id	sid	tid	token	lemma	pos
cana-1037	1	1	communications	communication	NOUN
cana-1037	1	2	on	on	ADP
cana-1037	1	3	applied	apply	VERB
cana-1037	1	4	nonlinear	nonlinear	ADJ
cana-1037	1	5	analysis	analysis	NOUN
cana-1037	1	6	issn	issn	NOUN
cana-1037	1	7	:	:	PUNCT
cana-1037	1	8	1074	1074	NUM
cana-1037	1	9	-	-	PUNCT
cana-1037	1	10	133x	133x	NUM
cana-1037	1	11	vol	vol	NOUN
cana-1037	1	12	31	31	NUM
cana-1037	1	13	no	no	NOUN
cana-1037	1	14	.	.	PUNCT
cana-1037	2	1	5s	5s	NUM
cana-1037	2	2	(	(	PUNCT
cana-1037	2	3	2024	2024	NUM
cana-1037	2	4	)	)	PUNCT
cana-1037	2	5	289	289	NUM
cana-1037	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1037	2	7	numerous	numerous	ADJ
cana-1037	2	8	determinants	determinant	NOUN
cana-1037	2	9	identities	identity	NOUN
cana-1037	2	10	involving	involve	VERB
cana-1037	2	11	jacobsthal	jacobsthal	ADJ
cana-1037	2	12	and	and	CCONJ
cana-1037	2	13	jacobsthal	jacobsthal	ADJ
cana-1037	2	14	lucas	lucas	PROPN
cana-1037	2	15	numbers	number	NOUN
cana-1037	2	16	t.ragunathan1	t.ragunathan1	PROPN
cana-1037	2	17	,	,	PUNCT
cana-1037	2	18	dr	dr	PROPN
cana-1037	2	19	.	.	PROPN
cana-1037	2	20	shweta	shweta	PROPN
cana-1037	2	21	choudhary2	choudhary2	PROPN
cana-1037	2	22	,	,	PUNCT
cana-1037	2	23	g.	g.	PROPN
cana-1037	2	24	venkat	venkat	PROPN
cana-1037	2	25	narayanan3	narayanan3	PROPN
cana-1037	2	26	,	,	PUNCT
cana-1037	2	27	s	s	PART
cana-1037	2	28	jagadeesh4	jagadeesh4	NOUN
cana-1037	2	29	1department	1department	NUM
cana-1037	2	30	of	of	ADP
cana-1037	2	31	basic	basic	ADJ
cana-1037	2	32	sciences	science	NOUN
cana-1037	2	33	,	,	PUNCT
cana-1037	2	34	srm	srm	PROPN
cana-1037	2	35	college	college	PROPN
cana-1037	2	36	of	of	ADP
cana-1037	2	37	agricultural	agricultural	ADJ
cana-1037	2	38	sciences	science	NOUN
cana-1037	2	39	,	,	PUNCT
cana-1037	2	40	srm	srm	PROPN
cana-1037	2	41	institute	institute	PROPN
cana-1037	2	42	of	of	ADP
cana-1037	2	43	science	science	NOUN
cana-1037	2	44	and	and	CCONJ
cana-1037	2	45	technology	technology	NOUN
cana-1037	2	46	,	,	PUNCT
cana-1037	2	47	baburayanpettai	baburayanpettai	PROPN
cana-1037	2	48	–	–	PUNCT
cana-1037	2	49	603201	603201	NUM
cana-1037	2	50	,	,	PUNCT
cana-1037	2	51	chengalpattu	chengalpattu	ADJ
cana-1037	2	52	dist	dist	NOUN
cana-1037	2	53	,	,	PUNCT
cana-1037	2	54	tamilnadu	tamilnadu	NOUN
cana-1037	2	55	,	,	PUNCT
cana-1037	2	56	india	india	PROPN
cana-1037	2	57	.	.	PUNCT
cana-1037	3	1	mail	mail	NOUN
cana-1037	4	1	i	i	PROPN
cana-1037	4	2	d	d	PROPN
cana-1037	4	3	:	:	PUNCT
cana-1037	5	1	math_ragu@rediffmail.com	math_ragu@rediffmail.com	X
cana-1037	5	2	.	.	PUNCT
cana-1037	6	1	orcid	orcid	PROPN
cana-1037	7	1	i	i	PRON
cana-1037	7	2	d	d	NOUN
cana-1037	7	3	:	:	PUNCT
cana-1037	7	4	https://orcid.org/0009-0004-4214-9686	https://orcid.org/0009-0004-4214-9686	PROPN
cana-1037	7	5	2associate	2associate	NUM
cana-1037	7	6	professor	professor	NOUN
cana-1037	7	7	,	,	PUNCT
cana-1037	7	8	as&h	as&h	PROPN
cana-1037	7	9	department	department	NOUN
cana-1037	7	10	,	,	PUNCT
cana-1037	7	11	abes	abes	PROPN
cana-1037	7	12	engineering	engineering	NOUN
cana-1037	7	13	college	college	PROPN
cana-1037	7	14	,	,	PUNCT
cana-1037	7	15	ghaziabad	ghaziabad	PROPN
cana-1037	7	16	,	,	PUNCT
cana-1037	7	17	shweta.chaudhary@abes.ac.in	shweta.chaudhary@abes.ac.in	NUM
cana-1037	7	18	3st.joseph	3st.joseph	PROPN
cana-1037	7	19	's	's	PART
cana-1037	7	20	college	college	NOUN
cana-1037	7	21	of	of	ADP
cana-1037	7	22	engineering	engineering	PROPN
cana-1037	7	23	,	,	PUNCT
cana-1037	7	24	chennai	chennai	NOUN
cana-1037	7	25	119	119	NUM
cana-1037	7	26	,	,	PUNCT
cana-1037	7	27	email	email	NOUN
cana-1037	7	28	i	i	PROPN
cana-1037	7	29	d	d	PROPN
cana-1037	7	30	:	:	PUNCT
cana-1037	7	31	gvenkatnarayanan@gmail.com	gvenkatnarayanan@gmail.com	PROPN
cana-1037	8	1	4associate	4associate	NUM
cana-1037	8	2	professor	professor	NOUN
cana-1037	8	3	,	,	PUNCT
cana-1037	8	4	department	department	NOUN
cana-1037	8	5	of	of	ADP
cana-1037	8	6	electronics	electronics	PROPN
cana-1037	8	7	&	&	CCONJ
cana-1037	8	8	communication	communication	PROPN
cana-1037	8	9	engineering	engineering	NOUN
cana-1037	8	10	,	,	PUNCT
cana-1037	8	11	aditya	aditya	PROPN
cana-1037	8	12	engineering	engineering	PROPN
cana-1037	8	13	college	college	PROPN
cana-1037	8	14	,	,	PUNCT
cana-1037	8	15	surampalem	surampalem	NOUN
cana-1037	8	16	,	,	PUNCT
cana-1037	8	17	india	india	PROPN
cana-1037	8	18	,	,	PUNCT
cana-1037	8	19	samudrala.jagadeesh@aec.edu.in	samudrala.jagadeesh@aec.edu.in	PROPN
cana-1037	8	20	article	article	NOUN
cana-1037	8	21	history	history	NOUN
cana-1037	8	22	:	:	PUNCT
cana-1037	8	23	received	receive	VERB
cana-1037	8	24	:	:	PUNCT
cana-1037	8	25	14	14	NUM
cana-1037	8	26	-	-	SYM
cana-1037	8	27	05	05	NUM
cana-1037	8	28	-	-	PUNCT
cana-1037	8	29	2024	2024	NUM
cana-1037	8	30	revised	revise	VERB
cana-1037	8	31	:	:	PUNCT
cana-1037	8	32	24	24	NUM
cana-1037	8	33	-	-	PUNCT
cana-1037	8	34	06	06	NUM
cana-1037	8	35	-	-	PUNCT
cana-1037	8	36	2024	2024	NUM
cana-1037	8	37	accepted	accept	VERB
cana-1037	8	38	:	:	PUNCT
cana-1037	8	39	08	08	NUM
cana-1037	8	40	-	-	SYM
cana-1037	8	41	07	07	NUM
cana-1037	8	42	-	-	PUNCT
cana-1037	8	43	2024	2024	NUM
cana-1037	8	44	abstract	abstract	NOUN
cana-1037	8	45	:	:	PUNCT
cana-1037	8	46	determinants	determinant	NOUN
cana-1037	8	47	have	have	AUX
cana-1037	8	48	played	play	VERB
cana-1037	8	49	an	an	DET
cana-1037	8	50	important	important	ADJ
cana-1037	8	51	role	role	NOUN
cana-1037	8	52	in	in	ADP
cana-1037	8	53	many	many	ADJ
cana-1037	8	54	areas	area	NOUN
cana-1037	8	55	of	of	ADP
cana-1037	8	56	mathematics	mathematic	NOUN
cana-1037	8	57	.	.	PUNCT
cana-1037	9	1	as	as	ADP
cana-1037	9	2	an	an	DET
cana-1037	9	3	example	example	NOUN
cana-1037	9	4	,	,	PUNCT
cana-1037	9	5	they	they	PRON
cana-1037	9	6	are	be	AUX
cana-1037	9	7	extremely	extremely	ADV
cana-1037	9	8	useful	useful	ADJ
cana-1037	9	9	in	in	ADP
cana-1037	9	10	the	the	DET
cana-1037	9	11	research	research	NOUN
cana-1037	9	12	and	and	CCONJ
cana-1037	9	13	resolution	resolution	NOUN
cana-1037	9	14	of	of	ADP
cana-1037	9	15	linear	linear	ADJ
cana-1037	9	16	equation	equation	NOUN
cana-1037	9	17	and	and	CCONJ
cana-1037	9	18	system	system	NOUN
cana-1037	9	19	problems	problem	NOUN
cana-1037	9	20	.	.	PUNCT
cana-1037	10	1	the	the	DET
cana-1037	10	2	study	study	NOUN
cana-1037	10	3	of	of	ADP
cana-1037	10	4	determinants	determinant	NOUN
cana-1037	10	5	can	can	AUX
cana-1037	10	6	be	be	AUX
cana-1037	10	7	approached	approach	VERB
cana-1037	10	8	from	from	ADP
cana-1037	10	9	several	several	ADJ
cana-1037	10	10	distinct	distinct	ADJ
cana-1037	10	11	angles	angle	NOUN
cana-1037	10	12	.	.	PUNCT
cana-1037	11	1	throughout	throughout	ADP
cana-1037	11	2	the	the	DET
cana-1037	11	3	course	course	NOUN
cana-1037	11	4	of	of	ADP
cana-1037	11	5	this	this	DET
cana-1037	11	6	inquiry	inquiry	NOUN
cana-1037	11	7	,	,	PUNCT
cana-1037	11	8	we	we	PRON
cana-1037	11	9	discover	discover	VERB
cana-1037	11	10	a	a	DET
cana-1037	11	11	large	large	ADJ
cana-1037	11	12	number	number	NOUN
cana-1037	11	13	of	of	ADP
cana-1037	11	14	determinant	determinant	ADJ
cana-1037	11	15	identities	identity	NOUN
cana-1037	11	16	involving	involve	VERB
cana-1037	11	17	jacobsthal	jacobsthal	ADJ
cana-1037	11	18	and	and	CCONJ
cana-1037	11	19	lucas	lucas	NOUN
cana-1037	11	20	numbers	number	NOUN
cana-1037	11	21	.	.	PUNCT
cana-1037	12	1	keywords	keyword	NOUN
cana-1037	12	2	:	:	PUNCT
cana-1037	12	3	jacobsthal	jacobsthal	ADJ
cana-1037	12	4	,	,	PUNCT
cana-1037	12	5	jacobsthal	jacobsthal	ADJ
cana-1037	12	6	lucas	lucas	PROPN
cana-1037	12	7	numbers	number	NOUN
cana-1037	12	8	,	,	PUNCT
cana-1037	12	9	determinants	determinant	NOUN
cana-1037	12	10	.	.	PUNCT
cana-1037	13	1	1	1	X
cana-1037	13	2	.	.	X
cana-1037	13	3	introduction	introduction	NOUN
cana-1037	13	4	the	the	DET
cana-1037	13	5	jacobsthal	jacobsthal	ADJ
cana-1037	13	6	numbers	number	NOUN
cana-1037	13	7	possess	possess	VERB
cana-1037	13	8	significant	significant	ADJ
cana-1037	13	9	features	feature	NOUN
cana-1037	13	10	and	and	CCONJ
cana-1037	13	11	are	be	AUX
cana-1037	13	12	applied	apply	VERB
cana-1037	13	13	in	in	ADP
cana-1037	13	14	number	number	NOUN
cana-1037	13	15	theory	theory	NOUN
cana-1037	13	16	,	,	PUNCT
cana-1037	13	17	graph	graph	NOUN
cana-1037	13	18	theory	theory	NOUN
cana-1037	13	19	,	,	PUNCT
cana-1037	13	20	combinatorics	combinatoric	NOUN
cana-1037	13	21	,	,	PUNCT
cana-1037	13	22	and	and	CCONJ
cana-1037	13	23	geometry	geometry	NOUN
cana-1037	13	24	.	.	PUNCT
cana-1037	14	1	while	while	SCONJ
cana-1037	14	2	analyzing	analyze	VERB
cana-1037	14	3	the	the	DET
cana-1037	14	4	relationship	relationship	NOUN
cana-1037	14	5	between	between	ADP
cana-1037	14	6	determinants	determinant	NOUN
cana-1037	14	7	and	and	CCONJ
cana-1037	14	8	permanents	permanent	NOUN
cana-1037	14	9	of	of	ADP
cana-1037	14	10	two	two	NUM
cana-1037	14	11	n	n	CCONJ
cana-1037	14	12	-	-	PUNCT
cana-1037	14	13	square	square	ADJ
cana-1037	14	14	upper	upper	ADJ
cana-1037	14	15	hessenberg	hessenberg	NOUN
cana-1037	14	16	matrices	matrix	NOUN
cana-1037	14	17	,	,	PUNCT
cana-1037	14	18	one	one	NUM
cana-1037	14	19	representing	represent	VERB
cana-1037	14	20	the	the	DET
cana-1037	14	21	adjacency	adjacency	NOUN
cana-1037	14	22	matrix	matrix	NOUN
cana-1037	14	23	of	of	ADP
cana-1037	14	24	a	a	DET
cana-1037	14	25	directed	direct	VERB
cana-1037	14	26	pseudograph	pseudograph	NOUN
cana-1037	14	27	,	,	PUNCT
cana-1037	14	28	and	and	CCONJ
cana-1037	14	29	derived	derived	ADJ
cana-1037	14	30	sum	sum	NOUN
cana-1037	14	31	formulas	formula	NOUN
cana-1037	14	32	for	for	ADP
cana-1037	14	33	the	the	DET
cana-1037	14	34	jacobsthal	jacobsthal	ADJ
cana-1037	14	35	sequence	sequence	NOUN
cana-1037	14	36	.	.	PUNCT
cana-1037	15	1	two	two	NUM
cana-1037	15	2	higher	high	ADJ
cana-1037	15	3	hessenberg	hessenberg	NOUN
cana-1037	15	4	matrices	matrix	NOUN
cana-1037	15	5	were	be	AUX
cana-1037	15	6	analyzed	analyze	VERB
cana-1037	15	7	,	,	PUNCT
cana-1037	15	8	demonstrating	demonstrate	VERB
cana-1037	15	9	that	that	SCONJ
cana-1037	15	10	their	their	PRON
cana-1037	15	11	permanents	permanent	NOUN
cana-1037	15	12	correspond	correspond	VERB
cana-1037	15	13	to	to	ADP
cana-1037	15	14	jacobsthal	jacobsthal	ADJ
cana-1037	15	15	numbers	number	NOUN
cana-1037	15	16	.	.	PUNCT
cana-1037	16	1	some	some	DET
cana-1037	16	2	families	family	NOUN
