id	sid	tid	token	lemma	pos
easat-5378	1	1	edelweiss	edelweiss	PROPN
easat-5378	1	2	applied	apply	VERB
easat-5378	1	3	science	science	NOUN
easat-5378	1	4	and	and	CCONJ
easat-5378	1	5	technology	technology	NOUN
easat-5378	1	6	issn	issn	PROPN
easat-5378	1	7	:	:	PUNCT
easat-5378	1	8	2576	2576	NUM
easat-5378	1	9	-	-	SYM
easat-5378	1	10	8484	8484	NUM
easat-5378	1	11	vol	vol	NOUN
easat-5378	1	12	.	.	PROPN
easat-5378	2	1	9	9	NUM
easat-5378	2	2	,	,	PUNCT
easat-5378	2	3	no	no	INTJ
easat-5378	2	4	.	.	NOUN
easat-5378	2	5	3	3	NUM
easat-5378	2	6	,	,	PUNCT
easat-5378	2	7	891	891	NUM
easat-5378	2	8	-	-	SYM
easat-5378	2	9	904	904	NUM
easat-5378	2	10	2025	2025	NUM
easat-5378	2	11	publisher	publisher	NOUN
easat-5378	2	12	:	:	PUNCT
easat-5378	2	13	learning	learn	VERB
easat-5378	2	14	gate	gate	NOUN
easat-5378	2	15	doi	doi	PROPN
easat-5378	2	16	:	:	PUNCT
easat-5378	2	17	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	2	18	©	©	PROPN
easat-5378	2	19	2025	2025	NUM
easat-5378	2	20	by	by	ADP
easat-5378	2	21	the	the	DET
easat-5378	2	22	author	author	NOUN
easat-5378	2	23	;	;	PUNCT
easat-5378	2	24	licensee	licensee	PROPN
easat-5378	2	25	learning	learning	NOUN
easat-5378	2	26	gate	gate	NOUN
easat-5378	3	1	©	©	PROPN
easat-5378	3	2	2025	2025	NUM
easat-5378	3	3	by	by	ADP
easat-5378	3	4	the	the	DET
easat-5378	3	5	author	author	NOUN
easat-5378	3	6	;	;	PUNCT
easat-5378	3	7	licensee	licensee	PROPN
easat-5378	3	8	learning	learning	NOUN
easat-5378	3	9	gate	gate	NOUN
easat-5378	3	10	history	history	NOUN
easat-5378	3	11	:	:	PUNCT
easat-5378	3	12	received	receive	VERB
easat-5378	3	13	:	:	PUNCT
easat-5378	3	14	1	1	NUM
easat-5378	3	15	january	january	NOUN
easat-5378	3	16	2025	2025	NUM
easat-5378	3	17	;	;	PUNCT
easat-5378	3	18	revised	revise	VERB
easat-5378	3	19	:	:	PUNCT
easat-5378	3	20	17	17	NUM
easat-5378	3	21	february	february	NOUN
easat-5378	3	22	2025	2025	NUM
easat-5378	3	23	;	;	PUNCT
easat-5378	3	24	accepted	accept	VERB
easat-5378	3	25	:	:	PUNCT
easat-5378	3	26	17	17	NUM
easat-5378	3	27	february	february	NOUN
easat-5378	3	28	2025	2025	NUM
easat-5378	3	29	;	;	PUNCT
easat-5378	3	30	published	publish	VERB
easat-5378	3	31	:	:	PUNCT
easat-5378	4	1	12	12	NUM
easat-5378	4	2	march	march	NOUN
easat-5378	4	3	2025	2025	NUM
easat-5378	4	4	*	*	PUNCT
easat-5378	4	5	correspondence	correspondence	NOUN
easat-5378	4	6	:	:	PUNCT
easat-5378	5	1	romer.castillo@g.batstate-u.edu.ph	romer.castillo@g.batstate-u.edu.ph	PROPN
easat-5378	5	2	mathematical	mathematical	ADJ
easat-5378	5	3	explorations	exploration	NOUN
easat-5378	5	4	on	on	ADP
easat-5378	5	5	the	the	DET
easat-5378	5	6	sequence	sequence	NOUN
easat-5378	5	7	of	of	ADP
easat-5378	5	8	factoriangular	factoriangular	ADJ
easat-5378	5	9	numbers	number	NOUN
easat-5378	5	10	:	:	PUNCT
easat-5378	5	11	extending	extend	VERB
easat-5378	5	12	the	the	DET
easat-5378	5	13	results	result	NOUN
easat-5378	5	14	on	on	ADP
easat-5378	5	15	generalizations	generalization	NOUN
easat-5378	5	16	romer	romer	PROPN
easat-5378	5	17	c.	c.	PROPN
easat-5378	5	18	castillo1	castillo1	PROPN
easat-5378	5	19	*	*	PUNCT
easat-5378	5	20	1batangas	1batangas	NUM
easat-5378	5	21	state	state	NOUN
easat-5378	5	22	university	university	NOUN
easat-5378	5	23	,	,	PUNCT
easat-5378	5	24	batangas	batangas	PROPN
easat-5378	5	25	city	city	PROPN
easat-5378	5	26	,	,	PUNCT
easat-5378	5	27	philippines	philippine	NOUN
easat-5378	5	28	;	;	PUNCT
easat-5378	5	29	romer.castillo@g.batstate-u.edu.ph	romer.castillo@g.batstate-u.edu.ph	PROPN
easat-5378	5	30	(	(	PUNCT
easat-5378	5	31	r.c.c	r.c.c	NOUN
easat-5378	5	32	.	.	PUNCT
easat-5378	5	33	)	)	PUNCT
easat-5378	5	34	.	.	PUNCT
easat-5378	6	1	abstract	abstract	ADV
easat-5378	6	2	:	:	PUNCT
easat-5378	6	3	a	a	DET
easat-5378	6	4	factoriangular	factoriangular	ADJ
easat-5378	6	5	number	number	NOUN
easat-5378	6	6	is	be	AUX
easat-5378	6	7	formed	form	VERB
easat-5378	6	8	by	by	ADP
easat-5378	6	9	adding	add	VERB
easat-5378	6	10	a	a	DET
easat-5378	6	11	factorial	factorial	NOUN
easat-5378	6	12	and	and	CCONJ
easat-5378	6	13	a	a	DET
easat-5378	6	14	triangular	triangular	NOUN
easat-5378	6	15	number	number	NOUN
easat-5378	6	16	.	.	PUNCT
easat-5378	7	1	if	if	SCONJ
easat-5378	7	2	corresponding	corresponding	ADJ
easat-5378	7	3	factorials	factorial	NOUN
easat-5378	7	4	and	and	CCONJ
easat-5378	7	5	triangular	triangular	NOUN
easat-5378	7	6	numbers	number	NOUN
easat-5378	7	7	are	be	AUX
easat-5378	7	8	added	add	VERB
easat-5378	7	9	,	,	PUNCT
