNCT
ejpam-6452	461	4	seán	seán	PROPN
ejpam-6452	461	5	m.	m.	PROPN
ejpam-6452	461	6	stewart	stewart	PROPN
ejpam-6452	461	7	.	.	PUNCT
ejpam-6452	462	1	107.01	107.01	NUM
ejpam-6452	462	2	a	a	DET
ejpam-6452	462	3	simple	simple	ADJ
ejpam-6452	462	4	integral	integral	ADJ
ejpam-6452	462	5	representation	representation	NOUN
ejpam-6452	462	6	of	of	ADP
ejpam-6452	462	7	the	the	DET
ejpam-6452	462	8	fibonacci	fibonacci	NOUN
ejpam-6452	462	9	numbers	number	NOUN
ejpam-6452	462	10	.	.	PUNCT
ejpam-6452	463	1	the	the	DET
ejpam-6452	463	2	mathematical	mathematical	ADJ
ejpam-6452	463	3	gazette	gazette	NOUN
ejpam-6452	463	4	,	,	PUNCT
ejpam-6452	463	5	107(568):120–123	107(568):120–123	NUM
ejpam-6452	463	6	,	,	PUNCT
ejpam-6452	463	7	2023	2023	NUM
ejpam-6452	463	8	.	.	PUNCT