cana-1037	16	3	of	of	ADP
cana-1037	16	4	toeplitz	toeplitz	NOUN
cana-1037	16	5	-	-	PUNCT
cana-1037	16	6	hessenberg	hessenberg	PROPN
cana-1037	16	7	matrices	matrix	NOUN
cana-1037	16	8	are	be	AUX
cana-1037	16	9	determined	determine	VERB
cana-1037	16	10	by	by	ADP
cana-1037	16	11	using	use	VERB
cana-1037	16	12	different	different	ADJ
cana-1037	16	13	translations	translation	NOUN
cana-1037	16	14	of	of	ADP
cana-1037	16	15	the	the	DET
cana-1037	16	16	jacobsthal	jacobsthal	ADJ
cana-1037	16	17	numbers	number	NOUN
cana-1037	16	18	as	as	ADP
cana-1037	16	19	the	the	DET
cana-1037	16	20	nonzero	nonzero	PROPN
cana-1037	16	21	entries	entry	NOUN
cana-1037	16	22	.	.	PUNCT
cana-1037	17	1	the	the	DET
cana-1037	17	2	determinant	determinant	ADJ
cana-1037	17	3	identities	identity	NOUN
cana-1037	17	4	can	can	AUX
cana-1037	17	5	be	be	AUX
cana-1037	17	6	expressed	express	VERB
cana-1037	17	7	using	use	VERB
cana-1037	17	8	the	the	DET
cana-1037	17	9	trudi	trudi	NOUN
cana-1037	17	10	formula	formula	NOUN
cana-1037	17	11	as	as	ADP
cana-1037	17	12	equations	equation	NOUN
cana-1037	17	13	that	that	PRON
cana-1037	17	14	consist	consist	VERB
cana-1037	17	15	of	of	ADP
cana-1037	17	16	combinations	combination	NOUN
cana-1037	17	17	of	of	ADP
cana-1037	17	18	jacobsthal	jacobsthal	ADJ
cana-1037	17	19	numbers	number	NOUN
cana-1037	17	20	and	and	CCONJ
cana-1037	17	21	multinomial	multinomial	ADJ
cana-1037	17	22	coefficients	coefficient	NOUN
cana-1037	17	23	.	.	PUNCT
cana-1037	18	1	here	here	ADV
cana-1037	18	2	,	,	PUNCT
cana-1037	18	3	we	we	PRON
cana-1037	18	4	present	present	VERB
cana-1037	18	5	analogous	analogous	ADJ
cana-1037	18	6	results	result	NOUN
cana-1037	18	7	using	use	VERB
cana-1037	18	8	the	the	DET
cana-1037	18	9	generalised	generalised	ADJ
cana-1037	18	10	trudi	trudi	NOUN
cana-1037	18	11	formula	formula	NOUN
cana-1037	18	12	.	.	PUNCT
cana-1037	19	1	the	the	DET
cana-1037	19	2	initial	initial	ADJ
cana-1037	19	3	column	column	NOUN
cana-1037	19	4	entries	entry	NOUN
cana-1037	19	5	are	be	AUX
cana-1037	19	6	adjusted	adjust	VERB
cana-1037	19	7	and	and	CCONJ
cana-1037	19	8	combinatorial	combinatorial	ADJ
cana-1037	19	9	proofs	proof	NOUN
cana-1037	19	10	are	be	AUX
cana-1037	19	11	offered	offer	VERB
cana-1037	19	12	in	in	ADP
cana-1037	19	13	various	various	ADJ
cana-1037	19	14	instances	instance	NOUN
cana-1037	19	15	.	.	PUNCT
cana-1037	20	1	as	as	ADP
cana-1037	20	2	a	a	DET
cana-1037	20	3	consequence	consequence	NOUN
cana-1037	20	4	of	of	ADP
cana-1037	20	5	this	this	PRON
cana-1037	20	6	,	,	PUNCT
cana-1037	20	7	the	the	DET
cana-1037	20	8	jacobsthal	jacobsthal	ADJ
cana-1037	20	9	polynomial	polynomial	ADJ
cana-1037	20	10	,	,	PUNCT
cana-1037	20	11	jn(1	jn(1	NOUN
cana-1037	20	12	)	)	PUNCT
cana-1037	20	13	=	=	SYM
cana-1037	20	14	fn	fn	NOUN
cana-1037	20	15	,	,	PUNCT
cana-1037	20	16	and	and	CCONJ
cana-1037	20	17	the	the	DET
cana-1037	20	18	jacobsthal	jacobsthal	ADJ
cana-1037	20	19	function	function	NOUN
cana-1037	20	20	,	,	PUNCT
cana-1037	20	21	jn(x	jn(x	NUM
cana-1037	20	22	)	)	PUNCT
cana-1037	20	23	,	,	PUNCT
cana-1037	20	24	are	be	AUX
cana-1037	20	25	closely	closely	ADV
cana-1037	20	26	related	relate	VERB
cana-1037	20	27	(	(	PUNCT
cana-1037	20	28	jhala	jhala	NOUN
cana-1037	20	29	et	et	PROPN
cana-1037	20	30	al	al	PROPN
cana-1037	20	31	.	.	PROPN
cana-1037	20	32	,	,	PUNCT
cana-1037	20	33	2013	2013	NUM
cana-1037	20	34	)	)	PUNCT
cana-1037	20	35	.	.	PUNCT
cana-1037	21	1	it	it	PRON
cana-1037	21	2	is	be	AUX
cana-1037	21	3	possible	possible	ADJ
cana-1037	21	4	to	to	PART
cana-1037	21	5	define	define	VERB
cana-1037	21	6	the	the	DET
cana-1037	21	7	polynomial	polynomial	ADJ
cana-1037	21	8	known	know	VERB
cana-1037	21	9	as	as	ADP
cana-1037	21	10	the	the	DET
cana-1037	21	11	jacobsthallucas	jacobsthalluca	NOUN
cana-1037	21	12	polynomial	polynomial	ADJ
cana-1037	21	13	for	for	ADP
cana-1037	21	14	the	the	DET
cana-1037	21	15	function	function	NOUN
cana-1037	21	16	jn(x	jn(x	NOUN
cana-1037	21	17	)	)	PUNCT
cana-1037	21	18	with	with	ADP
cana-1037	21	19	n	n	X
cana-1037	21	20	equal	equal	ADJ
cana-1037	21	21	to	to	ADP
cana-1037	21	22	zero	zero	NUM
cana-1037	21	23	,	,	PUNCT
cana-1037	21	24	provided	provide	VERB
cana-1037	21	25	that	that	SCONJ
cana-1037	21	26	j1	j1	PROPN
cana-1037	21	27	equals	equal	VERB
cana-1037	21	28	ln	ln	PROPN
cana-1037	21	29	.	.	PUNCT
cana-1037	21	30	a.	a.	PROPN
cana-1037	21	31	f.	f.	PROPN
cana-1037	21	32	horadam	horadam	PROPN
cana-1037	21	33	presented	present	VERB
cana-1037	21	34	the	the	DET
cana-1037	21	35	paper	paper	NOUN
cana-1037	21	36	"	"	PUNCT
cana-1037	21	37	horadam	horadam	X
cana-1037	21	38	,	,	PUNCT
cana-1037	21	39	jacobsthal	jacobsthal	ADJ
cana-1037	21	40	,	,	PUNCT
cana-1037	21	41	hoggatt	hoggatt	PROPN
cana-1037	21	42	,	,	PUNCT
cana-1037	21	43	and	and	CCONJ
cana-1037	21	44	bicknell	bicknell	NOUN
cana-1037	21	45	-	-	PUNCT
cana-1037	21	46	johnson	johnson	PROPN
cana-1037	21	47	representation	representation	NOUN
cana-1037	21	48	numbers	number	NOUN
cana-1037	21	49	"	"	PUNCT
cana-1037	21	50	in	in	ADP
cana-1037	21	51	may	may	PROPN
cana-1037	21	52	1994	1994	NUM
cana-1037	21	53	at	at	ADP
cana-1037	21	54	the	the	DET
cana-1037	21	55	university	university	NOUN
cana-1037	21	56	of	of	ADP
cana-1037	21	57	new	new	PROPN
cana-1037	21	58	england	england	PROPN
cana-1037	21	59	in	in	ADP
cana-1037	21	60	armidale	armidale	PROPN
cana-1037	21	61	,	,	PUNCT
cana-1037	21	62	which	which	PRON
cana-1037	21	63	is	be	AUX
cana-1037	21	64	in	in	ADP
cana-1037	21	65	australia	australia	PROPN
cana-1037	21	66	.	.	PUNCT
cana-1037	22	1	specifically	specifically	ADV
cana-1037	22	2	,	,	PUNCT
cana-1037	22	3	the	the	PRON
cana-1037	22	4	[	[	X
cana-1037	22	5	jn	jn	X
cana-1037	22	6	]	]	X
cana-1037	22	7	and	and	CCONJ
cana-1037	22	8	[	[	X
cana-1037	22	9	jn	jn	X
cana-1037	22	10	]	]	X
cana-1037	22	11	sequences	sequence	NOUN
cana-1037	22	12	are	be	AUX
cana-1037	22	13	of	of	ADP
cana-1037	22	14	importance	importance	NOUN
cana-1037	22	15	to	to	ADP
cana-1037	22	16	us	we	PRON
cana-1037	22	17	.	.	PUNCT
cana-1037	23	1	the	the	DET
cana-1037	23	2	recurrence	recurrence	NOUN
cana-1037	23	3	relations	relation	NOUN
cana-1037	23	4	jn+2	jn+2	X
cana-1037	23	5	=	=	SYM
cana-1037	23	6	jn+1	jn+1	PROPN
cana-1037	24	1	+	+	CCONJ
cana-1037	24	2	2jn	2jn	NOUN
cana-1037	24	3	,	,	PUNCT
cana-1037	24	4	j0	j0	PROPN
cana-1037	24	5	=	=	SYM
cana-1037	24	6	2	2	NUM
cana-1037	24	7	,	,	PUNCT
cana-1037	24	8	j1	j1	NOUN
cana-1037	24	9	=	=	SYM
cana-1037	24	10	1	1	NUM
cana-1037	24	11	,	,	PUNCT
cana-1037	24	12	n	n	CCONJ
cana-1037	24	13	>	>	X
cana-1037	24	14	0	0	NUM
cana-1037	24	15	,	,	PUNCT
cana-1037	24	16	and	and	CCONJ
cana-1037	24	17	jn+2	jn+2	PROPN
cana-1037	24	18	=	=	SYM
cana-1037	24	19	jn+1	jn+1	PROPN
cana-1037	25	1	+	+	CCONJ
cana-1037	25	2	2	2	NUM
cana-1037	25	3	jn	jn	PROPN
cana-1037	25	4	,	,	PUNCT
cana-1037	25	5	j0	j0	PROPN
cana-1037	25	6	=	=	SYM
cana-1037	25	7	2	2	NUM
cana-1037	25	8	,	,	PUNCT
cana-1037	25	9	j1	j1	NOUN
cana-1037	25	10	=	=	NOUN
cana-1037	25	11	1	1	NUM
cana-1037	25	12	respectively	respectively	ADV
cana-1037	25	13	,	,	PUNCT
cana-1037	25	14	define	define	VERB
cana-1037	25	15	them	they	PRON
cana-1037	25	16	so	so	SCONJ
cana-1037	25	17	that	that	SCONJ
cana-1037	25	18	they	they	PRON
cana-1037	25	19	may	may	AUX
cana-1037	25	20	be	be	AUX
cana-1037	25	21	differentiated	differentiate	VERB
cana-1037	25	22	from	from	ADP
cana-1037	25	23	one	one	NUM
cana-1037	25	24	another	another	DET
cana-1037	25	25	.	.	PUNCT
cana-1037	26	1	each	each	PRON
cana-1037	26	2	of	of	ADP
cana-1037	26	3	the	the	DET
cana-1037	26	4	sequences	sequence	NOUN
cana-1037	26	5	is	be	AUX
cana-1037	26	6	composed	compose	VERB
cana-1037	26	7	of	of	ADP
cana-1037	26	8	a	a	DET
cana-1037	26	9	number	number	NOUN
cana-1037	26	10	of	of	ADP
cana-1037	26	11	sets	set	NOUN
cana-1037	26	12	(	(	PUNCT
cana-1037	26	13	assaf	assaf	PROPN
cana-1037	26	14	e	e	PROPN
cana-1037	26	15	and	and	CCONJ
cana-1037	26	16	gueron	gueron	NOUN
cana-1037	26	17	s	s	NOUN
cana-1037	26	18	,	,	PUNCT
cana-1037	26	19	2001	2001	NUM
cana-1037	26	20	)	)	PUNCT
cana-1037	26	21	.	.	PUNCT
cana-1037	27	1	these	these	PRON
cana-1037	27	2	are	be	AUX
cana-1037	27	3	the	the	DET
cana-1037	27	4	jacobsthal	jacobsthal	ADJ
cana-1037	27	5	numbers	number	NOUN
cana-1037	27	6	in	in	ADP
cana-1037	27	7	the	the	DET
cana-1037	27	8	order	order	NOUN
cana-1037	27	9	that	that	SCONJ
cana-1037	27	10	they	they	PRON
cana-1037	27	11	are	be	AUX
cana-1037	27	12	presented	present	VERB
cana-1037	27	13	:	:	PUNCT
cana-1037	27	14	0	0	NUM
cana-1037	27	15	,	,	PUNCT
cana-1037	27	16	1	1	NUM
cana-1037	27	17	,	,	PUNCT
cana-1037	27	18	1	1	NUM
cana-1037	27	19	,	,	PUNCT
cana-1037	27	20	3	3	NUM
cana-1037	27	21	,	,	PUNCT
cana-1037	27	22	5	5	NUM
cana-1037	27	23	,	,	PUNCT
cana-1037	27	24	11	11	NUM
cana-1037	27	25	,	,	PUNCT
cana-1037	27	26	21	21	NUM
cana-1037	27	27	,	,	PUNCT
cana-1037	27	28	43	43	NUM
cana-1037	27	29	,	,	PUNCT
cana-1037	27	30	communications	communication	NOUN
cana-1037	27	31	on	on	ADP
cana-1037	27	32	applied	apply	VERB
cana-1037	27	33	nonlinear	nonlinear	ADJ
cana-1037	27	34	analysis	analysis	NOUN
cana-1037	27	35	issn	issn	NOUN
cana-1037	27	36	:	:	PUNCT
cana-1037	27	37	1074	1074	NUM
cana-1037	27	38	-	-	PUNCT
cana-1037	27	39	133x	133x	NUM
cana-1037	27	40	vol	vol	NOUN
cana-1037	27	41	31	31	NUM
cana-1037	27	42	no	no	NOUN
cana-1037	27	43	.	.	PUNCT
cana-1037	28	1	5s	5s	NUM
cana-1037	28	2	(	(	PUNCT
cana-1037	28	3	2024	2024	NUM
cana-1037	28	4	)	)	PUNCT
cana-1037	28	5	290	290	NUM
cana-1037	28	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1037	28	7	85	85	NUM
cana-1037	28	8	,	,	PUNCT
cana-1037	28	9	171	171	NUM
cana-1037	28	10	,	,	PUNCT
cana-1037	28	11	341	341	NUM
cana-1037	28	12	...	...	PUNCT
cana-1037	28	13	the	the	DET
cana-1037	28	14	series	series	NOUN
cana-1037	28	15	of	of	ADP
cana-1037	28	16	jl	jl	NOUN
cana-1037	28	17	numbers	number	NOUN
cana-1037	28	18	is	be	AUX
cana-1037	28	19	comprised	comprise	VERB
cana-1037	28	20	of	of	ADP
cana-1037	28	21	eight	eight	NUM
cana-1037	28	22	numbers	number	NOUN
cana-1037	28	23	:	:	PUNCT
cana-1037	28	24	2	2	NUM
cana-1037	28	25	,	,	PUNCT
cana-1037	28	26	1	1	NUM
cana-1037	28	27	,	,	PUNCT
cana-1037	28	28	5	5	NUM
cana-1037	28	29	,	,	PUNCT
cana-1037	28	30	7	7	NUM
cana-1037	28	31	,	,	PUNCT
cana-1037	28	32	17	17	NUM
cana-1037	28	33	,	,	PUNCT