easat-5378	7	10	the	the	DET
easat-5378	7	11	results	result	NOUN
easat-5378	7	12	are	be	AUX
easat-5378	7	13	n	n	PRON
easat-5378	7	14	-	-	PUNCT
easat-5378	7	15	factoriangular	factoriangular	ADJ
easat-5378	7	16	numbers	number	NOUN
easat-5378	7	17	.	.	PUNCT
easat-5378	8	1	other	other	ADJ
easat-5378	8	2	factoriangular	factoriangular	ADJ
easat-5378	8	3	numbers	number	NOUN
easat-5378	8	4	are	be	AUX
easat-5378	8	5	called	call	VERB
easat-5378	8	6	(	(	PUNCT
easat-5378	8	7	n	n	CCONJ
easat-5378	8	8	,	,	PUNCT
easat-5378	8	9	k)-factoriangular	k)-factoriangular	PUNCT
easat-5378	8	10	numbers	number	NOUN
easat-5378	8	11	,	,	PUNCT
easat-5378	8	12	n(m)-factoriangular	n(m)-factoriangular	ADJ
easat-5378	8	13	numbers	number	NOUN
easat-5378	8	14	,	,	PUNCT
easat-5378	8	15	(	(	PUNCT
easat-5378	8	16	n(m),k(m))-factoriangular	n(m),k(m))-factoriangular	ADJ
easat-5378	8	17	numbers	number	NOUN
easat-5378	8	18	,	,	PUNCT
easat-5378	8	19	and	and	CCONJ
easat-5378	8	20	(	(	PUNCT
easat-5378	8	21	n(a),k(b))-factoriangular	n(a),k(b))-factoriangular	ADJ
easat-5378	8	22	numbers	number	NOUN
easat-5378	8	23	.	.	PUNCT
easat-5378	9	1	the	the	DET
easat-5378	9	2	main	main	ADJ
easat-5378	9	3	objective	objective	NOUN
easat-5378	9	4	of	of	ADP
easat-5378	9	5	this	this	DET
easat-5378	9	6	study	study	NOUN
easat-5378	9	7	is	be	AUX
easat-5378	9	8	to	to	PART
easat-5378	9	9	explore	explore	VERB
easat-5378	9	10	the	the	DET
easat-5378	9	11	sequence	sequence	NOUN
easat-5378	9	12	of	of	ADP
easat-5378	9	13	(	(	PUNCT
easat-5378	9	14	n(m),k(m))-factoriangular	n(m),k(m))-factoriangular	ADJ
easat-5378	9	15	numbers	number	NOUN
easat-5378	9	16	and	and	CCONJ
easat-5378	9	17	the	the	DET
easat-5378	9	18	sequence	sequence	NOUN
easat-5378	9	19	of	of	ADP
easat-5378	9	20	(	(	PUNCT
easat-5378	9	21	n(a),k(b))-factoriangular	n(a),k(b))-factoriangular	ADJ
easat-5378	9	22	numbers	number	NOUN
easat-5378	9	23	as	as	ADP
easat-5378	9	24	generalizations	generalization	NOUN
easat-5378	9	25	of	of	ADP
easat-5378	9	26	the	the	DET
easat-5378	9	27	sequence	sequence	NOUN
easat-5378	9	28	of	of	ADP
easat-5378	9	29	n	n	CCONJ
easat-5378	9	30	-	-	PUNCT
easat-5378	9	31	factoriangular	factoriangular	ADJ
easat-5378	9	32	numbers	number	NOUN
easat-5378	9	33	.	.	PUNCT
easat-5378	10	1	this	this	DET
easat-5378	10	2	research	research	NOUN
easat-5378	10	3	is	be	AUX
easat-5378	10	4	a	a	DET
easat-5378	10	5	disciplinebased	disciplinebase	VERB
easat-5378	10	6	scholarship	scholarship	NOUN
easat-5378	10	7	of	of	ADP
easat-5378	10	8	discovery	discovery	NOUN
easat-5378	10	9	that	that	PRON
easat-5378	10	10	employs	employ	VERB
easat-5378	10	11	an	an	DET
easat-5378	10	12	exploratory	exploratory	ADJ
easat-5378	10	13	method	method	NOUN
easat-5378	10	14	involving	involve	VERB
easat-5378	10	15	the	the	DET
easat-5378	10	16	scientific	scientific	ADJ
easat-5378	10	17	approach	approach	NOUN
easat-5378	10	18	of	of	ADP
easat-5378	10	19	experimental	experimental	ADJ
easat-5378	10	20	mathematics	mathematic	NOUN
easat-5378	10	21	.	.	PUNCT
easat-5378	11	1	the	the	DET
easat-5378	11	2	mathematical	mathematical	ADJ
easat-5378	11	3	method	method	NOUN
easat-5378	11	4	was	be	AUX
easat-5378	11	5	used	use	VERB
easat-5378	11	6	in	in	ADP
easat-5378	11	7	doing	do	VERB
easat-5378	11	8	the	the	DET
easat-5378	11	9	explorations	exploration	NOUN
easat-5378	11	10	,	,	PUNCT
easat-5378	11	11	focusing	focus	VERB
easat-5378	11	12	on	on	ADP
easat-5378	11	13	the	the	DET
easat-5378	11	14	formulations	formulation	NOUN
easat-5378	11	15	and	and	CCONJ
easat-5378	11	16	proofs	proof	NOUN
easat-5378	11	17	of	of	ADP
easat-5378	11	18	theorems	theorem	NOUN
easat-5378	11	19	and	and	CCONJ
easat-5378	11	20	giving	give	VERB
easat-5378	11	21	some	some	DET
easat-5378	11	22	examples	example	NOUN
easat-5378	11	23	.	.	PUNCT
easat-5378	12	1	for	for	ADP
easat-5378	12	2	the	the	DET
easat-5378	12	3	main	main	ADJ
easat-5378	12	4	results	result	NOUN
easat-5378	12	5	,	,	PUNCT
easat-5378	12	6	ten	ten	NUM
easat-5378	12	7	theorems	theorem	NOUN
easat-5378	12	8	were	be	AUX
easat-5378	12	9	proven	prove	VERB
easat-5378	12	10	and	and	CCONJ
easat-5378	12	11	several	several	ADJ
easat-5378	12	12	examples	example	NOUN
easat-5378	12	13	of	of	ADP
easat-5378	12	14	sequences	sequence	NOUN
easat-5378	12	15	were	be	AUX
easat-5378	12	16	provided	provide	VERB
easat-5378	12	17	.	.	PUNCT
easat-5378	13	1	the	the	DET
easat-5378	13	2	theorems	theorem	NOUN
easat-5378	13	3	include	include	VERB
easat-5378	13	4	several	several	ADJ