cana-1037	28	34	31	31	NUM
cana-1037	28	35	,	,	PUNCT
cana-1037	28	36	65	65	NUM
cana-1037	28	37	,	,	PUNCT
cana-1037	28	38	127	127	NUM
cana-1037	28	39	,	,	PUNCT
cana-1037	28	40	257	257	NUM
cana-1037	28	41	,	,	PUNCT
cana-1037	28	42	511	511	NUM
cana-1037	28	43	,	,	PUNCT
cana-1037	28	44	and	and	CCONJ
cana-1037	28	45	1025	1025	NUM
cana-1037	28	46	.	.	PUNCT
cana-1037	29	1	these	these	DET
cana-1037	29	2	numbers	number	NOUN
cana-1037	29	3	are	be	AUX
cana-1037	29	4	all	all	PRON
cana-1037	29	5	integers	integer	NOUN
cana-1037	29	6	(	(	PUNCT
cana-1037	29	7	feng	feng	PROPN
cana-1037	29	8	,	,	PUNCT
cana-1037	29	9	2011	2011	NUM
cana-1037	29	10	)	)	PUNCT
cana-1037	29	11	.	.	PUNCT
cana-1037	30	1	2	2	X
cana-1037	30	2	.	.	X
cana-1037	30	3	fibonacci	fibonacci	NOUN
cana-1037	30	4	numbers	number	VERB
cana-1037	30	5	a	a	DET
cana-1037	30	6	sequence	sequence	NOUN
cana-1037	30	7	in	in	ADP
cana-1037	30	8	mathematics	mathematic	NOUN
cana-1037	30	9	known	know	VERB
cana-1037	30	10	as	as	ADP
cana-1037	30	11	the	the	DET
cana-1037	30	12	fibonacci	fibonacci	NOUN
cana-1037	30	13	numbers	number	NOUN
cana-1037	30	14	,	,	PUNCT
cana-1037	30	15	which	which	PRON
cana-1037	30	16	are	be	AUX
cana-1037	30	17	represented	represent	VERB
cana-1037	30	18	by	by	ADP
cana-1037	30	19	the	the	DET
cana-1037	30	20	symbol	symbol	NOUN
cana-1037	30	21	fn	fn	NOUN
cana-1037	30	22	,	,	PUNCT
cana-1037	30	23	is	be	AUX
cana-1037	30	24	characterized	characterize	VERB
cana-1037	30	25	by	by	ADP
cana-1037	30	26	the	the	DET
cana-1037	30	27	fact	fact	NOUN
cana-1037	30	28	that	that	SCONJ
cana-1037	30	29	each	each	DET
cana-1037	30	30	number	number	NOUN
cana-1037	30	31	is	be	AUX
cana-1037	30	32	equal	equal	ADJ
cana-1037	30	33	to	to	ADP
cana-1037	30	34	the	the	DET
cana-1037	30	35	sum	sum	NOUN
cana-1037	30	36	of	of	ADP
cana-1037	30	37	its	its	PRON
cana-1037	30	38	two	two	NUM
cana-1037	30	39	preceding	precede	VERB
cana-1037	30	40	integers	integer	NOUN
cana-1037	30	41	,	,	PUNCT
cana-1037	30	42	beginning	begin	VERB
cana-1037	30	43	with	with	ADP
cana-1037	30	44	0	0	NUM
cana-1037	30	45	and	and	CCONJ
cana-1037	30	46	1	1	NUM
cana-1037	30	47	.	.	X
cana-1037	31	1	it	it	PRON
cana-1037	31	2	is	be	AUX
cana-1037	31	3	known	know	VERB
cana-1037	31	4	that	that	SCONJ
cana-1037	31	5	this	this	DET
cana-1037	31	6	sequence	sequence	NOUN
cana-1037	31	7	is	be	AUX
cana-1037	31	8	unique	unique	ADJ
cana-1037	31	9	(	(	PUNCT
cana-1037	31	10	yogesh	yogesh	PROPN
cana-1037	31	11	kumar	kumar	PROPN
cana-1037	31	12	gupta	gupta	PROPN
cana-1037	31	13	and	and	CCONJ
cana-1037	31	14	omprakash	omprakash	NOUN
cana-1037	31	15	sikhwal	sikhwal	PROPN
cana-1037	31	16	mamta	mamta	PROPN
cana-1037	31	17	singh	singh	PROPN
cana-1037	31	18	,	,	PUNCT
cana-1037	31	19	2016	2016	NUM
cana-1037	31	20	)	)	PUNCT
cana-1037	31	21	.	.	PUNCT
cana-1037	32	1	as	as	ADP
cana-1037	32	2	the	the	DET
cana-1037	32	3	fibonacci	fibonacci	NOUN
cana-1037	32	4	sequence	sequence	NOUN
cana-1037	32	5	.	.	PUNCT
cana-1037	33	1	in	in	ADP
cana-1037	33	2	other	other	ADJ
cana-1037	33	3	words	word	NOUN
cana-1037	33	4	,	,	PUNCT
cana-1037	33	5	fn+2	fn+2	NOUN
cana-1037	33	6	=	=	SYM
cana-1037	33	7	fn+1	fn+1	PROPN
cana-1037	33	8	+	+	SYM
cana-1037	33	9	fn	fn	NOUN
cana-1037	33	10	,	,	PUNCT
cana-1037	33	11	f0	f0	PROPN
cana-1037	33	12	=	=	SYM
cana-1037	33	13	0	0	NUM
cana-1037	33	14	,	,	PUNCT
cana-1037	33	15	f1	f1	NOUN
cana-1037	33	16	=	=	SYM
cana-1037	33	17	1	1	NUM
cana-1037	33	18	and	and	CCONJ
cana-1037	33	19	binet	binet	NOUN
cana-1037	33	20	formula	formula	NOUN
cana-1037	33	21	for	for	ADP
cana-1037	33	22	fibonacci	fibonacci	NOUN
cana-1037	33	23	sequence	sequence	NOUN
cana-1037	33	24	𝐹𝑛	𝐹𝑛	PROPN
cana-1037	33	25	=	=	PUNCT
cana-1037	33	26	1	1	NUM
cana-1037	33	27	√5	√5	PROPN
cana-1037	33	28	(	(	PUNCT
cana-1037	33	29	1	1	NUM
cana-1037	33	30	+	+	CCONJ
cana-1037	33	31	√5	√5	ADP
cana-1037	33	32	2	2	NUM
cana-1037	33	33	)	)	PUNCT
cana-1037	33	34	𝑛	𝑛	NOUN
cana-1037	33	35	+	+	CCONJ
cana-1037	33	36	(	(	PUNCT
cana-1037	33	37	1	1	NUM
cana-1037	33	38	−	−	NOUN
cana-1037	33	39	√5	√5	ADV
cana-1037	33	40	2	2	NUM
cana-1037	33	41	)	)	PUNCT
cana-1037	33	42	𝑛	𝑛	PRON
cana-1037	33	43	the	the	DET
cana-1037	33	44	fibonacci	fibonacci	PROPN
cana-1037	33	45	series	series	PROPN
cana-1037	33	46	and	and	CCONJ
cana-1037	33	47	cassini	cassini	NOUN
cana-1037	33	48	's	's	PART
cana-1037	33	49	formula	formula	NOUN
cana-1037	33	50	fnfn+1−𝐹𝑛	fnfn+1−𝐹𝑛	PROPN
cana-1037	33	51	2	2	NUM
cana-1037	33	52	=	=	SYM
cana-1037	33	53	(	(	PUNCT
cana-1037	33	54	−1)𝑛	−1)𝑛	X
cana-1037	33	55	where	where	SCONJ
cana-1037	33	56	fn	fn	PROPN
cana-1037	33	57	nth	nth	PROPN
cana-1037	33	58	fibonacci	fibonacci	PROPN
cana-1037	33	59	number	number	NOUN
cana-1037	33	60	.	.	PUNCT
cana-1037	34	1	3	3	X
cana-1037	34	2	.	.	X
cana-1037	34	3	lucas	lucas	NOUN
cana-1037	34	4	numbers	number	NOUN
cana-1037	34	5	in	in	ADP
cana-1037	34	6	a	a	DET
cana-1037	34	7	manner	manner	NOUN
cana-1037	34	8	analogous	analogous	ADJ
cana-1037	34	9	to	to	ADP
cana-1037	34	10	that	that	PRON
cana-1037	34	11	of	of	ADP
cana-1037	34	12	the	the	DET
cana-1037	34	13	fibonacci	fibonacci	NOUN
cana-1037	34	14	numbers	number	NOUN
cana-1037	34	15	,	,	PUNCT
cana-1037	34	16	the	the	DET
cana-1037	34	17	lucas	lucas	ADJ
cana-1037	34	18	number	number	NOUN
cana-1037	34	19	is	be	AUX
cana-1037	34	20	defined	define	VERB
cana-1037	34	21	as	as	ADP
cana-1037	34	22	the	the	DET
cana-1037	34	23	product	product	NOUN
cana-1037	34	24	of	of	ADP
cana-1037	34	25	its	its	PRON
cana-1037	34	26	two	two	NUM
cana-1037	34	27	terms	term	NOUN
cana-1037	34	28	directly	directly	ADV
cana-1037	34	29	preceding	precede	VERB
cana-1037	34	30	it	it	PRON
cana-1037	34	31	.	.	PUNCT
cana-1037	35	1	upon	upon	SCONJ
cana-1037	35	2	completion	completion	NOUN
cana-1037	35	3	of	of	ADP
cana-1037	35	4	this	this	DET
cana-1037	35	5	process	process	NOUN
cana-1037	35	6	,	,	PUNCT
cana-1037	35	7	an	an	DET
cana-1037	35	8	integer	integer	NOUN
cana-1037	35	9	sequence	sequence	NOUN
cana-1037	35	10	in	in	ADP
cana-1037	35	11	the	the	DET
cana-1037	35	12	style	style	NOUN
cana-1037	35	13	of	of	ADP
cana-1037	35	14	fibonacci	fibonacci	NOUN
cana-1037	35	15	will	will	AUX
cana-1037	35	16	be	be	AUX
cana-1037	35	17	produced	produce	VERB
cana-1037	35	18	.	.	PUNCT
cana-1037	36	1	the	the	DET
cana-1037	36	2	first	first	ADJ
cana-1037	36	3	two	two	NUM
cana-1037	36	4	numbers	number	NOUN
cana-1037	36	5	in	in	ADP
cana-1037	36	6	the	the	DET
cana-1037	36	7	fibonacci	fibonacci	NOUN
cana-1037	36	8	sequence	sequence	NOUN
cana-1037	36	9	are	be	AUX
cana-1037	36	10	f0	f0	ADJ
cana-1037	36	11	=	=	SYM
cana-1037	36	12	0	0	NUM
cana-1037	36	13	and	and	CCONJ
cana-1037	36	14	f1	f1	NOUN
cana-1037	36	15	=	=	SYM
cana-1037	36	16	1	1	NUM
cana-1037	36	17	,	,	PUNCT
cana-1037	36	18	whereas	whereas	SCONJ
cana-1037	36	19	the	the	DET
cana-1037	36	20	first	first	ADJ
cana-1037	36	21	two	two	NUM
cana-1037	36	22	numbers	number	NOUN
cana-1037	36	23	in	in	ADP
cana-1037	36	24	the	the	DET
cana-1037	36	25	lucas	lucas	NOUN
cana-1037	36	26	sequence	sequence	NOUN
cana-1037	36	27	are	be	AUX
cana-1037	36	28	l0	l0	NOUN
cana-1037	36	29	=	=	SYM
cana-1037	36	30	2	2	NUM
cana-1037	36	31	and	and	CCONJ
cana-1037	36	32	l1	l1	PROPN
cana-1037	36	33	=	=	PROPN
cana-1037	37	1	1	1	X
cana-1037	37	2	.	.	PUNCT
cana-1037	38	1	in	in	ADP
cana-1037	38	2	spite	spite	NOUN
cana-1037	38	3	of	of	ADP
cana-1037	38	4	the	the	DET
cana-1037	38	5	fact	fact	NOUN
cana-1037	38	6	that	that	SCONJ
cana-1037	38	7	they	they	PRON
cana-1037	38	8	are	be	AUX
cana-1037	38	9	based	base	VERB
cana-1037	38	10	on	on	ADP
cana-1037	38	11	the	the	DET
cana-1037	38	12	same	same	ADJ
cana-1037	38	13	idea	idea	NOUN
cana-1037	38	14	,	,	PUNCT
cana-1037	38	15	the	the	DET
cana-1037	38	16	lucas	lucas	PROPN
cana-1037	38	17	numbers	number	NOUN
cana-1037	38	18	and	and	CCONJ
cana-1037	38	19	the	the	DET
cana-1037	38	20	fibonacci	fibonacci	NOUN
cana-1037	38	21	numbers	number	NOUN
cana-1037	38	22	are	be	AUX
cana-1037	38	23	very	very	ADV
cana-1037	38	24	different	different	ADJ
cana-1037	38	25	from	from	ADP
cana-1037	38	26	one	one	NUM
cana-1037	38	27	another	another	DET
cana-1037	38	28	(	(	PUNCT
cana-1037	38	29	horadam	horadam	PROPN
cana-1037	38	30	,	,	PUNCT
cana-1037	38	31	1971	1971	NUM
cana-1037	38	32	)	)	PUNCT
cana-1037	38	33	.	.	PUNCT
cana-1037	39	1	consequently	consequently	ADV
cana-1037	39	2	,	,	PUNCT
cana-1037	39	3	the	the	DET
cana-1037	39	4	following	follow	VERB
cana-1037	39	5	is	be	AUX
cana-1037	39	6	a	a	DET
cana-1037	39	7	description	description	NOUN
cana-1037	39	8	of	of	ADP
cana-1037	39	9	the	the	DET
cana-1037	39	10	lucas	lucas	NOUN
cana-1037	39	11	numbers	number	NOUN
cana-1037	39	12	:	:	PUNCT
cana-1037	40	1	𝐿𝑛	𝐿𝑛	PROPN
cana-1037	40	2	=	=	PRON
cana-1037	40	3	{	{	PUNCT
cana-1037	40	4	2	2	NUM
cana-1037	40	5	𝑖𝑓	𝑖𝑓	NOUN
cana-1037	40	6	𝑛	𝑛	PROPN
cana-1037	40	7	=	=	SYM
cana-1037	40	8	0	0	NUM
cana-1037	40	9	1	1	NUM
cana-1037	40	10	𝑖𝑓	𝑖𝑓	NUM
cana-1037	40	11	𝑛	𝑛	PROPN
cana-1037	40	12	=	=	SYM
cana-1037	40	13	1	1	NUM
cana-1037	40	14	𝐿𝑛−1	𝐿𝑛−1	PROPN
cana-1037	40	15	+	+	CCONJ
cana-1037	40	16	𝐿𝑛−2	𝐿𝑛−2	NOUN
cana-1037	40	17	𝑖𝑓	𝑖𝑓	ADP
cana-1037	40	18	𝑛	𝑛	NOUN
cana-1037	40	19	>	>	X
cana-1037	40	20	1	1	NUM
cana-1037	40	21	(	(	PUNCT
cana-1037	40	22	in	in	ADP
cana-1037	40	23	which	which	PRON
cana-1037	40	24	n	n	ADV
cana-1037	40	25	is	be	AUX
cana-1037	40	26	a	a	DET
cana-1037	40	27	member	member	NOUN
cana-1037	40	28	of	of	ADP
cana-1037	40	29	the	the	DET
cana-1037	40	30	natural	natural	ADJ
cana-1037	40	31	numbers	number	NOUN
cana-1037	40	32	)	)	PUNCT
cana-1037	40	33	numbers	number	NOUN
cana-1037	40	34	2	2	NUM
cana-1037	40	35	,	,	PUNCT
cana-1037	40	36	1	1	NUM
cana-1037	40	37	,	,	PUNCT
cana-1037	40	38	3	3	NUM
cana-1037	40	39	,	,	PUNCT
cana-1037	40	40	4	4	NUM
cana-1037	40	41	,	,	PUNCT
cana-1037	40	42	7	7	NUM
cana-1037	40	43	,	,	PUNCT
cana-1037	40	44	11	11	NUM
cana-1037	40	45	,	,	PUNCT
cana-1037	40	46	18	18	NUM
cana-1037	40	47	,	,	PUNCT
cana-1037	40	48	29	29	NUM
cana-1037	40	49	,	,	PUNCT
cana-1037	40	50	47	47	NUM
cana-1037	40	51	,	,	PUNCT
cana-1037	40	52	76	76	NUM
cana-1037	40	53	,	,	PUNCT
cana-1037	40	54	123	123	NUM
cana-1037	40	55	,	,	PUNCT
cana-1037	40	56	and	and	CCONJ
cana-1037	40	57	199	199	NUM
cana-1037	40	58	make	make	VERB
cana-1037	40	59	up	up	ADP
cana-1037	40	60	the	the	DET
cana-1037	40	61	first	first	ADJ