easat-5378	13	5	formulas	formula	NOUN
easat-5378	13	6	for	for	ADP
easat-5378	13	7	(	(	PUNCT
easat-5378	13	8	n(m),k(m))-factoriangular	n(m),k(m))-factoriangular	ADJ
easat-5378	13	9	numbers	number	NOUN
easat-5378	13	10	,	,	PUNCT
easat-5378	13	11	and	and	CCONJ
easat-5378	13	12	(	(	PUNCT
easat-5378	13	13	n(a),k(b))-factoriangular	n(a),k(b))-factoriangular	ADJ
easat-5378	13	14	numbers	number	NOUN
easat-5378	13	15	.	.	PUNCT
easat-5378	14	1	the	the	DET
easat-5378	14	2	proofs	proof	NOUN
easat-5378	14	3	for	for	ADP
easat-5378	14	4	theorems	theorem	NOUN
easat-5378	14	5	in	in	ADP
easat-5378	14	6	(	(	PUNCT
easat-5378	14	7	n(m),k(m))-factoriangular	n(m),k(m))-factoriangular	ADJ
easat-5378	14	8	numbers	number	NOUN
easat-5378	14	9	are	be	AUX
easat-5378	14	10	applicable	applicable	ADJ
easat-5378	14	11	for	for	ADP
easat-5378	14	12	similar	similar	ADJ
easat-5378	14	13	theorems	theorem	NOUN
easat-5378	14	14	in	in	ADP
easat-5378	14	15	(	(	PUNCT
easat-5378	14	16	n(a),k(b))-factoriangular	n(a),k(b))-factoriangular	ADJ
easat-5378	14	17	numbers	number	NOUN
easat-5378	14	18	.	.	PUNCT
easat-5378	15	1	specific	specific	ADJ
easat-5378	15	2	sequences	sequence	NOUN
easat-5378	15	3	of	of	ADP
easat-5378	15	4	some	some	DET
easat-5378	15	5	generalized	generalized	ADJ
easat-5378	15	6	factoriangular	factoriangular	ADJ
easat-5378	15	7	numbers	number	NOUN
easat-5378	15	8	were	be	AUX
easat-5378	15	9	presented	present	VERB
easat-5378	15	10	in	in	ADP
easat-5378	15	11	tables	table	NOUN
easat-5378	15	12	.	.	PUNCT
easat-5378	16	1	entries	entry	NOUN
easat-5378	16	2	of	of	ADP
easat-5378	16	3	numbers	number	NOUN
easat-5378	16	4	in	in	ADP
easat-5378	16	5	the	the	DET
easat-5378	16	6	tables	table	NOUN
easat-5378	16	7	may	may	AUX
easat-5378	16	8	lead	lead	VERB
easat-5378	16	9	to	to	ADP
easat-5378	16	10	the	the	DET
easat-5378	16	11	formation	formation	NOUN
easat-5378	16	12	of	of	ADP
easat-5378	16	13	triangular	triangular	NOUN
easat-5378	16	14	arrays	array	NOUN
easat-5378	16	15	of	of	ADP
easat-5378	16	16	factoriangular	factoriangular	ADJ
easat-5378	16	17	numbers	number	NOUN
easat-5378	16	18	that	that	PRON
easat-5378	16	19	may	may	AUX
easat-5378	16	20	be	be	AUX
easat-5378	16	21	further	far	ADV
easat-5378	16	22	explored	explore	VERB
easat-5378	16	23	by	by	ADP
easat-5378	16	24	other	other	ADJ
easat-5378	16	25	researchers	researcher	NOUN
easat-5378	16	26	,	,	PUNCT
easat-5378	16	27	especially	especially	ADV
easat-5378	16	28	those	those	PRON
easat-5378	16	29	mostly	mostly	ADV
easat-5378	16	30	interested	interested	ADJ
easat-5378	16	31	in	in	ADP
easat-5378	16	32	recreational	recreational	ADJ
easat-5378	16	33	mathematics	mathematic	NOUN
easat-5378	16	34	.	.	PUNCT
easat-5378	17	1	keywords	keyword	NOUN
easat-5378	17	2	:	:	PUNCT
easat-5378	17	3	factorial	factorial	ADJ
easat-5378	17	4	,	,	PUNCT
easat-5378	17	5	factoriangular	factoriangular	ADJ
easat-5378	17	6	number	number	NOUN
easat-5378	17	7	,	,	PUNCT
easat-5378	17	8	generalization	generalization	NOUN
easat-5378	17	9	,	,	PUNCT
easat-5378	17	10	integer	integer	NOUN
easat-5378	17	11	sequence	sequence	NOUN
easat-5378	17	12	,	,	PUNCT
easat-5378	17	13	number	number	NOUN
easat-5378	17	14	theory	theory	NOUN
easat-5378	17	15	,	,	PUNCT
easat-5378	17	16	triangular	triangular	ADJ
easat-5378	17	17	number	number	NOUN
easat-5378	17	18	.	.	PUNCT
easat-5378	18	1	1	1	X
easat-5378	18	2	.	.	X
easat-5378	18	3	introduction	introduction	NOUN
easat-5378	18	4	mathematics	mathematics	PROPN
easat-5378	18	5	literature	literature	PROPN
easat-5378	18	6	provide	provide	VERB
easat-5378	18	7	a	a	DET
easat-5378	18	8	long	long	ADJ
easat-5378	18	9	history	history	NOUN
easat-5378	18	10	of	of	ADP
easat-5378	18	11	research	research	NOUN
easat-5378	18	12	in	in	ADP
easat-5378	18	13	triangular	triangular	NOUN
easat-5378	18	14	numbers	number	NOUN
easat-5378	18	15	and	and	CCONJ
easat-5378	18	16	factorials	factorial	NOUN
easat-5378	18	17	.	.	PUNCT
easat-5378	19	1	it	it	PRON
easat-5378	19	2	is	be	AUX
easat-5378	19	3	commonly	commonly	ADV
easat-5378	19	4	argued	argue	VERB
easat-5378	19	5	that	that	SCONJ
easat-5378	19	6	triangular	triangular	NOUN
easat-5378	19	7	numbers	number	NOUN
easat-5378	19	8	were	be	AUX
easat-5378	19	9	already	already	ADV
easat-5378	19	10	known	know	VERB
easat-5378	19	11	to	to	ADP
easat-5378	19	12	the	the	DET
easat-5378	19	13	ancient	ancient	ADJ
easat-5378	19	14	greeks	greeks	PROPN
easat-5378	19	15	,	,	PUNCT
easat-5378	19	16	who	who	PRON