cana-1037	40	62	twelve	twelve	NUM
cana-1037	40	63	lucas	lucas	NOUN
cana-1037	40	64	numbers	number	NOUN
cana-1037	40	65	.	.	PUNCT
cana-1037	41	1	binet	binet	PROPN
cana-1037	41	2	's	's	PART
cana-1037	41	3	formula	formula	NOUN
cana-1037	41	4	for	for	ADP
cana-1037	41	5	the	the	DET
cana-1037	41	6	lucas	lucas	NOUN
cana-1037	41	7	number	number	NOUN
cana-1037	41	8	𝐿𝑛	𝐿𝑛	PROPN
cana-1037	41	9	=	=	PRON
cana-1037	41	10	(	(	PUNCT
cana-1037	41	11	1	1	NUM
cana-1037	42	1	+	+	CCONJ
cana-1037	42	2	√5	√5	ADV
cana-1037	42	3	2	2	NUM
cana-1037	42	4	)	)	PUNCT
cana-1037	42	5	𝑛	𝑛	NOUN
cana-1037	43	1	+	+	CCONJ
cana-1037	43	2	(	(	PUNCT
cana-1037	43	3	1	1	NUM
cana-1037	43	4	−	−	NOUN
cana-1037	43	5	√5	√5	ADV
cana-1037	43	6	2	2	NUM
cana-1037	43	7	)	)	PUNCT
cana-1037	43	8	𝑛	𝑛	PRON
cana-1037	43	9	4	4	NUM
cana-1037	43	10	.	.	NOUN
cana-1037	43	11	jacobsthal	jacobsthal	ADJ
cana-1037	43	12	and	and	CCONJ
cana-1037	43	13	jacobsthal	jacobsthal	PROPN
cana-1037	43	14	lucas	lucas	PROPN
cana-1037	43	15	it	it	PRON
cana-1037	43	16	is	be	AUX
cana-1037	43	17	necessary	necessary	ADJ
cana-1037	43	18	to	to	PART
cana-1037	43	19	make	make	VERB
cana-1037	43	20	use	use	NOUN
cana-1037	43	21	of	of	ADP
cana-1037	43	22	the	the	DET
cana-1037	43	23	following	follow	VERB
cana-1037	43	24	equations	equation	NOUN
cana-1037	43	25	in	in	ADP
cana-1037	43	26	order	order	NOUN
cana-1037	43	27	to	to	PART
cana-1037	43	28	establish	establish	VERB
cana-1037	43	29	the	the	DET
cana-1037	43	30	second	second	ADJ
cana-1037	43	31	-	-	PUNCT
cana-1037	43	32	order	order	NOUN
cana-1037	43	33	recurrence	recurrence	NOUN
cana-1037	43	34	relations	relation	NOUN
cana-1037	43	35	and	and	CCONJ
cana-1037	43	36	initial	initial	ADJ
cana-1037	43	37	conditions	condition	NOUN
cana-1037	43	38	for	for	ADP
cana-1037	43	39	the	the	DET
cana-1037	43	40	jacobsthal	jacobsthal	ADJ
cana-1037	43	41	numbers	number	NOUN
cana-1037	43	42	,	,	PUNCT
cana-1037	43	43	jn	jn	PROPN
cana-1037	43	44	,	,	PUNCT
cana-1037	43	45	n	n	PROPN
cana-1037	43	46	>	>	X
cana-1037	43	47	0	0	NUM
cana-1037	43	48	,	,	PUNCT
cana-1037	43	49	as	as	ADV
cana-1037	43	50	well	well	ADV
cana-1037	43	51	as	as	ADP
cana-1037	43	52	the	the	DET
cana-1037	43	53	jl	jl	PROPN
cana-1037	43	54	numbers	number	NOUN
cana-1037	43	55	,	,	PUNCT
cana-1037	43	56	jn	jn	PROPN
cana-1037	43	57	,	,	PUNCT
cana-1037	43	58	n	n	PROPN
cana-1037	43	59	>	>	X
cana-1037	43	60	0	0	X
cana-1037	43	61	.	.	PUNCT
cana-1037	44	1	for	for	ADP
cana-1037	44	2	jacobsthal	jacobsthal	ADJ
cana-1037	44	3	numbers	number	NOUN
cana-1037	44	4	,	,	PUNCT
cana-1037	44	5	jn+2	jn+2	PROPN
cana-1037	44	6	=	=	SYM
cana-1037	44	7	jn+1	jn+1	PROPN
cana-1037	44	8	+	+	CCONJ
cana-1037	44	9	2jn	2jn	NOUN
cana-1037	44	10	,	,	PUNCT
cana-1037	44	11	j0	j0	PROPN
cana-1037	44	12	=	=	SYM
cana-1037	44	13	0	0	NUM
cana-1037	44	14	,	,	PUNCT
cana-1037	44	15	and	and	CCONJ
cana-1037	44	16	j1	j1	PROPN
cana-1037	44	17	=	=	NOUN
cana-1037	44	18	1	1	NUM
cana-1037	44	19	one	one	NUM
cana-1037	44	20	.	.	PUNCT
cana-1037	45	1	for	for	ADP
cana-1037	45	2	jl	jl	NOUN
cana-1037	45	3	numbers	number	NOUN
cana-1037	45	4	,	,	PUNCT
cana-1037	45	5	jn+2	jn+2	PROPN
cana-1037	45	6	=	=	SYM
cana-1037	45	7	jn+1	jn+1	PROPN
cana-1037	45	8	+	+	CCONJ
cana-1037	45	9	2jn	2jn	NOUN
cana-1037	45	10	,	,	PUNCT
cana-1037	45	11	j0	j0	PROPN
cana-1037	45	12	=	=	SYM
cana-1037	45	13	2	2	NUM
cana-1037	45	14	,	,	PUNCT
cana-1037	45	15	and	and	CCONJ
cana-1037	45	16	j1	j1	PROPN
cana-1037	45	17	=	=	SYM
cana-1037	45	18	1	1	X
cana-1037	45	19	.	.	PUNCT
cana-1037	46	1	communications	communication	NOUN
cana-1037	46	2	on	on	ADP
cana-1037	46	3	applied	apply	VERB
cana-1037	46	4	nonlinear	nonlinear	ADJ
cana-1037	46	5	analysis	analysis	NOUN
cana-1037	46	6	issn	issn	NOUN
cana-1037	46	7	:	:	PUNCT
cana-1037	46	8	1074	1074	NUM
cana-1037	46	9	-	-	PUNCT
cana-1037	46	10	133x	133x	NUM
cana-1037	46	11	vol	vol	NOUN
cana-1037	46	12	31	31	NUM
cana-1037	46	13	no	no	NOUN
cana-1037	46	14	.	.	PUNCT
cana-1037	47	1	5s	5s	NUM
cana-1037	47	2	(	(	PUNCT
cana-1037	47	3	2024	2024	NUM
cana-1037	47	4	)	)	PUNCT
cana-1037	47	5	291	291	NUM
cana-1037	47	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1037	48	1	the	the	DET
cana-1037	48	2	following	follow	VERB
cana-1037	48	3	is	be	AUX
cana-1037	48	4	a	a	DET
cana-1037	48	5	list	list	NOUN
cana-1037	48	6	of	of	ADP
cana-1037	48	7	the	the	DET
cana-1037	48	8	formulas	formula	NOUN
cana-1037	48	9	that	that	PRON
cana-1037	48	10	govern	govern	VERB
cana-1037	48	11	these	these	DET
cana-1037	48	12	sequences	sequence	NOUN
cana-1037	48	13	,	,	PUNCT
cana-1037	48	14	which	which	PRON
cana-1037	48	15	are	be	AUX
cana-1037	48	16	referred	refer	VERB
cana-1037	48	17	to	to	ADP
cana-1037	48	18	as	as	ADP
cana-1037	48	19	the	the	DET
cana-1037	48	20	binet	binet	NOUN
cana-1037	48	21	formulas	formula	NOUN
cana-1037	48	22	:	:	PUNCT
cana-1037	48	23	jn=	jn=	PROPN
cana-1037	48	24	2n	2n	NUM
cana-1037	48	25	(	(	PUNCT
cana-1037	48	26	-1)n	-1)n	NOUN
cana-1037	48	27	/	/	SYM
cana-1037	48	28	3	3	NUM
cana-1037	48	29	,	,	PUNCT
cana-1037	48	30	and	and	CCONJ
cana-1037	48	31	jn	jn	PROPN
cana-1037	48	32	=	=	PROPN
cana-1037	48	33	2n	2n	PROPN
cana-1037	49	1	+	+	CCONJ
cana-1037	49	2	(	(	PUNCT
cana-1037	49	3	-1)n	-1)n	NOUN
cana-1037	49	4	.	.	PUNCT
cana-1037	50	1	the	the	DET
cana-1037	50	2	following	follow	VERB
cana-1037	50	3	is	be	AUX
cana-1037	50	4	a	a	DET
cana-1037	50	5	list	list	NOUN
cana-1037	50	6	of	of	ADP
cana-1037	50	7	the	the	DET
cana-1037	50	8	roots	root	NOUN
cana-1037	50	9	of	of	ADP
cana-1037	50	10	a	a	DET
cana-1037	50	11	characteristic	characteristic	ADJ
cana-1037	50	12	equation	equation	NOUN
cana-1037	50	13	that	that	PRON
cana-1037	50	14	is	be	AUX
cana-1037	50	15	associated	associate	VERB
cana-1037	50	16	with	with	ADP
cana-1037	50	17	the	the	DET
cana-1037	50	18	values	value	NOUN
cana-1037	50	19	2	2	NUM
cana-1037	50	20	and	and	CCONJ
cana-1037	50	21	-1	-1	ADP
cana-1037	50	22	:	:	PUNCT
cana-1037	50	23	the	the	DET
cana-1037	50	24	recursive	recursive	ADJ
cana-1037	50	25	formulas	formula	NOUN
cana-1037	50	26	that	that	PRON
cana-1037	50	27	are	be	AUX
cana-1037	50	28	presented	present	VERB
cana-1037	50	29	here	here	ADV
cana-1037	50	30	are	be	AUX
cana-1037	50	31	as	as	SCONJ
cana-1037	50	32	follows	follow	VERB
cana-1037	50	33	:	:	PUNCT
cana-1037	50	34	if	if	SCONJ
cana-1037	50	35	j0	j0	PROPN
cana-1037	50	36	=	=	SYM
cana-1037	50	37	0	0	PROPN
cana-1037	50	38	and	and	CCONJ
cana-1037	50	39	j1=	j1=	PROPN
cana-1037	50	40	1	1	NUM
cana-1037	50	41	,	,	PUNCT
cana-1037	50	42	then	then	ADV
cana-1037	50	43	jn+2	jn+2	PROPN
cana-1037	50	44	=	=	SYM
cana-1037	50	45	jn+1	jn+1	PROPN
cana-1037	50	46	+	+	CCONJ
cana-1037	50	47	2jn	2jn	NOUN
cana-1037	50	48	.	.	PUNCT
cana-1037	51	1	on	on	ADP
cana-1037	51	2	the	the	DET
cana-1037	51	3	other	other	ADJ
cana-1037	51	4	hand	hand	NOUN
cana-1037	51	5	,	,	PUNCT
cana-1037	51	6	if	if	SCONJ
cana-1037	51	7	j0	j0	PROPN
cana-1037	51	8	=	=	SYM
cana-1037	51	9	2	2	NUM
cana-1037	51	10	and	and	CCONJ
cana-1037	51	11	j1	j1	PROPN
cana-1037	51	12	=	=	SYM
cana-1037	51	13	1	1	NUM
cana-1037	51	14	,	,	PUNCT
cana-1037	51	15	then	then	ADV
cana-1037	51	16	jn+2	jn+2	PROPN
cana-1037	51	17	=	=	SYM
cana-1037	51	18	jn+1	jn+1	PROPN
cana-1037	51	19	+2jn	+2jn	PROPN
cana-1037	51	20	.	.	PUNCT
cana-1037	52	1	a	a	DET
cana-1037	52	2	representation	representation	NOUN
cana-1037	52	3	of	of	ADP
cana-1037	52	4	these	these	DET
cana-1037	52	5	sequences	sequence	NOUN
cana-1037	52	6	can	can	AUX
cana-1037	52	7	be	be	AUX
cana-1037	52	8	found	find	VERB
cana-1037	52	9	in	in	ADP
cana-1037	52	10	the	the	DET
cana-1037	52	11	binet	binet	NOUN
cana-1037	52	12	formulas	formula	NOUN
cana-1037	52	13	,	,	PUNCT
cana-1037	52	14	which	which	PRON
cana-1037	52	15	are	be	AUX
cana-1037	52	16	as	as	SCONJ
cana-1037	52	17	follows	follow	VERB
cana-1037	52	18	:	:	PUNCT
cana-1037	52	19	a	a	DET
cana-1037	52	20	second	second	ADJ
cana-1037	52	21	way	way	NOUN
cana-1037	52	22	to	to	PART
cana-1037	52	23	write	write	VERB
cana-1037	52	24	jn=	jn=	PROPN
cana-1037	52	25	2n	2n	NUM
cana-1037	52	26	(	(	PUNCT
cana-1037	52	27	-1)n	-1)n	NOUN
cana-1037	52	28	/	/	SYM
cana-1037	52	29	3	3	NUM
cana-1037	52	30	,	,	PUNCT
cana-1037	52	31	and	and	CCONJ
cana-1037	52	32	the	the	DET
cana-1037	52	33	first	first	ADJ
cana-1037	52	34	way	way	NOUN
cana-1037	52	35	is	be	AUX
cana-1037	52	36	as	as	ADP
cana-1037	52	37	2n	2n	NUM
cana-1037	52	38	+	+	CCONJ
cana-1037	52	39	(	(	PUNCT
cana-1037	52	40	-1)n	-1)n	NOUN
cana-1037	52	41	/	/	SYM
cana-1037	52	42	3	3	NUM
cana-1037	52	43	.	.	X
cana-1037	52	44	formulas	formula	NOUN
cana-1037	52	45	like	like	ADP
cana-1037	52	46	this	this	PRON
cana-1037	52	47	are	be	AUX
cana-1037	52	48	derived	derive	VERB
cana-1037	52	49	from	from	ADP
cana-1037	52	50	the	the	DET
cana-1037	52	51	characteristic	characteristic	ADJ
cana-1037	52	52	equation	equation	NOUN
cana-1037	52	53	,	,	PUNCT
cana-1037	52	54	which	which	PRON
cana-1037	52	55	is	be	AUX
cana-1037	52	56	a	a	DET
cana-1037	52	57	mathematical	mathematical	ADJ
cana-1037	52	58	expression	expression	NOUN
cana-1037	52	59	that	that	PRON
cana-1037	52	60	describes	describe	VERB
cana-1037	52	61	the	the	DET
cana-1037	52	62	relationship	relationship	NOUN
cana-1037	52	63	between	between	ADP
cana-1037	52	64	the	the	DET
cana-1037	52	65	numbers	number	NOUN
cana-1037	52	66	2	2	NUM
cana-1037	52	67	and	and	CCONJ
cana-1037	52	68	-1	-1	PUNCT
cana-1037	52	69	(	(	PUNCT
cana-1037	52	70	campos	campos	PROPN
cana-1037	52	71	et	et	PROPN
cana-1037	52	72	al	al	PROPN
cana-1037	52	73	.	.	PROPN
cana-1037	52	74	,	,	PUNCT
cana-1037	52	75	2014	2014	NUM
cana-1037	52	76	)	)	PUNCT
cana-1037	52	77	.	.	PUNCT
cana-1037	53	1	identity	identity	NOUN
cana-1037	53	2	1.1	1.1	NUM
cana-1037	53	3	:	:	PUNCT
cana-1037	53	4	show	show	VERB
cana-1037	53	5	that	that	SCONJ
cana-1037	53	6	,	,	PUNCT
cana-1037	53	7	for	for	ADP
cana-1037	53	8	any	any	DET
cana-1037	53	9	number	number	NOUN
cana-1037	53	10	n≥0	n≥0	NOUN
cana-1037	53	11	,	,	PUNCT
cana-1037	53	12	|	|	ADV
cana-1037	53	13	ɉn+1	ɉn+1	NUM
cana-1037	53	14	ɉn+2	ɉn+2	NUM
cana-1037	53	15	ɉn+3	ɉn+3	NOUN
cana-1037	54	1	ɉn+4	ɉn+4	NUM
cana-1037	54	2	ɉn+5	ɉn+5	INTJ
cana-1037	54	3	ɉn+6	ɉn+6	PROPN
cana-1037	54	4	ɉn+7	ɉn+7	PROPN
cana-1037	54	5	ɉn+8	ɉn+8	PROPN
cana-1037	54	6	ɉn+9	ɉn+9	NOUN
cana-1037	54	7	|	|	NOUN
cana-1037	54	8	=	=	SYM