easat-5378	19	17	viewed	view	VERB
easat-5378	19	18	them	they	PRON
easat-5378	19	19	with	with	ADP
easat-5378	19	20	reverence	reverence	NOUN
easat-5378	20	1	[	[	X
easat-5378	20	2	1	1	NUM
easat-5378	20	3	]	]	PUNCT
easat-5378	20	4	and	and	CCONJ
easat-5378	20	5	most	most	ADV
easat-5378	20	6	especially	especially	ADV
easat-5378	20	7	to	to	ADP
easat-5378	20	8	the	the	DET
easat-5378	20	9	pythagoreans	pythagorean	NOUN
easat-5378	20	10	who	who	PRON
easat-5378	20	11	discoursed	discourse	VERB
easat-5378	20	12	on	on	ADP
easat-5378	20	13	the	the	DET
easat-5378	20	14	number	number	NOUN
easat-5378	20	15	of	of	ADP
easat-5378	20	16	dots	dot	NOUN
easat-5378	20	17	or	or	CCONJ
easat-5378	20	18	pebbles	pebble	NOUN
easat-5378	20	19	that	that	PRON
easat-5378	20	20	could	could	AUX
easat-5378	20	21	form	form	VERB
easat-5378	20	22	geometrical	geometrical	ADJ
easat-5378	20	23	figures	figure	NOUN
easat-5378	20	24	,	,	PUNCT
easat-5378	20	25	such	such	ADJ
easat-5378	20	26	as	as	ADP
easat-5378	20	27	a	a	DET
easat-5378	20	28	triangle	triangle	NOUN
easat-5378	21	1	[	[	X
easat-5378	21	2	2	2	NUM
easat-5378	21	3	]	]	PUNCT
easat-5378	21	4	.	.	PUNCT
easat-5378	22	1	while	while	SCONJ
easat-5378	22	2	the	the	DET
easat-5378	22	3	triangular	triangular	NOUN
easat-5378	22	4	numbers	number	NOUN
easat-5378	22	5	were	be	AUX
easat-5378	22	6	known	know	VERB
easat-5378	22	7	to	to	ADP
easat-5378	22	8	the	the	DET
easat-5378	22	9	pythagoreans	pythagorean	NOUN
easat-5378	22	10	of	of	ADP
easat-5378	22	11	ancient	ancient	ADJ
easat-5378	22	12	greece	greece	PROPN
easat-5378	22	13	,	,	PUNCT
easat-5378	22	14	the	the	DET
easat-5378	22	15	factorials	factorial	NOUN
easat-5378	22	16	were	be	AUX
easat-5378	22	17	known	know	VERB
easat-5378	22	18	to	to	ADP
easat-5378	22	19	the	the	DET
easat-5378	22	20	jains	jains	PROPN
easat-5378	22	21	of	of	ADP
easat-5378	22	22	ancient	ancient	ADJ
easat-5378	22	23	india	india	PROPN
easat-5378	22	24	and	and	CCONJ
easat-5378	22	25	to	to	ADP
easat-5378	22	26	the	the	DET
easat-5378	22	27	hebrews	hebrew	NOUN
easat-5378	22	28	of	of	ADP
easat-5378	22	29	ancient	ancient	ADJ
easat-5378	22	30	middle	middle	PROPN
easat-5378	22	31	east	east	PROPN
easat-5378	22	32	.	.	PUNCT
easat-5378	23	1	although	although	SCONJ
easat-5378	23	2	greek	greek	ADJ
easat-5378	23	3	mathematics	mathematic	NOUN
easat-5378	23	4	included	include	VERB
easat-5378	23	5	combinatorics	combinatoric	NOUN
easat-5378	23	6	,	,	PUNCT
easat-5378	23	7	there	there	PRON
easat-5378	23	8	is	be	VERB
easat-5378	23	9	no	no	DET
easat-5378	23	10	direct	direct	ADJ
easat-5378	23	11	evidence	evidence	NOUN
easat-5378	23	12	of	of	ADP
easat-5378	23	13	ancient	ancient	ADJ
easat-5378	23	14	greek	greek	ADJ
easat-5378	23	15	study	study	NOUN
easat-5378	23	16	of	of	ADP
easat-5378	23	17	factorials	factorial	NOUN
easat-5378	23	18	.	.	PUNCT
easat-5378	24	1	it	it	PRON
easat-5378	24	2	is	be	AUX
easat-5378	24	3	in	in	ADP
easat-5378	24	4	about	about	ADP
easat-5378	24	5	mid-17th	mid-17th	NOUN
easat-5378	24	6	to	to	ADP
easat-5378	24	7	the	the	DET
easat-5378	24	8	early	early	ADJ
easat-5378	24	9	18th	18th	ADJ
easat-5378	24	10	century	century	NOUN
easat-5378	24	11	that	that	SCONJ
easat-5378	24	12	the	the	DET
easat-5378	24	13	factorial	factorial	ADJ
easat-5378	24	14	function	function	NOUN
easat-5378	24	15	was	be	AUX
easat-5378	24	16	intensively	intensively	ADV
easat-5378	24	17	studied	study	VERB
easat-5378	24	18	by	by	ADP
easat-5378	24	19	leading	lead	VERB
easat-5378	24	20	mathematicians	mathematician	NOUN
easat-5378	24	21	of	of	ADP
easat-5378	24	22	the	the	DET
easat-5378	24	23	period	period	NOUN
easat-5378	24	24	including	include	VERB
easat-5378	24	25	[	[	X
easat-5378	24	26	3	3	NUM
easat-5378	24	27	-	-	SYM
easat-5378	24	28	5	5	NUM
easat-5378	24	29	]	]	PUNCT
easat-5378	24	30	.	.	PUNCT
easat-5378	25	1	the	the	DET
easat-5378	25	2	literature	literature	NOUN
easat-5378	25	3	also	also	ADV
easat-5378	25	4	provides	provide	VERB
easat-5378	25	5	some	some	DET
easat-5378	25	6	expositions	exposition	NOUN
easat-5378	25	7	on	on	ADP
easat-5378	25	8	triangular	triangular	NOUN
easat-5378	25	9	numbers	number	NOUN
easat-5378	25	10	[	[	X
easat-5378	25	11	1	1	X
easat-5378	25	12	]	]	X
easat-5378	25	13	early	early	ADJ
easat-5378	25	14	works	work	NOUN
easat-5378	25	15	on	on	ADP
easat-5378	25	16	factorial	factorial	ADJ
easat-5378	25	17	function	function	NOUN
easat-5378	25	18	[	[	X
easat-5378	25	19	3	3	NUM
easat-5378	25	20	]	]	PUNCT
easat-5378	25	21	and	and	CCONJ
easat-5378	25	22	some	some	DET
easat-5378	25	23	early	early	ADJ
easat-5378	25	24	and	and	CCONJ
easat-5378	25	25	recent	recent	ADJ
easat-5378	25	26	studies	study	NOUN
easat-5378	25	27	on	on	ADP
easat-5378	25	28	triangular	triangular	NOUN
easat-5378	25	29	numbers	number	NOUN
easat-5378	25	30	,	,	PUNCT
easat-5378	25	31	factorials	factorial	NOUN
easat-5378	25	32	,	,	PUNCT
easat-5378	25	33	and	and	CCONJ
easat-5378	25	34	other	other	ADJ
easat-5378	25	35	related	related	ADJ
easat-5378	25	36	numbers	number	NOUN
easat-5378	25	37	[	[	X
easat-5378	25	38	6	6	NUM
easat-5378	25	39	]	]	PUNCT
easat-5378	25	40	.	.	PUNCT
easat-5378	26	1	generalization	generalization	NOUN
easat-5378	26	2	is	be	AUX
easat-5378	26	3	an	an	DET
easat-5378	26	4	important	important	ADJ
easat-5378	26	5	part	part	NOUN
easat-5378	26	6	of	of	ADP
easat-5378	26	7	mathematics	mathematic	NOUN
easat-5378	26	8	and	and	CCONJ
easat-5378	26	9	it	it	PRON
easat-5378	26	10	serves	serve	VERB
easat-5378	26	11	as	as	ADP
easat-5378	26	12	a	a	DET
easat-5378	26	13	tool	tool	NOUN
easat-5378	26	14	in	in	ADP
easat-5378	26	15	constructing	construct	VERB
easat-5378	26	16	new	new	ADJ
easat-5378	26	17	knowledge	knowledge	NOUN
easat-5378	27	1	[	[	X
easat-5378	27	2	7	7	NUM
easat-5378	27	3	]	]	PUNCT
easat-5378	27	4	.	.	PUNCT
easat-5378	28	1	mathematical	mathematical	ADJ
easat-5378	28	2	generalization	generalization	NOUN
easat-5378	28	3	encompasses	encompass	VERB
easat-5378	28	4	a	a	DET
easat-5378	28	5	claim	claim	NOUN
easat-5378	28	6	that	that	SCONJ
easat-5378	28	7	some	some	DET
easat-5378	28	8	property	property	NOUN
easat-5378	28	9	or	or	CCONJ
easat-5378	28	10	techniques	technique	NOUN
easat-5378	28	11	holds	hold	VERB
easat-5378	28	12	for	for	ADP
easat-5378	28	13	a	a	DET
easat-5378	28	14	set	set	NOUN
easat-5378	28	15	of	of	ADP
easat-5378	28	16	mathematical	mathematical	ADJ
easat-5378	28	17	objects	object	NOUN
easat-5378	28	18	or	or	CCONJ
easat-5378	28	19	conditions	condition	NOUN
easat-5378	28	20	,	,	PUNCT
easat-5378	28	21	the	the	DET
easat-5378	28	22	scope	scope	NOUN
easat-5378	28	23	of	of	ADP
easat-5378	28	24	which	which	PRON
easat-5378	28	25	is	be	AUX
easat-5378	28	26	always	always	ADV
easat-5378	28	27	larger	large	ADJ
easat-5378	28	28	than	than	ADP
easat-5378	28	29	the	the	DET
easat-5378	28	30	set	set	NOUN
easat-5378	28	31	of	of	ADP
easat-5378	28	32	individually	individually	ADV
easat-5378	28	33	verified	verify	VERB
easat-5378	28	34	cases	case	NOUN
easat-5378	28	35	[	[	X
easat-5378	28	36	8	8	NUM
easat-5378	28	37	]	]	PUNCT
easat-5378	28	38	.	.	PUNCT
easat-5378	29	1	like	like	ADP
easat-5378	29	2	any	any	DET
easat-5378	29	3	other	other	ADJ
easat-5378	29	4	theorem	theorem	NOUN
easat-5378	29	5	,	,	PUNCT
easat-5378	29	6	a	a	DET
easat-5378	29	7	generalization	generalization	NOUN
easat-5378	29	8	is	be	AUX
easat-5378	29	9	accepted	accept	VERB
easat-5378	29	10	to	to	PART
easat-5378	29	11	be	be	AUX
easat-5378	29	12	true	true	ADJ
easat-5378	29	13	if	if	SCONJ
easat-5378	29	14	and	and	CCONJ
easat-5378	29	15	only	only	ADV
easat-5378	29	16	if	if	SCONJ
easat-5378	29	17	it	it	PRON
easat-5378	29	18	is	be	AUX
easat-5378	29	19	supported	support	VERB
easat-5378	29	20	by	by	ADP
easat-5378	29	21	a	a	DET
easat-5378	29	22	valid	valid	ADJ
easat-5378	29	23	proof	proof	NOUN
easat-5378	29	24	.	.	PUNCT
easat-5378	30	1	generalizations	generalization	NOUN
easat-5378	30	2	have	have	AUX
easat-5378	30	3	been	be	AUX
easat-5378	30	4	applied	apply	VERB
easat-5378	30	5	to	to	ADP
easat-5378	30	6	a	a	DET
easat-5378	30	7	variety	variety	NOUN
easat-5378	30	8	of	of	ADP
easat-5378	30	9	numbertheoretic	numbertheoretic	ADJ
easat-5378	30	10	problems	problem	NOUN
easat-5378	30	11	.	.	PUNCT
easat-5378	31	1	the	the	DET
easat-5378	31	2	triangular	triangular	NOUN
easat-5378	31	3	numbers	number	NOUN
easat-5378	31	4	,	,	PUNCT
easat-5378	31	5	the	the	DET
easat-5378	31	6	factorials	factorial	NOUN
easat-5378	31	7	,	,	PUNCT
easat-5378	31	8	and	and	CCONJ
easat-5378	31	9	many	many	ADJ
easat-5378	31	10	number	number	NOUN
easat-5378	31	11	-	-	PUNCT
easat-5378	31	12	theoretic	theoretic	NOUN
easat-5378	31	13	theorems	theorem	NOUN
easat-5378	31	14	https://orcid.org/0000-0002-5797-7854	https://orcid.org/0000-0002-5797-7854	PROPN
easat-5378	31	15	mailto:romer.castillo@g.batstate-u.edu.ph	mailto:romer.castillo@g.batstate-u.edu.ph	PROPN
easat-5378	31	16	892	892	NUM
easat-5378	31	17	edelweiss	edelweiss	PROPN
easat-5378	31	18	applied	apply	VERB