cana-1037	54	9	0	0	NUM
cana-1037	54	10	identity	identity	NOUN
cana-1037	54	11	1.2	1.2	NUM
cana-1037	54	12	:	:	PUNCT
cana-1037	54	13	show	show	VERB
cana-1037	54	14	that	that	SCONJ
cana-1037	54	15	,	,	PUNCT
cana-1037	54	16	for	for	ADP
cana-1037	54	17	any	any	DET
cana-1037	54	18	number	number	NOUN
cana-1037	54	19	n≥0	n≥0	NOUN
cana-1037	54	20	,	,	PUNCT
cana-1037	54	21	|	|	ADV
cana-1037	54	22	.	.	PUNCT
cana-1037	55	1	ɉn	ɉn	ADP
cana-1037	55	2	−	−	PROPN
cana-1037	55	3	ɉn+1	ɉn+1	NUM
cana-1037	55	4	ɉn+1	ɉn+1	NUM
cana-1037	55	5	−	−	NOUN
cana-1037	55	6	ɉn+2	ɉn+2	NUM
cana-1037	55	7	ɉn+2	ɉn+2	NUM
cana-1037	55	8	−	−	NOUN
cana-1037	55	9	ɉn	ɉn	ADP
cana-1037	55	10	ɉn+1	ɉn+1	NOUN
cana-1037	55	11	−	−	NOUN
cana-1037	55	12	ɉn+2	ɉn+2	NUM
cana-1037	55	13	ɉn+2	ɉn+2	NUM
cana-1037	55	14	−	−	NOUN
cana-1037	55	15	ɉn	ɉn	ADP
cana-1037	55	16	ɉn	ɉn	PUNCT
cana-1037	55	17	−	−	PROPN
cana-1037	55	18	ɉn+1	ɉn+1	NUM
cana-1037	55	19	ɉn+2	ɉn+2	NUM
cana-1037	55	20	−	−	PROPN
cana-1037	55	21	ɉn	ɉn	ADP
cana-1037	55	22	ɉn	ɉn	PUNCT
cana-1037	55	23	−	−	PROPN
cana-1037	55	24	ɉn+1	ɉn+1	NUM
cana-1037	55	25	ɉn+1	ɉn+1	NUM
cana-1037	55	26	−	−	NOUN
cana-1037	55	27	ɉn+2	ɉn+2	NUM
cana-1037	55	28	|	|	NOUN
cana-1037	55	29	=	=	SYM
cana-1037	55	30	0	0	NUM
cana-1037	55	31	identity1.3	identity1.3	NUM
cana-1037	55	32	:	:	PUNCT
cana-1037	55	33	show	show	VERB
cana-1037	55	34	that	that	SCONJ
cana-1037	55	35	,	,	PUNCT
cana-1037	55	36	for	for	ADP
cana-1037	55	37	any	any	DET
cana-1037	55	38	number	number	NOUN
cana-1037	55	39	n≥0	n≥0	NOUN
cana-1037	55	40	,	,	PUNCT
cana-1037	55	41	|	|	ADV
cana-1037	55	42	1	1	NUM
cana-1037	55	43	1	1	NUM
cana-1037	55	44	1	1	NUM
cana-1037	55	45	ɉn	ɉn	ADP
cana-1037	55	46	ɉn+1	ɉn+1	NUM
cana-1037	55	47	ɉn+2	ɉn+2	NUM
cana-1037	55	48	ɉn+1	ɉn+1	NUM
cana-1037	55	49	+	+	NUM
cana-1037	55	50	ɉn+2	ɉn+2	NUM
cana-1037	55	51	ɉn	ɉn	X
cana-1037	55	52	+	+	NUM
cana-1037	55	53	ɉn+2	ɉn+2	NUM
cana-1037	55	54	ɉn	ɉn	PRON
cana-1037	55	55	+	+	CCONJ
cana-1037	55	56	ɉn+1	ɉn+1	NOUN
cana-1037	55	57	|	|	NOUN
cana-1037	55	58	=	=	SYM
cana-1037	55	59	0	0	NUM
cana-1037	56	1	identity1.4	identity1.4	NOUN
cana-1037	56	2	:	:	PUNCT
cana-1037	56	3	show	show	VERB
cana-1037	56	4	that	that	SCONJ
cana-1037	56	5	,	,	PUNCT
cana-1037	56	6	for	for	ADP
cana-1037	56	7	any	any	DET
cana-1037	56	8	number	number	NOUN
cana-1037	56	9	n≥0	n≥0	NOUN
cana-1037	56	10	,	,	PUNCT
cana-1037	56	11	|	|	ADV
cana-1037	56	12	ɉn	ɉn	PRON
cana-1037	56	13	ɉn	ɉn	X
cana-1037	56	14	+	+	CCONJ
cana-1037	56	15	ɉn+1	ɉn+1	NOUN
cana-1037	56	16	ɉn	ɉn	X
cana-1037	56	17	+	+	CCONJ
cana-1037	56	18	ɉn+1	ɉn+1	NUM
cana-1037	56	19	+	+	NUM
cana-1037	56	20	ɉn+2	ɉn+2	NUM
cana-1037	56	21	2ɉn	2ɉn	ADJ
cana-1037	56	22	2ɉn	2ɉn	NOUN
cana-1037	56	23	+	+	CCONJ
cana-1037	56	24	3ɉn+1	3ɉn+1	NUM
cana-1037	56	25	2ɉn	2ɉn	NOUN
cana-1037	56	26	+	+	CCONJ
cana-1037	56	27	3	3	NUM
cana-1037	56	28	ɉn+1	ɉn+1	NUM
cana-1037	56	29	+	+	NUM
cana-1037	56	30	4ɉn+2	4ɉn+2	NUM
cana-1037	56	31	3ɉn	3ɉn	ADJ
cana-1037	56	32	3ɉn	3ɉn	NOUN
cana-1037	56	33	+	+	CCONJ
cana-1037	56	34	6ɉn+1	6ɉn+1	NUM
cana-1037	56	35	3ɉn	3ɉn	NOUN
cana-1037	56	36	+	+	CCONJ
cana-1037	56	37	6ɉn+1	6ɉn+1	NOUN
cana-1037	56	38	+	+	CCONJ
cana-1037	56	39	12ɉn+2	12ɉn+2	NUM
cana-1037	56	40	|	|	NOUN
cana-1037	56	41	=	=	SYM
cana-1037	56	42	3ɉnɉn+1ɉn+2	3ɉnɉn+1ɉn+2	NUM
cana-1037	56	43	identity1.5	identity1.5	NOUN
cana-1037	56	44	:	:	PUNCT
cana-1037	56	45	show	show	VERB
cana-1037	56	46	that	that	SCONJ
cana-1037	56	47	,	,	PUNCT
cana-1037	56	48	for	for	ADP
cana-1037	56	49	any	any	DET
cana-1037	56	50	number	number	NOUN
cana-1037	56	51	n≥0	n≥0	NOUN
cana-1037	56	52	,	,	PUNCT
cana-1037	56	53	|	|	ADV
cana-1037	56	54	0	0	SYM
cana-1037	56	55	ɉnɉn+1	ɉnɉn+1	ADJ
cana-1037	56	56	2	2	NUM
cana-1037	56	57	ɉnɉn+2	ɉnɉn+2	NOUN
cana-1037	56	58	2	2	NUM
cana-1037	56	59	ɉn	ɉn	ADP
cana-1037	56	60	2ɉn+1	2ɉn+1	NUM
cana-1037	56	61	0	0	NUM
cana-1037	56	62	ɉn+1ɉn+2	ɉn+1ɉn+2	NOUN
cana-1037	56	63	2	2	NUM
cana-1037	56	64	ɉn	ɉn	PART
cana-1037	56	65	2ɉn+2	2ɉn+2	NUM
cana-1037	56	66	ɉn+2ɉn+1	ɉn+2ɉn+1	NOUN
cana-1037	56	67	2	2	NUM
cana-1037	56	68	0	0	NUM
cana-1037	56	69	|	|	NOUN
cana-1037	56	70	=	=	SYM
cana-1037	57	1	2ɉn	2ɉn	NOUN
cana-1037	58	1	3ɉn+1	3ɉn+1	NUM
cana-1037	58	2	3	3	NUM
cana-1037	58	3	ɉn+2	ɉn+2	NUM
cana-1037	58	4	3	3	NUM
cana-1037	58	5	communications	communication	NOUN
cana-1037	58	6	on	on	ADP
cana-1037	58	7	applied	apply	VERB
cana-1037	58	8	nonlinear	nonlinear	ADJ
cana-1037	58	9	analysis	analysis	NOUN
cana-1037	58	10	issn	issn	NOUN
cana-1037	58	11	:	:	PUNCT
cana-1037	58	12	1074	1074	NUM
cana-1037	58	13	-	-	PUNCT
cana-1037	58	14	133x	133x	NUM
cana-1037	58	15	vol	vol	NOUN
cana-1037	58	16	31	31	NUM
cana-1037	58	17	no	no	NOUN
cana-1037	58	18	.	.	PUNCT
cana-1037	59	1	5s	5s	NUM
cana-1037	59	2	(	(	PUNCT
cana-1037	59	3	2024	2024	NUM
cana-1037	59	4	)	)	PUNCT
cana-1037	59	5	292	292	NUM
cana-1037	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1037	59	7	identity	identity	NOUN
cana-1037	59	8	1.6	1.6	NUM
cana-1037	59	9	:	:	PUNCT
cana-1037	59	10	|	|	ADV
cana-1037	59	11	2ɉn	2ɉn	ADJ
cana-1037	59	12	2ɉn	2ɉn	NOUN
cana-1037	59	13	1	1	NUM
cana-1037	59	14	ɉn+1	ɉn+1	NUM
cana-1037	59	15	ɉn+1	ɉn+1	NUM
cana-1037	59	16	1	1	NUM
cana-1037	59	17	ɉn+2	ɉn+2	NUM
cana-1037	59	18	ɉn+2	ɉn+2	NUM
cana-1037	59	19	1	1	NUM
cana-1037	59	20	|	|	NOUN
cana-1037	59	21	=	=	SYM
cana-1037	59	22	2[ɉnɉn+1	2[ɉnɉn+1	PROPN
cana-1037	60	1	−	−	ADP
cana-1037	60	2	ɉnɉn+1	ɉnɉn+1	ADJ
cana-1037	60	3	]	]	PUNCT
cana-1037	60	4	identity	identity	NOUN
cana-1037	60	5	1.7	1.7	NUM
cana-1037	60	6	:	:	PUNCT
cana-1037	60	7	|	|	ADV
cana-1037	60	8	ɉn	ɉn	X
cana-1037	60	9	ɉn	ɉn	X
cana-1037	60	10	ɉn	ɉn	X
cana-1037	60	11	+	+	X
cana-1037	60	12	ɉn	ɉn	PRON
cana-1037	60	13	ɉn+1	ɉn+1	NUM
cana-1037	60	14	ɉn+1	ɉn+1	X
cana-1037	60	15	ɉn+1	ɉn+1	NUM
cana-1037	60	16	+	+	CCONJ
cana-1037	60	17	ɉn+1	ɉn+1	NUM
cana-1037	60	18	ɉn+2	ɉn+2	NUM
cana-1037	60	19	ɉn+2	ɉn+2	NUM
cana-1037	60	20	ɉn+2	ɉn+2	NUM
cana-1037	61	1	+	+	NUM
cana-1037	61	2	ɉn+2	ɉn+2	NUM
cana-1037	61	3	|	|	ADV
cana-1037	61	4	=	=	SYM
cana-1037	61	5	0	0	NUM
cana-1037	61	6	identity	identity	NOUN
cana-1037	61	7	1.8	1.8	NUM
cana-1037	61	8	:	:	PUNCT
cana-1037	61	9	|	|	ADV
cana-1037	61	10	ɉn+1	ɉn+1	VERB
cana-1037	62	1	+	+	CCONJ
cana-1037	62	2	ɉn+2	ɉn+2	NUM
cana-1037	62	3	ɉn	ɉn	PRON
cana-1037	62	4	ɉn	ɉn	ADP
cana-1037	62	5	ɉn+1	ɉn+1	NUM
cana-1037	62	6	ɉn+2	ɉn+2	NUM
cana-1037	62	7	+	+	CCONJ
cana-1037	62	8	ɉn	ɉn	DET
cana-1037	62	9	ɉn+1	ɉn+1	VERB
cana-1037	62	10	+	+	CCONJ
cana-1037	62	11	ɉn+1	ɉn+1	NUM
cana-1037	62	12	ɉn+2	ɉn+2	NUM
cana-1037	62	13	ɉn+2	ɉn+2	NUM
cana-1037	62	14	ɉn	ɉn	PRON
cana-1037	62	15	+	+	CCONJ
cana-1037	62	16	ɉn+1	ɉn+1	NOUN
cana-1037	62	17	|	|	ADV
cana-1037	62	18	=	=	SYM
cana-1037	62	19	4ɉnɉn+1	4ɉnɉn+1	NOUN
cana-1037	62	20	ɉn+2	ɉn+2	NUM
cana-1037	62	21	identity1.9	identity1.9	VERB
cana-1037	62	22	:	:	PUNCT
cana-1037	62	23	show	show	VERB
cana-1037	62	24	that	that	SCONJ
cana-1037	62	25	,	,	PUNCT
cana-1037	62	26	for	for	ADP
cana-1037	62	27	any	any	DET
cana-1037	62	28	number	number	NOUN
cana-1037	62	29	n≥0	n≥0	NOUN
cana-1037	62	30	,	,	PUNCT
cana-1037	62	31	|	|	ADV
cana-1037	62	32	1	1	NUM
cana-1037	62	33	+	+	CCONJ
cana-1037	62	34	ɉn	ɉn	ADP
cana-1037	62	35	ɉn+1	ɉn+1	NUM
cana-1037	62	36	ɉn+2	ɉn+2	NUM
cana-1037	62	37	ɉn	ɉn	ADP
cana-1037	62	38	1	1	NUM
cana-1037	62	39	+	+	CCONJ
cana-1037	62	40	ɉn+1	ɉn+1	NUM
cana-1037	62	41	ɉn+2	ɉn+2	NUM
cana-1037	62	42	ɉn	ɉn	PART
cana-1037	62	43	ɉn+1	ɉn+1	NUM
cana-1037	62	44	1	1	NUM
cana-1037	62	45	+	+	CCONJ
cana-1037	62	46	ɉn+2	ɉn+2	NUM
cana-1037	62	47	|	|	ADV
cana-1037	62	48	=	=	NOUN
cana-1037	62	49	1	1	NUM
cana-1037	62	50	+	+	CCONJ
cana-1037	62	51	ɉn	ɉn	ADP
cana-1037	62	52	+	+	CCONJ
cana-1037	62	53	ɉn+1	ɉn+1	NUM
cana-1037	62	54	+	+	CCONJ
cana-1037	62	55	ɉn+2	ɉn+2	NUM
cana-1037	62	56	identity1.10	identity1.10	NOUN
cana-1037	62	57	:	:	PUNCT
cana-1037	62	58	show	show	VERB
cana-1037	62	59	that	that	SCONJ
cana-1037	62	60	,	,	PUNCT
cana-1037	62	61	for	for	ADP
cana-1037	62	62	any	any	DET
cana-1037	62	63	number	number	NOUN
cana-1037	62	64	n≥0	n≥0	NOUN
cana-1037	62	65	,	,	PUNCT
cana-1037	62	66	|	|	ADV
cana-1037	62	67	2	2	NUM
cana-1037	62	68	ɉn	ɉn	PRON
cana-1037	62	69	ɉn	ɉn	X
cana-1037	62	70	+	+	NOUN
cana-1037	62	71	ɉn+1	ɉn+1	NOUN
cana-1037	62	72	ɉn	ɉn	X
cana-1037	62	73	+	+	CCONJ
cana-1037	62	74	ɉn+1	ɉn+1	NUM
cana-1037	63	1	+	+	NUM
cana-1037	63	2	ɉn+2	ɉn+2	NUM
cana-1037	63	3	ɉn	ɉn	ADP
cana-1037	63	4	3ɉn	3ɉn	ADJ
cana-1037	63	5	+	+	CCONJ
cana-1037	63	6	2ɉn+1	2ɉn+1	NUM
cana-1037	64	1	4ɉn	4ɉn	NOUN
cana-1037	65	1	+	+	CCONJ
cana-1037	66	1	3ɉn+1	3ɉn+1	NUM
cana-1037	66	2	+	+	NUM
cana-1037	66	3	2ɉn+2	2ɉn+2	NUM
cana-1037	66	4	3ɉn	3ɉn	NOUN
cana-1037	66	5	6ɉn	6ɉn	NOUN
cana-1037	67	1	+	+	CCONJ
cana-1037	67	2	3ɉn+1	3ɉn+1	NUM
cana-1037	67	3	10ɉn	10ɉn	NOUN
cana-1037	68	1	+	+	CCONJ
cana-1037	68	2	6ɉn+1	6ɉn+1	NOUN
cana-1037	68	3	+	+	CCONJ
cana-1037	69	1	3ɉn+2	3ɉn+2	CCONJ
cana-1037	69	2	|	|	ADV
cana-1037	70	1	=	=	NOUN
cana-1037	70	2	ɉ𝑛	ɉ𝑛	NOUN
cana-1037	70	3	3	3	NUM
cana-1037	70	4	5	5	NUM
cana-1037	70	5	.	.	PUNCT
cana-1037	70	6	jacobsthal	jacobsthal	PROPN
cana-1037	70	7	lucas	lucas	PROPN
cana-1037	70	8	numbers	number	NOUN
cana-1037	70	9	jn+2	jn+2	PROPN
cana-1037	70	10	=	=	SYM
cana-1037	70	11	jn+1	jn+1	PROPN
cana-1037	71	1	+	+	CCONJ
cana-1037	71	2	2jn	2jn	NOUN
cana-1037	71	3	is	be	AUX
cana-1037	71	4	the	the	DET
cana-1037	71	5	recurrence	recurrence	NOUN
cana-1037	71	6	relation	relation	NOUN
cana-1037	71	7	that	that	PRON
cana-1037	71	8	defines	define	VERB
cana-1037	71	9	the	the	DET
cana-1037	71	10	jl	jl	PROPN
cana-1037	71	11	numbers	number	NOUN
cana-1037	71	12	.	.	PUNCT
cana-1037	72	1	in	in	ADP
cana-1037	72	2	this	this	DET
cana-1037	72	3	equation	equation	NOUN
cana-1037	72	4	,	,	PUNCT
cana-1037	72	5	n	n	PRON
cana-1037	72	6	is	be	AUX
cana-1037	72	7	a	a	DET
cana-1037	72	8	number	number	NOUN
cana-1037	72	9	that	that	PRON
cana-1037	72	10	is	be	AUX
cana-1037	72	11	higher	high	ADJ
cana-1037	72	12	than	than	ADP
cana-1037	72	13	or	or	CCONJ
cana-1037	72	14	equal	equal	ADJ
cana-1037	72	15	to	to	ADP
cana-1037	72	16	zero	zero	NUM
cana-1037	72	17	,	,	PUNCT
cana-1037	72	18	and	and	CCONJ
cana-1037	72	19	the	the	DET
cana-1037	72	20	beginning	beginning	ADJ
cana-1037	72	21	values	value	NOUN
cana-1037	72	22	are	be	AUX
cana-1037	72	23	j0	j0	PROPN
cana-1037	72	24	=	=	SYM
cana-1037	72	25	2	2	NUM