easat-5378	31	19	science	science	NOUN
easat-5378	31	20	and	and	CCONJ
easat-5378	31	21	technology	technology	NOUN
easat-5378	31	22	issn	issn	PROPN
easat-5378	31	23	:	:	PUNCT
easat-5378	31	24	2576	2576	NUM
easat-5378	31	25	-	-	SYM
easat-5378	31	26	8484	8484	NUM
easat-5378	31	27	vol	vol	NOUN
easat-5378	31	28	.	.	PROPN
easat-5378	32	1	9	9	NUM
easat-5378	32	2	,	,	PUNCT
easat-5378	32	3	no	no	INTJ
easat-5378	32	4	.	.	NOUN
easat-5378	33	1	3	3	NUM
easat-5378	33	2	:	:	PUNCT
easat-5378	33	3	891	891	NUM
easat-5378	33	4	-	-	SYM
easat-5378	33	5	904	904	NUM
easat-5378	33	6	,	,	PUNCT
easat-5378	33	7	2025	2025	NUM
easat-5378	33	8	doi	doi	NOUN
easat-5378	33	9	:	:	PUNCT
easat-5378	33	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	33	11	©	©	PROPN
easat-5378	33	12	2025	2025	NUM
easat-5378	33	13	by	by	ADP
easat-5378	33	14	the	the	DET
easat-5378	33	15	author	author	NOUN
easat-5378	33	16	;	;	PUNCT
easat-5378	33	17	licensee	licensee	PROPN
easat-5378	33	18	learning	learning	NOUN
easat-5378	33	19	gate	gate	NOUN
easat-5378	33	20	related	relate	VERB
easat-5378	33	21	to	to	ADP
easat-5378	33	22	them	they	PRON
easat-5378	33	23	have	have	AUX
easat-5378	33	24	been	be	AUX
easat-5378	33	25	generalized	generalize	VERB
easat-5378	33	26	in	in	ADP
easat-5378	33	27	several	several	ADJ
easat-5378	33	28	ways	way	NOUN
easat-5378	33	29	in	in	ADP
easat-5378	33	30	the	the	DET
easat-5378	33	31	previous	previous	ADJ
easat-5378	33	32	studies	study	NOUN
easat-5378	33	33	.	.	PUNCT
easat-5378	34	1	for	for	ADP
easat-5378	34	2	instance	instance	NOUN
easat-5378	34	3	,	,	PUNCT
easat-5378	34	4	an	an	DET
easat-5378	34	5	expository	expository	ADJ
easat-5378	34	6	article	article	NOUN
easat-5378	34	7	discusses	discuss	VERB
easat-5378	34	8	some	some	DET
easat-5378	34	9	generalized	generalized	ADJ
easat-5378	34	10	factorials	factorial	NOUN
easat-5378	34	11	[	[	X
easat-5378	34	12	9	9	NUM
easat-5378	34	13	]	]	PUNCT
easat-5378	34	14	.	.	PUNCT
easat-5378	35	1	a	a	DET
easat-5378	35	2	more	more	ADV
easat-5378	35	3	recent	recent	ADJ
easat-5378	35	4	article	article	NOUN
easat-5378	35	5	generalizes	generalize	VERB
easat-5378	35	6	triangular	triangular	NOUN
easat-5378	35	7	numbers	number	NOUN
easat-5378	35	8	to	to	ADP
easat-5378	35	9	arbitrary	arbitrary	ADJ
easat-5378	35	10	higher	higher	ADV
easat-5378	35	11	-	-	PUNCT
easat-5378	35	12	dimensional	dimensional	ADJ
easat-5378	35	13	spaces	space	NOUN
easat-5378	35	14	[	[	X
easat-5378	35	15	10	10	NUM
easat-5378	35	16	]	]	PUNCT
easat-5378	35	17	.	.	PUNCT
easat-5378	36	1	triangular	triangular	NOUN
easat-5378	36	2	numbers	number	NOUN
easat-5378	36	3	and	and	CCONJ
easat-5378	36	4	factorials	factorial	NOUN
easat-5378	36	5	are	be	AUX
easat-5378	36	6	associated	associate	VERB
easat-5378	36	7	in	in	ADP
easat-5378	36	8	such	such	DET
easat-5378	36	9	a	a	DET
easat-5378	36	10	way	way	NOUN
easat-5378	36	11	that	that	PRON
easat-5378	36	12	triangular	triangular	NOUN
easat-5378	36	13	numbers	number	NOUN
easat-5378	36	14	are	be	AUX
easat-5378	36	15	the	the	DET
easat-5378	36	16	additive	additive	ADJ
easat-5378	36	17	analogs	analog	NOUN
easat-5378	36	18	of	of	ADP
easat-5378	36	19	the	the	DET
easat-5378	36	20	factorials	factorial	NOUN
easat-5378	36	21	[	[	X
easat-5378	36	22	11	11	NUM
easat-5378	36	23	]	]	PUNCT
easat-5378	36	24	.	.	PUNCT
easat-5378	37	1	the	the	DET
easat-5378	37	2	apparent	apparent	ADJ
easat-5378	37	3	natural	natural	ADJ
easat-5378	37	4	connection	connection	NOUN
easat-5378	37	5	between	between	ADP
easat-5378	37	6	these	these	DET
easat-5378	37	7	two	two	NUM
easat-5378	37	8	sequences	sequence	NOUN
easat-5378	37	9	of	of	ADP
easat-5378	37	10	numbers	number	NOUN
easat-5378	37	11	contributed	contribute	VERB
easat-5378	37	12	to	to	ADP
easat-5378	37	13	the	the	DET
easat-5378	37	14	interest	interest	NOUN
easat-5378	37	15	of	of	ADP
easat-5378	37	16	adding	add	VERB
easat-5378	37	17	corresponding	correspond	VERB
easat-5378	37	18	factorials	factorial	NOUN
easat-5378	37	19	and	and	CCONJ
easat-5378	37	20	triangular	triangular	NOUN
easat-5378	37	21	numbers	number	NOUN
easat-5378	37	22	to	to	PART
easat-5378	37	23	form	form	VERB
easat-5378	37	24	a	a	DET
easat-5378	37	25	new	new	ADJ
easat-5378	37	26	sequence	sequence	NOUN
easat-5378	37	27	of	of	ADP
easat-5378	37	28	integers	integer	NOUN
easat-5378	37	29	,	,	PUNCT
easat-5378	37	30	which	which	PRON
easat-5378	37	31	is	be	AUX
easat-5378	37	32	called	call	VERB
easat-5378	37	33	factoriangular	factoriangular	ADJ
easat-5378	37	34	numbers	number	NOUN
easat-5378	37	35	[	[	X
easat-5378	37	36	12	12	NUM
easat-5378	37	37	]	]	PUNCT
easat-5378	37	38	.	.	PUNCT
easat-5378	38	1	this	this	DET
easat-5378	38	2	relatively	relatively	ADV
easat-5378	38	3	new	new	ADJ
easat-5378	38	4	sequence	sequence	NOUN
easat-5378	38	5	is	be	AUX
easat-5378	38	6	included	include	VERB
easat-5378	38	7	in	in	ADP
easat-5378	38	8	the	the	DET
easat-5378	38	9	online	online	ADJ
easat-5378	38	10	encyclopedia	encyclopedia	NOUN
easat-5378	38	11	of	of	ADP
easat-5378	38	12	integer	integer	NOUN
easat-5378	38	13	sequences	sequence	NOUN
easat-5378	38	14	(	(	PUNCT
easat-5378	38	15	oeis	oeis	NOUN
easat-5378	38	16	)	)	PUNCT
easat-5378	38	17	as	as	ADP
easat-5378	38	18	entry	entry	NOUN
easat-5378	38	19	a101292	a101292	ADP
easat-5378	38	20	[	[	X
easat-5378	38	21	13	13	NUM
easat-5378	38	22	]	]	PUNCT
easat-5378	38	23	.	.	PUNCT
easat-5378	39	1	with	with	ADP
easat-5378	39	2	the	the	DET
easat-5378	39	3	introduction	introduction	NOUN
easat-5378	39	4	of	of	ADP
easat-5378	39	5	factoriangular	factoriangular	ADJ
easat-5378	39	6	numbers	number	NOUN
easat-5378	39	7	,	,	PUNCT
easat-5378	39	8	the	the	DET
easat-5378	39	9	literature	literature	NOUN
easat-5378	39	10	now	now	ADV
easat-5378	39	11	provides	provide	VERB
easat-5378	39	12	fibonacci	fibonacci	NOUN
easat-5378	39	13	factoriangular	factoriangular	ADJ
easat-5378	39	14	numbers	number	NOUN
easat-5378	39	15	[	[	X
easat-5378	39	16	14	14	NUM
easat-5378	39	17	]	]	X
easat-5378	39	18	pell	pell	VERB
easat-5378	39	19	factoriangular	factoriangular	ADJ
easat-5378	39	20	numbers	number	NOUN
easat-5378	39	21	[	[	X
easat-5378	39	22	15	15	NUM
easat-5378	39	23	]	]	X
easat-5378	39	24	lucas	lucas	ADJ
easat-5378	39	25	factoriangular	factoriangular	ADJ
easat-5378	39	26	numbers	number	NOUN
easat-5378	39	27	[	[	X
easat-5378	39	28	16	16	NUM
easat-5378	39	29	]	]	X
easat-5378	39	30	factoriangular	factoriangular	ADJ
easat-5378	39	31	numbers	number	NOUN
easat-5378	39	32	in	in	ADP
easat-5378	39	33	balancing	balancing	NOUN
easat-5378	39	34	and	and	CCONJ
easat-5378	39	35	lucas	lucas	NOUN
easat-5378	39	36	-	-	PUNCT
easat-5378	39	37	balancing	balance	VERB
easat-5378	39	38	sequence	sequence	NOUN
easat-5378	39	39	[	[	X
easat-5378	39	40	17	17	NUM
easat-5378	39	41	]	]	PUNCT
easat-5378	39	42	and	and	CCONJ
easat-5378	39	43	multiple	multiple	ADJ
easat-5378	39	44	factoriangular	factoriangular	ADJ
easat-5378	39	45	numbers	number	NOUN
easat-5378	39	46	[	[	X
easat-5378	39	47	18	18	NUM
easat-5378	39	48	]	]	PUNCT
easat-5378	39	49	.	.	PUNCT
easat-5378	40	1	the	the	DET
easat-5378	40	2	multiple	multiple	ADJ
easat-5378	40	3	factoriangular	factoriangular	ADJ
easat-5378	40	4	number	number	NOUN
easat-5378	40	5	is	be	AUX
easat-5378	40	6	a	a	DET
easat-5378	40	7	generalization	generalization	NOUN
easat-5378	40	8	of	of	ADP
easat-5378	40	9	the	the	DET
easat-5378	40	10	factoriangular	factoriangular	ADJ
easat-5378	40	11	number	number	NOUN
easat-5378	40	12	.	.	PUNCT
easat-5378	41	1	several	several	ADJ
easat-5378	41	2	articles	article	NOUN
easat-5378	41	3	also	also	ADV
easat-5378	41	4	discuss	discuss	VERB
easat-5378	41	5	some	some	DET
easat-5378	41	6	other	other	ADJ
easat-5378	41	7	generalizations	generalization	NOUN
easat-5378	41	8	of	of	ADP
easat-5378	41	9	factoriangular	factoriangular	ADJ
easat-5378	41	10	numbers	number	NOUN
easat-5378	41	11	[	[	X
easat-5378	41	12	19	19	NUM
easat-5378	41	13	,	,	PUNCT
easat-5378	41	14	20	20	NUM
easat-5378	41	15	]	]	PUNCT
easat-5378	41	16	.	.	PUNCT
easat-5378	42	1	in	in	ADP
easat-5378	42	2	this	this	DET
easat-5378	42	3	expository	expository	ADJ
easat-5378	42	4	paper	paper	NOUN
easat-5378	42	5	,	,	PUNCT
easat-5378	42	6	we	we	PRON
easat-5378	42	7	provide	provide	VERB
easat-5378	42	8	some	some	DET
easat-5378	42	9	explorations	exploration	NOUN
easat-5378	42	10	on	on	ADP
easat-5378	42	11	further	further	ADJ
easat-5378	42	12	generalizations	generalization	NOUN
easat-5378	42	13	of	of	ADP
easat-5378	42	14	factoriangular	factoriangular	ADJ
easat-5378	42	15	numbers	number	NOUN
easat-5378	42	16	.	.	PUNCT
easat-5378	43	1	2	2	X
easat-5378	43	2	.	.	X
easat-5378	43	3	methodology	methodology	NOUN
easat-5378	43	4	this	this	DET
easat-5378	43	5	expository	expository	ADJ
easat-5378	43	6	article	article	NOUN
easat-5378	43	7	is	be	AUX
easat-5378	43	8	a	a	DET
easat-5378	43	9	result	result	NOUN
easat-5378	43	10	of	of	ADP
easat-5378	43	11	discipline	discipline	NOUN