cana-1037	72	26	and	and	CCONJ
cana-1037	72	27	j1	j1	PROPN
cana-1037	72	28	=	=	SYM
cana-1037	72	29	1	1	X
cana-1037	72	30	.	.	X
cana-1037	73	1	for	for	ADP
cana-1037	73	2	example	example	NOUN
cana-1037	73	3	,	,	PUNCT
cana-1037	73	4	2	2	NUM
cana-1037	73	5	,	,	PUNCT
cana-1037	73	6	1	1	NUM
cana-1037	73	7	,	,	PUNCT
cana-1037	73	8	5	5	NUM
cana-1037	73	9	,	,	PUNCT
cana-1037	73	10	7	7	NUM
cana-1037	73	11	,	,	PUNCT
cana-1037	73	12	17	17	NUM
cana-1037	73	13	,	,	PUNCT
cana-1037	73	14	31	31	NUM
cana-1037	73	15	,	,	PUNCT
cana-1037	73	16	65	65	NUM
cana-1037	73	17	,	,	PUNCT
cana-1037	73	18	127	127	NUM
cana-1037	73	19	,	,	PUNCT
cana-1037	73	20	257	257	NUM
cana-1037	73	21	...	...	PUNCT
cana-1037	73	22	jl	jl	NOUN
cana-1037	73	23	numbers	number	NOUN
cana-1037	73	24	were	be	AUX
cana-1037	73	25	the	the	DET
cana-1037	73	26	subject	subject	NOUN
cana-1037	73	27	of	of	ADP
cana-1037	73	28	discussion	discussion	NOUN
cana-1037	73	29	in	in	ADP
cana-1037	73	30	this	this	DET
cana-1037	73	31	chapter	chapter	NOUN
cana-1037	73	32	,	,	PUNCT
cana-1037	73	33	where	where	SCONJ
cana-1037	73	34	we	we	PRON
cana-1037	73	35	talked	talk	VERB
cana-1037	73	36	about	about	ADP
cana-1037	73	37	certain	certain	ADJ
cana-1037	73	38	determinants	determinant	NOUN
cana-1037	73	39	and	and	CCONJ
cana-1037	73	40	results	result	NOUN
cana-1037	73	41	.	.	PUNCT
cana-1037	74	1	identity1.11	identity1.11	NOUN
cana-1037	74	2	:	:	PUNCT
cana-1037	74	3	for	for	ADP
cana-1037	74	4	each	each	DET
cana-1037	74	5	integer𝑛	integer𝑛	NOUN
cana-1037	74	6	≥	≥	X
cana-1037	74	7	0	0	NUM
cana-1037	74	8	prove	prove	VERB
cana-1037	74	9	that	that	SCONJ
cana-1037	74	10	|	|	ADV
cana-1037	74	11	𝔧n	𝔧n	ADV
cana-1037	74	12	𝔧n+1	𝔧n+1	PRON
cana-1037	74	13	𝔧n+2	𝔧n+2	NOUN
cana-1037	74	14	𝔧n+3	𝔧n+3	NOUN
cana-1037	74	15	𝔧n+4	𝔧n+4	INTJ
cana-1037	74	16	𝔧n+5	𝔧n+5	NUM
cana-1037	74	17	𝔧n+6	𝔧n+6	NUM
cana-1037	74	18	𝔧n+7	𝔧n+7	NOUN
cana-1037	74	19	𝔧n+8	𝔧n+8	NOUN
cana-1037	74	20	|	|	ADV
cana-1037	74	21	=	=	SYM
cana-1037	74	22	0	0	NUM
cana-1037	74	23	communications	communication	NOUN
cana-1037	74	24	on	on	ADP
cana-1037	74	25	applied	apply	VERB
cana-1037	74	26	nonlinear	nonlinear	ADJ
cana-1037	74	27	analysis	analysis	NOUN
cana-1037	74	28	issn	issn	NOUN
cana-1037	74	29	:	:	PUNCT
cana-1037	74	30	1074	1074	NUM
cana-1037	74	31	-	-	PUNCT
cana-1037	74	32	133x	133x	NUM
cana-1037	74	33	vol	vol	NOUN
cana-1037	74	34	31	31	NUM
cana-1037	74	35	no	no	NOUN
cana-1037	74	36	.	.	PUNCT
cana-1037	75	1	5s	5s	NUM
cana-1037	75	2	(	(	PUNCT
cana-1037	75	3	2024	2024	NUM
cana-1037	75	4	)	)	PUNCT
cana-1037	75	5	293	293	NUM
cana-1037	75	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1037	75	7	identity1.12	identity1.12	NOUN
cana-1037	75	8	:	:	PUNCT
cana-1037	75	9	show	show	VERB
cana-1037	75	10	that	that	SCONJ
cana-1037	75	11	,	,	PUNCT
cana-1037	75	12	for	for	ADP
cana-1037	75	13	any	any	DET
cana-1037	75	14	number	number	NOUN
cana-1037	75	15	n≥0	n≥0	NOUN
cana-1037	75	16	,	,	PUNCT
cana-1037	75	17	for	for	ADP
cana-1037	75	18	each	each	DET
cana-1037	75	19	integer	integer	NOUN
cana-1037	75	20	n	n	PRON
cana-1037	75	21	≥	≥	NOUN
cana-1037	75	22	0	0	NUM
cana-1037	76	1	|	|	ADV
cana-1037	76	2	𝔧n	𝔧n	ADV
cana-1037	76	3	−	−	PROPN
cana-1037	76	4	𝔧n+1	𝔧n+1	NUM
cana-1037	76	5	𝔧n+1	𝔧n+1	NOUN
cana-1037	76	6	−	−	NOUN
cana-1037	76	7	𝔧n+2	𝔧n+2	INTJ
cana-1037	76	8	𝔧n+2	𝔧n+2	VERB
cana-1037	76	9	−	−	PROPN
cana-1037	76	10	𝔧n	𝔧n	ADP
cana-1037	76	11	𝔧n+1	𝔧n+1	NOUN
cana-1037	76	12	−	−	NOUN
cana-1037	76	13	𝔧n+2	𝔧n+2	INTJ
cana-1037	76	14	𝔧n+2	𝔧n+2	NOUN
cana-1037	76	15	−	−	NOUN
cana-1037	76	16	𝔧n	𝔧n	ADV
cana-1037	76	17	𝔧n	𝔧n	ADP
cana-1037	76	18	−	−	PROPN
cana-1037	76	19	𝔧n+1	𝔧n+1	NUM
cana-1037	76	20	𝔧n+2	𝔧n+2	NOUN
cana-1037	76	21	−	−	NOUN
cana-1037	76	22	𝔧n	𝔧n	ADV
cana-1037	76	23	𝔧n	𝔧n	ADP
cana-1037	76	24	−	−	PROPN
cana-1037	76	25	𝔧n+1	𝔧n+1	NUM
cana-1037	76	26	𝔧n+1	𝔧n+1	NOUN
cana-1037	76	27	−	−	NOUN
cana-1037	76	28	𝔧n+2	𝔧n+2	ADV
cana-1037	76	29	|	|	ADV
cana-1037	76	30	=	=	SYM
cana-1037	76	31	0	0	NUM
cana-1037	77	1	identity	identity	NOUN
cana-1037	77	2	1.13	1.13	NUM
cana-1037	77	3	:	:	PUNCT
cana-1037	77	4	show	show	VERB
cana-1037	77	5	that	that	SCONJ
cana-1037	77	6	,	,	PUNCT
cana-1037	77	7	for	for	ADP
cana-1037	77	8	any	any	DET
cana-1037	77	9	number	number	NOUN
cana-1037	77	10	n≥0	n≥0	NOUN
cana-1037	77	11	,	,	PUNCT
cana-1037	77	12	for	for	ADP
cana-1037	77	13	each	each	DET
cana-1037	77	14	integer	integer	NOUN
cana-1037	77	15	𝑛	𝑛	PRON
cana-1037	77	16	≥	≥	NUM
cana-1037	77	17	0	0	NUM
cana-1037	77	18	prove	prove	VERB
cana-1037	77	19	that	that	SCONJ
cana-1037	77	20	|	|	ADV
cana-1037	77	21	1	1	NUM
cana-1037	77	22	1	1	NUM
cana-1037	77	23	1	1	NUM
cana-1037	77	24	𝔧n	𝔧n	ADV
cana-1037	77	25	𝔧n+1	𝔧n+1	NUM
cana-1037	77	26	𝔧n+2	𝔧n+2	NOUN
cana-1037	77	27	𝔧n+1	𝔧n+1	VERB
cana-1037	77	28	+	+	SYM
cana-1037	77	29	𝔧n+2	𝔧n+2	X
cana-1037	77	30	𝔧n	𝔧n	ADV
cana-1037	77	31	+	+	CCONJ
cana-1037	77	32	𝔧n+2	𝔧n+2	X
cana-1037	77	33	𝔧n	𝔧n	CCONJ
cana-1037	77	34	+	+	CCONJ
cana-1037	77	35	𝔧n+1	𝔧n+1	NUM
cana-1037	77	36	|	|	ADV
cana-1037	77	37	=	=	SYM
cana-1037	77	38	0	0	NUM
cana-1037	77	39	identity1.14	identity1.14	NOUN
cana-1037	77	40	:	:	PUNCT
cana-1037	77	41	show	show	VERB
cana-1037	77	42	that	that	SCONJ
cana-1037	77	43	,	,	PUNCT
cana-1037	77	44	for	for	ADP
cana-1037	77	45	any	any	DET
cana-1037	77	46	number	number	NOUN
cana-1037	77	47	n≥0	n≥0	NOUN
cana-1037	77	48	,	,	PUNCT
cana-1037	77	49	|	|	ADV
cana-1037	77	50	𝔧n	𝔧n	ADV
cana-1037	77	51	𝔧n	𝔧n	ADV
cana-1037	77	52	+	+	CCONJ
cana-1037	77	53	𝔧n+1	𝔧n+1	NUM
cana-1037	77	54	𝔧n	𝔧n	ADP
cana-1037	77	55	+	+	CCONJ
cana-1037	77	56	𝔧n+1	𝔧n+1	NUM
cana-1037	77	57	+	+	NUM
cana-1037	77	58	𝔧n+2	𝔧n+2	NOUN
cana-1037	77	59	2𝔧n	2𝔧n	NOUN
cana-1037	77	60	2𝔧n	2𝔧n	NOUN
cana-1037	78	1	+	+	CCONJ
cana-1037	79	1	3𝔧n+1	3𝔧n+1	NUM
cana-1037	79	2	2𝔧n	2𝔧n	NOUN
cana-1037	79	3	+	+	CCONJ
cana-1037	80	1	3𝔧n+1	3𝔧n+1	NUM
cana-1037	80	2	+	+	NUM
cana-1037	80	3	4𝔧n+2	4𝔧n+2	NUM
cana-1037	80	4	3𝔧n	3𝔧n	ADJ
cana-1037	80	5	3𝔧n	3𝔧n	NOUN
cana-1037	80	6	+	+	CCONJ
cana-1037	80	7	6𝔧n+1	6𝔧n+1	NUM
cana-1037	80	8	3𝔧n	3𝔧n	ADJ
cana-1037	80	9	+	+	CCONJ
cana-1037	80	10	6𝔧n+1	6𝔧n+1	NOUN
cana-1037	81	1	+	+	CCONJ
cana-1037	81	2	12𝔧n+2	12𝔧n+2	NUM
cana-1037	82	1	|	|	NOUN
cana-1037	83	1	=	=	SYM
cana-1037	84	1	3𝔧n𝔧n+1𝔧n+2	3𝔧n𝔧n+1𝔧n+2	NUM
cana-1037	84	2	identity	identity	NOUN
cana-1037	84	3	1.15	1.15	NUM
cana-1037	84	4	:	:	PUNCT
cana-1037	84	5	show	show	VERB
cana-1037	84	6	that	that	SCONJ
cana-1037	84	7	,	,	PUNCT
cana-1037	84	8	for	for	ADP
cana-1037	84	9	any	any	DET
cana-1037	84	10	number	number	NOUN
cana-1037	84	11	n≥0	n≥0	NOUN
cana-1037	84	12	,	,	PUNCT
cana-1037	84	13	|	|	ADV
cana-1037	84	14	0	0	SYM
cana-1037	84	15	𝔧n𝔧n+1	𝔧n𝔧n+1	PROPN
cana-1037	84	16	2	2	NUM
cana-1037	84	17	𝔧n𝔧n+2	𝔧n𝔧n+2	NOUN
cana-1037	84	18	2	2	NUM
cana-1037	84	19	𝔧n	𝔧n	ADP
cana-1037	84	20	2𝔧n+1	2𝔧n+1	NUM
cana-1037	84	21	0	0	NUM
cana-1037	84	22	𝔧n+1𝔧n+2	𝔧n+1𝔧n+2	NOUN
cana-1037	84	23	2	2	NUM
cana-1037	84	24	𝔧n	𝔧n	ADP
cana-1037	84	25	2𝔧n+2	2𝔧n+2	NUM
cana-1037	84	26	𝔧n+2𝔧n+1	𝔧n+2𝔧n+1	NOUN
cana-1037	84	27	2	2	NUM
cana-1037	84	28	0	0	NUM
cana-1037	85	1	|	|	NOUN
cana-1037	85	2	=	=	SYM
cana-1037	85	3	2𝔧𝑛	2𝔧𝑛	ADJ
cana-1037	85	4	3	3	NUM
cana-1037	85	5	𝔧𝑛+1	𝔧𝑛+1	SYM
cana-1037	85	6	3	3	NUM
cana-1037	85	7	𝔧𝑛+2	𝔧𝑛+2	SYM
cana-1037	85	8	3	3	NUM
cana-1037	85	9	identity1.16	identity1.16	NOUN
cana-1037	85	10	:	:	PUNCT
cana-1037	85	11	show	show	VERB
cana-1037	85	12	that	that	SCONJ
cana-1037	85	13	,	,	PUNCT
cana-1037	85	14	for	for	ADP
cana-1037	85	15	any	any	DET
cana-1037	85	16	number	number	NOUN
cana-1037	85	17	n≥0	n≥0	NOUN
cana-1037	85	18	,	,	PUNCT
cana-1037	85	19	|	|	ADV
cana-1037	85	20	𝔧n+1	𝔧n+1	PUNCT
cana-1037	85	21	+	+	SYM
cana-1037	85	22	𝔧n+2	𝔧n+2	X
cana-1037	85	23	𝔧n	𝔧n	ADV
cana-1037	85	24	𝔧n	𝔧n	ADP
cana-1037	85	25	𝔧n+1	𝔧n+1	DET
cana-1037	85	26	𝔧n+2	𝔧n+2	NOUN
cana-1037	85	27	+	+	CCONJ
cana-1037	85	28	𝔧n	𝔧n	ADP
cana-1037	85	29	𝔧n+1	𝔧n+1	NUM
cana-1037	85	30	𝔧n+2	𝔧n+2	NOUN
cana-1037	85	31	𝔧n+2	𝔧n+2	X
cana-1037	85	32	𝔧n	𝔧n	CCONJ
cana-1037	85	33	+	+	CCONJ
cana-1037	85	34	𝔧n+1	𝔧n+1	NUM
cana-1037	85	35	|	|	ADV
cana-1037	85	36	=	=	SYM
cana-1037	85	37	4𝔧n𝔧n+1𝔧n+2	4𝔧n𝔧n+1𝔧n+2	NUM
cana-1037	85	38	identity	identity	NOUN
cana-1037	85	39	1.17	1.17	NUM
cana-1037	85	40	:	:	PUNCT
cana-1037	85	41	show	show	VERB
cana-1037	85	42	that	that	SCONJ
cana-1037	85	43	,	,	PUNCT
cana-1037	85	44	for	for	ADP
cana-1037	85	45	any	any	DET
cana-1037	85	46	number	number	NOUN
cana-1037	85	47	n≥0	n≥0	NOUN
cana-1037	85	48	,	,	PUNCT
cana-1037	85	49	|	|	ADV
cana-1037	85	50	1	1	NUM
cana-1037	85	51	+	+	CCONJ
cana-1037	85	52	𝔧n	𝔧n	ADV
cana-1037	85	53	𝔧n	𝔧n	ADV
cana-1037	85	54	𝔧n+2	𝔧n+2	ADJ
cana-1037	85	55	𝔧n	𝔧n	ADV
cana-1037	85	56	1	1	NUM
cana-1037	85	57	+	+	NUM
cana-1037	85	58	𝔧n+1	𝔧n+1	NUM
cana-1037	85	59	𝔧n+2	𝔧n+2	X
cana-1037	85	60	𝔧n	𝔧n	ADV
cana-1037	85	61	𝔧n+1	𝔧n+1	PRON
cana-1037	85	62	1	1	NUM
cana-1037	85	63	+	+	CCONJ
cana-1037	85	64	𝔧n+2	𝔧n+2	AUX
cana-1037	85	65	|	|	ADV
cana-1037	85	66	=	=	SYM
cana-1037	85	67	1+𝔧n	1+𝔧n	NUM
cana-1037	85	68	+	+	NUM
cana-1037	85	69	𝔧n+1	𝔧n+1	NUM
cana-1037	85	70	+	+	NUM
cana-1037	85	71	𝔧n+2	𝔧n+2	VERB
cana-1037	85	72	communications	communication	NOUN
cana-1037	85	73	on	on	ADP
cana-1037	85	74	applied	apply	VERB
cana-1037	85	75	nonlinear	nonlinear	ADJ
cana-1037	85	76	analysis	analysis	NOUN
cana-1037	85	77	issn	issn	NOUN
cana-1037	85	78	:	:	PUNCT
cana-1037	85	79	1074	1074	NUM
cana-1037	85	80	-	-	PUNCT
cana-1037	85	81	133x	133x	NUM
cana-1037	85	82	vol	vol	NOUN
cana-1037	85	83	31	31	NUM
cana-1037	85	84	no	no	NOUN
cana-1037	85	85	.	.	PUNCT
cana-1037	86	1	5s	5s	NUM
cana-1037	86	2	(	(	PUNCT
cana-1037	86	3	2024	2024	NUM
cana-1037	86	4	)	)	PUNCT
cana-1037	86	5	294	294	NUM
cana-1037	86	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1037	86	7	identity	identity	NOUN
cana-1037	86	8	1.18	1.18	NUM
cana-1037	86	9	:	:	PUNCT
cana-1037	86	10	show	show	VERB
cana-1037	86	11	that	that	SCONJ
cana-1037	86	12	,	,	PUNCT
cana-1037	86	13	for	for	ADP
cana-1037	86	14	any	any	DET
cana-1037	86	15	number	number	NOUN
cana-1037	86	16	n≥0	n≥0	NOUN
cana-1037	86	17	,	,	PUNCT
cana-1037	86	18	|	|	ADV
cana-1037	86	19	𝔧n	𝔧n	ADV
cana-1037	86	20	𝔧n	𝔧n	ADV
cana-1037	86	21	+	+	CCONJ
cana-1037	86	22	𝔧n+1	𝔧n+1	NUM
cana-1037	86	23	𝔧n	𝔧n	ADP
cana-1037	86	24	+	+	CCONJ
cana-1037	86	25	𝔧n+1	𝔧n+1	NUM
cana-1037	86	26	+	+	SYM
cana-1037	86	27	𝔧n+2	𝔧n+2	NOUN
cana-1037	86	28	2𝔧n	2𝔧n	ADJ
cana-1037	86	29	3𝔧n	3𝔧n	NOUN
cana-1037	86	30	+	+	CCONJ
cana-1037	86	31	2𝔧n+1	2𝔧n+1	NUM
cana-1037	86	32	4𝔧n	4𝔧n	NOUN
cana-1037	86	33	+	+	CCONJ
cana-1037	86	34	3𝔧n+1	3𝔧n+1	NUM
cana-1037	86	35	+	+	CCONJ
cana-1037	86	36	2𝔧n+2	2𝔧n+2	NUM
cana-1037	86	37	3𝔧n	3𝔧n	NOUN
cana-1037	86	38	6𝔧n	6𝔧n	ADJ
cana-1037	87	1	+	+	CCONJ
cana-1037	88	1	3𝔧n+1	3𝔧n+1	NUM
cana-1037	88	2	10𝔧n	10𝔧n	ADJ
cana-1037	88	3	+	+	CCONJ
cana-1037	88	4	6𝔧n+1	6𝔧n+1	NOUN
cana-1037	88	5	+	+	NUM
cana-1037	88	6	3𝔧n+2	3𝔧n+2	NUM
cana-1037	89	1	|	|	ADV
cana-1037	89	2	=	=	SYM
cana-1037	89	3	𝔧𝑛	𝔧𝑛	PART
cana-1037	89	4	3	3	NUM
cana-1037	89	5	identity1.19	identity1.19	NOUN
cana-1037	89	6	:	:	PUNCT
cana-1037	89	7	show	show	VERB
cana-1037	89	8	that	that	SCONJ
cana-1037	89	9	,	,	PUNCT
cana-1037	89	10	for	for	ADP
cana-1037	89	11	any	any	DET
cana-1037	89	12	number	number	NOUN
cana-1037	89	13	n≥0	n≥0	NOUN
cana-1037	89	14	,	,	PUNCT
cana-1037	89	15	|	|	ADV
cana-1037	89	16	𝔧n	𝔧n	ADV
cana-1037	89	17	−	−	PROPN
cana-1037	89	18	𝔧n+1	𝔧n+1	NUM
cana-1037	89	19	−	−	NOUN
cana-1037	89	20	𝔧n+2	𝔧n+2	PART
cana-1037	89	21	2𝔧n	2𝔧n	NOUN
cana-1037	89	22	2𝔧n	2𝔧n	NOUN
cana-1037	89	23	2𝔧n+1	2𝔧n+1	NUM
cana-1037	89	24	𝔧n+1	𝔧n+1	NUM
cana-1037	89	25	−	−	NOUN
cana-1037	89	26	𝔧n+2	𝔧n+2	NOUN
cana-1037	89	27	−	−	PUNCT
cana-1037	89	28	𝔧n	𝔧n	ADP
cana-1037	89	29	2𝔧n+1	2𝔧n+1	NUM
cana-1037	90	1	2𝔧n+2	2𝔧n+2	NUM
cana-1037	91	1	2𝔧n+2	2𝔧n+2	NUM
cana-1037	91	2	𝔧n+2	𝔧n+2	NOUN
cana-1037	91	3	−	−	PUNCT
cana-1037	91	4	𝔧n	𝔧n	ADP
cana-1037	91	5	−	−	PROPN
cana-1037	91	6	𝔧n+1	𝔧n+1	NUM
cana-1037	91	7	|	|	ADV
cana-1037	91	8	=	=	SYM
cana-1037	91	9	(	(	PUNCT
cana-1037	91	10	𝔧n	𝔧n	X
cana-1037	91	11	+	+	CCONJ
cana-1037	91	12	𝔧n+1	𝔧n+1	NUM
cana-1037	91	13	+	+	NUM
cana-1037	91	14	𝔧n+2)3	𝔧n+2)3	PUNCT
cana-1037	91	15	identity	identity	NOUN
cana-1037	91	16	1.20	1.20	NUM
cana-1037	91	17	:	:	PUNCT
cana-1037	91	18	show	show	VERB
cana-1037	91	19	that	that	SCONJ
cana-1037	91	20	,	,	PUNCT
cana-1037	91	21	for	for	ADP
cana-1037	91	22	any	any	DET
cana-1037	91	23	number	number	NOUN
cana-1037	91	24	n≥0	n≥0	NOUN
cana-1037	91	25	,	,	PUNCT
cana-1037	91	26	|	|	ADV
cana-1037	91	27	1	1	NUM
cana-1037	91	28	𝔧n	𝔧n	ADP
cana-1037	91	29	𝔧n	𝔧n	ADV
cana-1037	91	30	2	2	NUM
cana-1037	91	31	𝔧n	𝔧n	ADP
cana-1037	91	32	2	2	NUM
cana-1037	91	33	1	1	NUM
cana-1037	91	34	𝔧n	𝔧n	ADP
cana-1037	91	35	𝔧n	𝔧n	ADV
cana-1037	91	36	𝔧n	𝔧n	ADV
cana-1037	91	37	2	2	NUM
cana-1037	91	38	1	1	NUM
cana-1037	91	39	|	|	ADV
cana-1037	91	40	=	=	PUNCT
cana-1037	91	41	(	(	PUNCT
cana-1037	91	42	1	1	NUM
cana-1037	91	43	−	−	NUM
cana-1037	91	44	𝔧𝑛	𝔧𝑛	PART
cana-1037	91	45	3	3	NUM
cana-1037	91	46	)	)	SYM
cana-1037	91	47	2	2	NUM
cana-1037	91	48	6	6	NUM
cana-1037	91	49	.	.	PUNCT
cana-1037	91	50	j	j	PROPN
cana-1037	91	51	&	&	CCONJ
cana-1037	91	52	jl	jl	PROPN
cana-1037	91	53	numbers	number	NOUN
cana-1037	91	54	using	use	VERB
cana-1037	91	55	of	of	ADP
cana-1037	91	56	the	the	DET
cana-1037	91	57	matrix	matrix	NOUN
cana-1037	91	58	method	method	NOUN
cana-1037	91	59	,	,	PUNCT
cana-1037	91	60	the	the	DET
cana-1037	91	61	jacobsthal	jacobsthal	ADJ
cana-1037	91	62	sequence	sequence	NOUN
cana-1037	91	63	jn	jn	PROPN
cana-1037	91	64	is	be	AUX
cana-1037	91	65	comprised	comprise	VERB
cana-1037	91	66	of	of	ADP
cana-1037	91	67	fifteen	fifteen	NUM
cana-1037	91	68	different	different	ADJ
cana-1037	91	69	elements	element	NOUN
cana-1037	91	70	,	,	PUNCT
cana-1037	91	71	which	which	PRON
cana-1037	91	72	are	be	AUX
cana-1037	91	73	as	as	SCONJ
cana-1037	91	74	follows	follow	VERB
cana-1037	91	75	:	:	PUNCT
cana-1037	91	76	0	0	NUM
cana-1037	91	77	,	,	PUNCT
cana-1037	91	78	1	1	NUM
cana-1037	91	79	,	,	PUNCT
cana-1037	91	80	3	3	NUM
cana-1037	91	81	,	,	PUNCT
cana-1037	91	82	5	5	NUM
cana-1037	91	83	,	,	PUNCT
cana-1037	91	84	11	11	NUM
cana-1037	91	85	,	,	PUNCT
cana-1037	91	86	21	21	NUM
cana-1037	91	87	,	,	PUNCT
cana-1037	91	88	43	43	NUM
cana-1037	91	89	,	,	PUNCT
cana-1037	91	90	171	171	NUM
cana-1037	91	91	,	,	PUNCT
cana-1037	91	92	341	341	NUM
cana-1037	91	93	,	,	PUNCT
cana-1037	91	94	683	683	NUM
cana-1037	91	95	,	,	PUNCT
cana-1037	91	96	1365	1365	NUM
cana-1037	91	97	,	,	PUNCT
cana-1037	91	98	2731	2731	NUM
cana-1037	91	99	,	,	PUNCT
cana-1037	91	100	5461	5461	NUM
cana-1037	91	101	...	...	PUNCT
cana-1037	92	1	jn	jn	PROPN
cana-1037	92	2	=	=	PROPN
cana-1037	92	3	2n	2n	NUM
cana-1037	92	4	(	(	PUNCT
cana-1037	92	5	-1)n	-1)n	NOUN
cana-1037	92	6	/	/	SYM
cana-1037	92	7	3	3	NUM
cana-1037	92	8	is	be	AUX
cana-1037	92	9	one	one	NUM
cana-1037	92	10	mathematical	mathematical	ADJ
cana-1037	92	11	representation	representation	NOUN
cana-1037	92	12	of	of	ADP
cana-1037	92	13	the	the	DET
cana-1037	92	14	sequence	sequence	NOUN
cana-1037	92	15	that	that	PRON
cana-1037	92	16	can	can	AUX
cana-1037	92	17	be	be	AUX
cana-1037	92	18	used	use	VERB
cana-1037	92	19	to	to	PART
cana-1037	92	20	represent	represent	VERB
cana-1037	92	21	it	it	PRON
cana-1037	92	22	.	.	PUNCT
cana-1037	93	1	the	the	DET
cana-1037	93	2	first	first	ADJ
cana-1037	93	3	ten	ten	NUM
cana-1037	93	4	terms	term	NOUN
cana-1037	93	5	in	in	ADP
cana-1037	93	6	the	the	DET
cana-1037	93	7	jl	jl	PROPN
cana-1037	93	8	numbers	number	NOUN
cana-1037	93	9	,	,	PUNCT
cana-1037	93	10	denoted	denote	VERB
cana-1037	93	11	by	by	ADP
cana-1037	93	12	the	the	DET
cana-1037	93	13	symbol	symbol	NOUN
cana-1037	93	14	"	"	PUNCT
cana-1037	93	15	jn	jn	PROPN
cana-1037	93	16	,	,	PUNCT
cana-1037	93	17	"	"	PUNCT
cana-1037	93	18	are	be	AUX
cana-1037	93	19	as	as	SCONJ
cana-1037	93	20	follows	follow	VERB
cana-1037	93	21	:	:	PUNCT
cana-1037	93	22	2	2	NUM
cana-1037	93	23	,	,	PUNCT
cana-1037	93	24	1	1	NUM
cana-1037	93	25	,	,	PUNCT
cana-1037	93	26	5	5	NUM
cana-1037	93	27	,	,	PUNCT
cana-1037	93	28	7	7	NUM
cana-1037	93	29	,	,	PUNCT
cana-1037	93	30	17	17	NUM
cana-1037	93	31	,	,	PUNCT
cana-1037	93	32	31	31	NUM
cana-1037	93	33	,	,	PUNCT
cana-1037	93	34	65	65	NUM
cana-1037	93	35	,	,	PUNCT
cana-1037	93	36	127	127	NUM
cana-1037	93	37	,	,	PUNCT
cana-1037	93	38	257	257	NUM
cana-1037	93	39	,	,	PUNCT
cana-1037	93	40	and	and	CCONJ
cana-1037	93	41	511	511	NUM
cana-1037	93	42	.	.	PUNCT
cana-1037	94	1	for	for	ADP
cana-1037	94	2	the	the	DET
cana-1037	94	3	purpose	purpose	NOUN
cana-1037	94	4	of	of	ADP
cana-1037	94	5	defining	define	VERB
cana-1037	94	6	the	the	DET
cana-1037	94	7	sequence	sequence	NOUN
cana-1037	94	8	,	,	PUNCT
cana-1037	94	9	the	the	DET
cana-1037	94	10	formula	formula	NOUN
cana-1037	94	11	jn	jn	NOUN
cana-1037	94	12	=	=	PUNCT
cana-1037	94	13	2n	2n	PROPN
cana-1037	94	14	+	+	CCONJ
cana-1037	94	15	(	(	PUNCT
cana-1037	94	16	-1)n	-1)n	NOUN
cana-1037	94	17	is	be	AUX
cana-1037	94	18	used	use	VERB
cana-1037	94	19	.	.	PUNCT
cana-1037	95	1	the	the	DET
cana-1037	95	2	following	follow	VERB
cana-1037	95	3	is	be	AUX
cana-1037	95	4	an	an	DET
cana-1037	95	5	expression	expression	NOUN
cana-1037	95	6	of	of	ADP
cana-1037	95	7	the	the	DET
cana-1037	95	8	formulas	formula	NOUN
cana-1037	95	9	with	with	ADP
cana-1037	95	10	regard	regard	NOUN
cana-1037	95	11	to	to	ADP
cana-1037	95	12	j	j	PROPN
cana-1037	95	13	and	and	CCONJ
cana-1037	95	14	jl	jl	PROPN
cana-1037	95	15	sequences	sequence	NOUN
cana-1037	95	16	:	:	PUNCT
cana-1037	95	17	ɉn+1	ɉn+1	NUM
cana-1037	95	18	ɉn-1	ɉn-1	X
cana-1037	95	19	ɉn	ɉn	PRON
cana-1037	95	20	2	2	NUM
cana-1037	95	21	=	=	SYM
cana-1037	95	22	(	(	PUNCT
cana-1037	95	23	-1)n	-1)n	NOUN
cana-1037	95	24	2n-1	2n-1	PROPN
cana-1037	95	25	𝔧n+1𝔧n-1𝔧n	𝔧n+1𝔧n-1𝔧n	PRON
cana-1037	95	26	2=	2=	NUM
cana-1037	95	27	9(-1)n+12n-1	9(-1)n+12n-1	NUM
cana-1037	95	28	we	we	PRON
cana-1037	95	29	define	define	VERB
cana-1037	95	30	the	the	DET
cana-1037	95	31	jacobsthal	jacobsthal	ADJ
cana-1037	95	32	k	k	NOUN
cana-1037	95	33	-	-	NOUN
cana-1037	95	34	matrix	matrix	NOUN
cana-1037	95	35	as	as	ADP
cana-1037	95	36	k=	k=	X
cana-1037	95	37	[	[	PUNCT
cana-1037	95	38	1	1	NUM
cana-1037	95	39	2	2	NUM
cana-1037	95	40	1	1	NUM
cana-1037	95	41	0	0	NUM
cana-1037	95	42	]	]	PUNCT
cana-1037	95	43	it	it	PRON
cana-1037	95	44	is	be	AUX
cana-1037	95	45	evident	evident	ADJ
cana-1037	95	46	.	.	PUNCT
cana-1037	96	1	[	[	PUNCT
cana-1037	96	2	ɉn+1	ɉn+1	X
cana-1037	96	3	ɉn	ɉn	X
cana-1037	96	4	]	]	PUNCT
cana-1037	96	5	=	=	SYM
cana-1037	96	6	k	k	X
cana-1037	96	7	[	[	PUNCT
cana-1037	96	8	ɉn	ɉn	X
cana-1037	96	9	ɉn−1	ɉn−1	PROPN
cana-1037	96	10	]	]	PUNCT
cana-1037	96	11	and	and	CCONJ
cana-1037	96	12	[	[	PUNCT
cana-1037	96	13	𝔧n+1	𝔧n+1	NUM
cana-1037	96	14	𝔧n	𝔧n	ADP
cana-1037	96	15	]	]	PUNCT
cana-1037	96	16	=	=	SYM
cana-1037	96	17	k	k	X
cana-1037	96	18	[	[	PUNCT
cana-1037	96	19	𝔧n	𝔧n	ADP
cana-1037	96	20	𝔧n−1	𝔧n−1	PROPN
cana-1037	96	21	]	]	SYM
cana-1037	96	22	where	where	SCONJ
cana-1037	96	23	ɉn+1	ɉn+1	NOUN
cana-1037	96	24	denotes	denote	VERB
cana-1037	96	25	the	the	DET
cana-1037	96	26	n+1	n+1	PROPN
cana-1037	96	27	th	th	NOUN
cana-1037	96	28	jl	jl	NOUN
cana-1037	96	29	number	number	NOUN
cana-1037	96	30	and	and	CCONJ
cana-1037	96	31	j_(n+1	j_(n+1	PROPN
cana-1037	96	32	)	)	PUNCT
cana-1037	96	33	stands	stand	VERB
cana-1037	96	34	for	for	ADP
cana-1037	96	35	the	the	DET
cana-1037	96	36	jl	jl	ADJ
cana-1037	96	37	number	number	NOUN
cana-1037	96	38	.	.	PUNCT
cana-1037	97	1	communications	communication	NOUN
cana-1037	97	2	on	on	ADP
cana-1037	97	3	applied	apply	VERB
cana-1037	97	4	nonlinear	nonlinear	ADJ
cana-1037	97	5	analysis	analysis	NOUN
cana-1037	97	6	issn	issn	NOUN
cana-1037	97	7	:	:	PUNCT
cana-1037	97	8	1074	1074	NUM
cana-1037	97	9	-	-	PUNCT
cana-1037	97	10	133x	133x	NUM
cana-1037	97	11	vol	vol	NOUN
cana-1037	97	12	31	31	NUM
cana-1037	97	13	no	no	NOUN
cana-1037	97	14	.	.	PUNCT
cana-1037	98	1	5s	5s	NUM
cana-1037	98	2	(	(	PUNCT
cana-1037	98	3	2024	2024	NUM
cana-1037	98	4	)	)	PUNCT
cana-1037	98	5	295	295	NUM
cana-1037	99	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1037	99	2	identity2.1	identity2.1	NUM
cana-1037	99	3	:	:	PUNCT
cana-1037	99	4	let	let	VERB
cana-1037	99	5	k	k	PRON
cana-1037	99	6	be	be	AUX
cana-1037	99	7	a	a	DET
cana-1037	99	8	2	2	NUM
cana-1037	99	9	x	x	SYM
cana-1037	99	10	2	2	NUM
cana-1037	99	11	matrix	matrix	NOUN
cana-1037	99	12	in	in	ADP
cana-1037	99	13	(	(	PUNCT
cana-1037	99	14	1	1	NUM
cana-1037	99	15	)	)	PUNCT
cana-1037	99	16	then	then	ADV
cana-1037	99	17	𝐾𝑛	𝐾𝑛	PROPN
cana-1037	99	18	=	=	PUNCT
cana-1037	100	1	[	[	PUNCT
cana-1037	100	2	ɉn+1	ɉn+1	NUM
cana-1037	100	3	2ɉn	2ɉn	NOUN
cana-1037	100	4	ɉn	ɉn	ADP
cana-1037	100	5	2ɉn−1	2ɉn−1	PROPN
cana-1037	100	6	]	]	PUNCT
cana-1037	100	7	------------------(a	------------------(a	NOUN
cana-1037	100	8	)	)	PUNCT
cana-1037	100	9	proof	proof	NOUN
cana-1037	100	10	:	:	PUNCT
cana-1037	100	11	we	we	PRON
cana-1037	100	12	prove	prove	VERB
cana-1037	100	13	the	the	DET
cana-1037	100	14	identity	identity	NOUN
cana-1037	100	15	by	by	ADP
cana-1037	100	16	mathematical	mathematical	ADJ
cana-1037	100	17	induction	induction	NOUN
cana-1037	100	18	let	let	VERB
cana-1037	100	19	n	n	PRON
cana-1037	100	20	=	=	NOUN
cana-1037	100	21	1	1	NUM
cana-1037	100	22	then	then	ADV
cana-1037	100	23	k	k	PROPN
cana-1037	101	1	=	=	PUNCT
cana-1037	102	1	[	[	PUNCT
cana-1037	102	2	1	1	NUM
cana-1037	102	3	2	2	NUM
cana-1037	102	4	1	1	NUM
cana-1037	102	5	0	0	NUM
cana-1037	102	6	]	]	PUNCT
cana-1037	103	1	=	=	PUNCT
cana-1037	103	2	[	[	PUNCT
cana-1037	103	3	ɉ2	ɉ2	NOUN
cana-1037	103	4	2ɉ1	2ɉ1	NUM
cana-1037	103	5	ɉ1	ɉ1	NOUN
cana-1037	103	6	2ɉ0	2ɉ0	NUM
cana-1037	103	7	]	]	PUNCT
cana-1037	104	1	so	so	CCONJ
cana-1037	104	2	the	the	DET
cana-1037	104	3	above	above	ADJ
cana-1037	104	4	result	result	NOUN
cana-1037	104	5	is	be	AUX
cana-1037	104	6	true	true	ADJ
cana-1037	104	7	assuming	assume	VERB
cana-1037	104	8	the	the	DET
cana-1037	104	9	outcome	outcome	NOUN
cana-1037	104	10	is	be	AUX
cana-1037	104	11	valid	valid	ADJ
cana-1037	104	12	for	for	ADP
cana-1037	104	13	n	n	CCONJ
cana-1037	104	14	,	,	PUNCT
cana-1037	104	15	ie𝐾𝑛	ie𝐾𝑛	ADV
cana-1037	104	16	=	=	PUNCT
cana-1037	105	1	[	[	PUNCT
cana-1037	105	2	ɉn+1	ɉn+1	NUM
cana-1037	105	3	2ɉn	2ɉn	NOUN
cana-1037	105	4	ɉn	ɉn	ADP
cana-1037	105	5	2ɉn−1	2ɉn−1	NUM
cana-1037	105	6	]	]	PUNCT
cana-1037	105	7	we	we	PRON
cana-1037	105	8	will	will	AUX
cana-1037	105	9	now	now	ADV
cana-1037	105	10	demonstrate	demonstrate	VERB
cana-1037	105	11	that	that	SCONJ
cana-1037	105	12	the	the	DET
cana-1037	105	13	outcome	outcome	NOUN
cana-1037	105	14	is	be	AUX
cana-1037	105	15	valid	valid	ADJ
cana-1037	105	16	for	for	ADP
cana-1037	105	17	n+1	n+1	NUM
cana-1037	105	18	𝐾𝑛+1	𝐾𝑛+1	NOUN
cana-1037	105	19	=	=	PUNCT
cana-1037	105	20	𝐾𝑛.	𝐾𝑛.	PROPN
cana-1037	105	21	𝐾	𝐾	NOUN
cana-1037	105	22	=	=	SYM
cana-1037	105	23	[	[	PUNCT
cana-1037	105	24	ɉn+1	ɉn+1	NUM
cana-1037	105	25	2ɉn	2ɉn	NOUN
cana-1037	105	26	ɉn	ɉn	ADP
cana-1037	105	27	2ɉn−1	2ɉn−1	PROPN
cana-1037	105	28	]	]	PUNCT
cana-1037	105	29	.	.	PUNCT
cana-1037	106	1	[	[	PUNCT
cana-1037	106	2	1	1	NUM
cana-1037	106	3	2	2	NUM
cana-1037	106	4	1	1	NUM
cana-1037	106	5	0	0	NUM
cana-1037	106	6	]	]	PUNCT
cana-1037	106	7	=	=	PUNCT
cana-1037	106	8	[	[	PUNCT
cana-1037	106	9	ɉn+1	ɉn+1	NUM
cana-1037	106	10	+	+	NUM
cana-1037	106	11	2ɉn	2ɉn	NOUN
cana-1037	106	12	2ɉn+1	2ɉn+1	ADJ
cana-1037	106	13	ɉn	ɉn	X
cana-1037	106	14	+	+	CCONJ
cana-1037	106	15	2ɉn−1	2ɉn−1	NUM
cana-1037	106	16	2ɉn	2ɉn	NOUN
cana-1037	106	17	]	]	PUNCT
cana-1037	106	18	=	=	PUNCT
cana-1037	107	1	[	[	PUNCT
cana-1037	107	2	ɉn+2	ɉn+2	NUM
cana-1037	107	3	2ɉn+1	2ɉn+1	NUM
cana-1037	107	4	ɉn+1	ɉn+1	NUM
cana-1037	107	5	2ɉn	2ɉn	NOUN
cana-1037	107	6	]	]	PUNCT
cana-1037	107	7	and	and	CCONJ
cana-1037	107	8	the	the	DET
cana-1037	107	9	result	result	NOUN
cana-1037	107	10	follows	follow	VERB
cana-1037	107	11	.	.	PUNCT
cana-1037	108	1	corollary	corollary	ADJ
cana-1037	108	2	:	:	PUNCT
cana-1037	108	3	for	for	ADP
cana-1037	108	4	all	all	DET
cana-1037	108	5	positive	positive	ADJ
cana-1037	108	6	integers	integer	NOUN
cana-1037	108	7	n	n	CCONJ
cana-1037	108	8	,	,	PUNCT
cana-1037	108	9	(	(	PUNCT
cana-1037	108	10	i	i	NOUN
cana-1037	108	11	)	)	PUNCT
cana-1037	108	12	𝑑𝑒𝑡(𝐾𝑛	𝑑𝑒𝑡(𝐾𝑛	PROPN
cana-1037	108	13	)	)	PUNCT
cana-1037	108	14	=	=	SYM
cana-1037	108	15	(	(	PUNCT
cana-1037	108	16	−2)𝑛	−2)𝑛	PROPN
cana-1037	108	17	(	(	PUNCT
cana-1037	108	18	ii	ii	NOUN
cana-1037	108	19	)	)	PUNCT
cana-1037	108	20	ɉn+1ɉn−1	ɉn+1ɉn−1	PROPN
cana-1037	108	21	−	−	PROPN
cana-1037	108	22	ɉn	ɉn	X
cana-1037	108	23	2	2	NUM
cana-1037	108	24	=	=	SYM
cana-1037	108	25	(	(	PUNCT
cana-1037	108	26	−1)𝑛2𝑛−1	−1)𝑛2𝑛−1	PROPN
cana-1037	108	27	cassini	cassini	PROPN
cana-1037	108	28	’s	’s	PART
cana-1037	108	29	like	like	ADP
cana-1037	108	30	formula	formula	NOUN
cana-1037	108	31	references	reference	NOUN
cana-1037	108	32	[	[	X
cana-1037	108	33	1	1	NUM
cana-1037	108	34	]	]	PUNCT
cana-1037	108	35	assaf	assaf	PROPN
cana-1037	108	36	e	e	PROPN
cana-1037	108	37	and	and	CCONJ
cana-1037	108	38	gueron	gueron	PROPN
cana-1037	108	39	s.	s.	PROPN
cana-1037	108	40	characterization	characterization	NOUN
cana-1037	108	41	of	of	ADP
cana-1037	108	42	regular	regular	ADJ
cana-1037	108	43	diophantine	diophantine	NOUN
cana-1037	108	44	quadruples	quadruple	NOUN
cana-1037	108	45	,	,	PUNCT
cana-1037	108	46	elem	elem	NOUN
cana-1037	108	47	.	.	PUNCT
cana-1037	109	1	math	math	NOUN
cana-1037	109	2	.	.	PUNCT
cana-1037	110	1	2001	2001	NUM
cana-1037	110	2	;	;	PUNCT
cana-1037	110	3	56:71	56:71	NUM
cana-1037	110	4	-	-	SYM
cana-1037	110	5	81	81	NUM
cana-1037	110	6	.	.	PUNCT
cana-1037	111	1	[	[	X
cana-1037	111	2	2	2	NUM
cana-1037	111	3	]	]	PUNCT
cana-1037	111	4	campos	campos	NOUN
cana-1037	111	5	.	.	PUNCT
cana-1037	112	1	h	h	PROPN
cana-1037	112	2	,	,	PUNCT
cana-1037	112	3	catarino	catarino	PROPN
cana-1037	112	4	.	.	PUNCT
cana-1037	113	1	p	p	X
cana-1037	113	2	,	,	PUNCT
cana-1037	113	3	a.p	a.p	PROPN
cana-1037	113	4	.	.	PROPN
cana-1037	113	5	aires	aires	PROPN
cana-1037	113	6	,	,	PUNCT
cana-1037	113	7	p.	p.	PROPN
cana-1037	113	8	vasco	vasco	PROPN
cana-1037	113	9	and	and	CCONJ
cana-1037	113	10	a.	a.	NOUN
cana-1037	113	11	borges	borge	NOUN
cana-1037	113	12	,	,	PUNCT
cana-1037	113	13	on	on	ADP
cana-1037	113	14	some	some	DET
cana-1037	113	15	identities	identity	NOUN
cana-1037	113	16	of	of	ADP
cana-1037	113	17	k	k	NOUN
cana-1037	113	18	-	-	PUNCT
cana-1037	113	19	jacobsthal	jacobsthal	ADJ
cana-1037	113	20	-	-	PUNCT
cana-1037	113	21	lucas	lucas	NOUN
cana-1037	113	22	numbers	number	NOUN
cana-1037	113	23	,	,	PUNCT
cana-1037	113	24	int	int	NOUN
cana-1037	113	25	.	.	PUNCT
cana-1037	114	1	journal	journal	PROPN
cana-1037	114	2	of	of	ADP
cana-1037	114	3	math	math	NOUN
cana-1037	114	4	.	.	PUNCT
cana-1037	115	1	analysis	analysis	NOUN
cana-1037	115	2	,	,	PUNCT
cana-1037	115	3	2014	2014	NUM
cana-1037	115	4	:	:	PUNCT
cana-1037	115	5	8(10	8(10	NUM
cana-1037	115	6	)	)	PUNCT
cana-1037	115	7	;	;	PUNCT
cana-1037	115	8	489	489	NUM
cana-1037	115	9	-	-	SYM
cana-1037	115	10	494	494	NUM
cana-1037	115	11	.	.	PUNCT
cana-1037	116	1	[	[	X
cana-1037	116	2	3	3	X
cana-1037	116	3	]	]	X
cana-1037	116	4	feng	feng	PROPN
cana-1037	116	5	a.	a.	PROPN
cana-1037	116	6	fibonacci	fibonacci	PROPN
cana-1037	116	7	identities	identity	NOUN
cana-1037	116	8	via	via	ADP
cana-1037	116	9	determinant	determinant	ADJ
cana-1037	116	10	of	of	ADP
cana-1037	116	11	tridiagonal	tridiagonal	ADJ
cana-1037	116	12	matrix	matrix	NOUN
cana-1037	116	13	,	,	PUNCT
cana-1037	116	14	applied	apply	VERB
cana-1037	116	15	mathematics	mathematic	NOUN
cana-1037	116	16	and	and	CCONJ
cana-1037	116	17	computation	computation	NOUN
cana-1037	116	18	,	,	PUNCT
cana-1037	116	19	2011	2011	NUM
cana-1037	116	20	:	:	PUNCT
cana-1037	116	21	217	217	NUM
cana-1037	116	22	,	,	PUNCT
cana-1037	116	23	pp	pp	ADJ
cana-1037	116	24	.	.	PUNCT
cana-1037	117	1	5978—5981	5978—5981	NOUN
cana-1037	117	2	.	.	PUNCT
cana-1037	118	1	[	[	X
cana-1037	118	2	4	4	X
cana-1037	118	3	]	]	X
cana-1037	118	4	jhala	jhala	NOUN
cana-1037	118	5	.	.	PUNCT
cana-1037	119	1	d	d	X
cana-1037	119	2	,	,	PUNCT
cana-1037	119	3	sisodiya	sisodiya	ADJ
cana-1037	119	4	.k	.k	PROPN
cana-1037	119	5	and	and	CCONJ
cana-1037	120	1	g.p.s	g.p.s	PROPN
cana-1037	120	2	.	.	PUNCT
cana-1037	120	3	rathore	rathore	PROPN
cana-1037	120	4	,	,	PUNCT
cana-1037	120	5	on	on	ADP
cana-1037	120	6	some	some	DET
cana-1037	120	7	identities	identity	NOUN
cana-1037	120	8	for	for	ADP
cana-1037	120	9	k	k	ADJ
cana-1037	120	10	-	-	ADJ
cana-1037	120	11	jacobsthal	jacobsthal	ADJ
cana-1037	120	12	numbers	number	NOUN
cana-1037	120	13	,	,	PUNCT
cana-1037	120	14	int	int	NOUN
cana-1037	120	15	.	.	PUNCT
cana-1037	121	1	journal	journal	PROPN
cana-1037	121	2	of	of	ADP
cana-1037	121	3	math	math	NOUN
cana-1037	121	4	.	.	PUNCT
cana-1037	122	1	analysis	analysis	NOUN
cana-1037	122	2	,	,	PUNCT
cana-1037	122	3	2013	2013	NUM
cana-1037	122	4	:	:	PUNCT
cana-1037	122	5	7(12	7(12	NUM
cana-1037	122	6	)	)	PUNCT
cana-1037	122	7	;	;	PUNCT
cana-1037	122	8	551	551	NUM
cana-1037	122	9	-	-	SYM
cana-1037	122	10	556	556	NUM
cana-1037	122	11	[	[	SYM
cana-1037	122	12	5	5	NUM
cana-1037	122	13	]	]	PUNCT
cana-1037	122	14	a.	a.	NOUN
cana-1037	122	15	f.	f.	PROPN
cana-1037	122	16	horadam	horadam	PROPN
cana-1037	122	17	,	,	PUNCT
cana-1037	122	18	pell	pell	PROPN
cana-1037	122	19	identities	identity	NOUN
cana-1037	122	20	.	.	PUNCT
cana-1037	123	1	fibonacci	fibonacci	PROPN
cana-1037	123	2	quart	quart	PROPN
cana-1037	123	3	.	.	PUNCT
cana-1037	124	1	1971	1971	NUM
cana-1037	124	2	:	:	PUNCT
cana-1037	124	3	9	9	NUM
cana-1037	124	4	(	(	PUNCT
cana-1037	124	5	3);245	3);245	NUM
cana-1037	124	6	-	-	SYM
cana-1037	124	7	263	263	NUM
cana-1037	124	8	.	.	PUNCT
cana-1037	125	1	[	[	X
cana-1037	125	2	6	6	NUM
cana-1037	125	3	]	]	PUNCT
cana-1037	125	4	yogesh	yogesh	PROPN
cana-1037	125	5	kumar	kumar	PROPN
cana-1037	125	6	gupta	gupta	PROPN
cana-1037	125	7	and	and	CCONJ
cana-1037	125	8	omprakash	omprakash	NOUN
cana-1037	125	9	sikhwal	sikhwal	PROPN
cana-1037	125	10	mamta	mamta	PROPN
cana-1037	125	11	singh	singh	PROPN
cana-1037	125	12	.	.	PUNCT
cana-1037	126	1	determinantal	determinantal	ADJ
cana-1037	126	2	identities	identity	NOUN
cana-1037	126	3	of	of	ADP
cana-1037	126	4	fibonacci	fibonacci	PROPN
cana-1037	126	5	lucas	lucas	PROPN
cana-1037	126	6	and	and	CCONJ
cana-1037	126	7	generalized	generalized	ADJ
cana-1037	126	8	fibonacci	fibonacci	NOUN
cana-1037	126	9	lucas	lucas	PROPN
cana-1037	126	10	sequence	sequence	PROPN
cana-1037	126	11	,	,	PUNCT
cana-1037	126	12	mayfeb	mayfeb	PROPN
cana-1037	126	13	journal	journal	NOUN
cana-1037	126	14	of	of	ADP
cana-1037	126	15	mathematics	mathematic	NOUN
cana-1037	126	16	,	,	PUNCT
cana-1037	126	17	2016	2016	NUM
cana-1037	126	18	:	:	PUNCT
cana-1037	126	19	vol	vol	NOUN
cana-1037	126	20	2;pg	2;pg	NUM
cana-1037	126	21	no.17	no.17	PROPN
cana-1037	126	22	-	-	PUNCT
cana-1037	126	23	23	23	NUM