easat-5378	43	12	-	-	PUNCT
easat-5378	43	13	based	base	VERB
easat-5378	43	14	scholarship	scholarship	NOUN
easat-5378	43	15	of	of	ADP
easat-5378	43	16	discovery	discovery	NOUN
easat-5378	43	17	particularly	particularly	ADV
easat-5378	43	18	,	,	PUNCT
easat-5378	43	19	basic	basic	ADJ
easat-5378	43	20	research	research	NOUN
easat-5378	43	21	in	in	ADP
easat-5378	43	22	number	number	NOUN
easat-5378	43	23	theory	theory	NOUN
easat-5378	43	24	.	.	PUNCT
easat-5378	44	1	we	we	PRON
easat-5378	44	2	employ	employ	VERB
easat-5378	44	3	an	an	DET
easat-5378	44	4	exploratory	exploratory	ADJ
easat-5378	44	5	method	method	NOUN
easat-5378	44	6	involving	involve	VERB
easat-5378	44	7	the	the	DET
easat-5378	44	8	scientific	scientific	ADJ
easat-5378	44	9	approach	approach	NOUN
easat-5378	44	10	of	of	ADP
easat-5378	44	11	experimental	experimental	ADJ
easat-5378	44	12	mathematics	mathematic	NOUN
easat-5378	44	13	.	.	PUNCT
easat-5378	45	1	experimental	experimental	ADJ
easat-5378	45	2	mathematics	mathematics	PROPN
easat-5378	45	3	is	be	AUX
easat-5378	45	4	the	the	DET
easat-5378	45	5	methodology	methodology	NOUN
easat-5378	45	6	of	of	ADP
easat-5378	45	7	doing	do	VERB
easat-5378	45	8	mathematics	mathematic	NOUN
easat-5378	45	9	that	that	PRON
easat-5378	45	10	includes	include	VERB
easat-5378	45	11	the	the	DET
easat-5378	45	12	use	use	NOUN
easat-5378	45	13	of	of	ADP
easat-5378	45	14	computations	computation	NOUN
easat-5378	45	15	for	for	ADP
easat-5378	45	16	gaining	gain	VERB
easat-5378	45	17	insight	insight	NOUN
easat-5378	45	18	and	and	CCONJ
easat-5378	45	19	intuition	intuition	NOUN
easat-5378	45	20	,	,	PUNCT
easat-5378	45	21	discovering	discover	VERB
easat-5378	45	22	new	new	ADJ
easat-5378	45	23	patterns	pattern	NOUN
easat-5378	45	24	and	and	CCONJ
easat-5378	45	25	relationships	relationship	NOUN
easat-5378	45	26	,	,	PUNCT
easat-5378	45	27	using	use	VERB
easat-5378	45	28	graphical	graphical	ADJ
easat-5378	45	29	displays	display	NOUN
easat-5378	45	30	to	to	PART
easat-5378	45	31	suggest	suggest	VERB
easat-5378	45	32	underlying	underlying	ADJ
easat-5378	45	33	mathematical	mathematical	ADJ
easat-5378	45	34	principles	principle	NOUN
easat-5378	45	35	,	,	PUNCT
easat-5378	45	36	testing	testing	NOUN
easat-5378	45	37	and	and	CCONJ
easat-5378	45	38	especially	especially	ADV
easat-5378	45	39	falsifying	falsify	VERB
easat-5378	45	40	conjectures	conjecture	NOUN
easat-5378	45	41	,	,	PUNCT
easat-5378	45	42	exploring	explore	VERB
easat-5378	45	43	a	a	DET
easat-5378	45	44	possible	possible	ADJ
easat-5378	45	45	result	result	NOUN
easat-5378	45	46	to	to	PART
easat-5378	45	47	see	see	VERB
easat-5378	45	48	if	if	SCONJ
easat-5378	45	49	it	it	PRON
easat-5378	45	50	is	be	AUX
easat-5378	45	51	worth	worth	ADJ
easat-5378	45	52	formal	formal	ADJ
easat-5378	45	53	proof	proof	NOUN
easat-5378	45	54	,	,	PUNCT
easat-5378	45	55	suggesting	suggest	VERB
easat-5378	45	56	approaches	approach	NOUN
easat-5378	45	57	for	for	ADP
easat-5378	45	58	formal	formal	ADJ
easat-5378	45	59	proof	proof	NOUN
easat-5378	45	60	,	,	PUNCT
easat-5378	45	61	replacing	replace	VERB
easat-5378	45	62	lengthy	lengthy	ADJ
easat-5378	45	63	hand	hand	NOUN
easat-5378	45	64	derivations	derivation	NOUN
easat-5378	45	65	with	with	ADP
easat-5378	45	66	computer	computer	NOUN
easat-5378	45	67	-	-	PUNCT
easat-5378	45	68	based	base	VERB
easat-5378	45	69	derivations	derivation	NOUN
easat-5378	45	70	,	,	PUNCT
easat-5378	45	71	and	and	CCONJ
easat-5378	45	72	confirming	confirm	VERB
easat-5378	45	73	analytically	analytically	ADV
easat-5378	45	74	derived	derive	VERB
easat-5378	45	75	results	result	NOUN
easat-5378	45	76	[	[	X
easat-5378	45	77	21	21	NUM
easat-5378	45	78	,	,	PUNCT
easat-5378	45	79	22	22	NUM
easat-5378	45	80	]	]	PUNCT
easat-5378	45	81	.	.	PUNCT
easat-5378	46	1	more	more	ADV
easat-5378	46	2	particularly	particularly	ADV
easat-5378	46	3	,	,	PUNCT
easat-5378	46	4	we	we	PRON
easat-5378	46	5	use	use	VERB
easat-5378	46	6	the	the	DET
easat-5378	46	7	mathematical	mathematical	ADJ
easat-5378	46	8	method	method	NOUN
easat-5378	47	1	[	[	X
easat-5378	47	2	22	22	NUM
easat-5378	47	3	]	]	PUNCT
easat-5378	47	4	presented	present	VERB
easat-5378	47	5	below	below	ADV
easat-5378	47	6	:	:	PUNCT
easat-5378	47	7	figure	figure	VERB
easat-5378	47	8	1	1	NUM
easat-5378	47	9	.	.	PUNCT
easat-5378	47	10	mathematical	mathematical	ADJ
easat-5378	47	11	method	method	NOUN
easat-5378	47	12	.	.	PUNCT
easat-5378	48	1	893	893	NUM
easat-5378	48	2	edelweiss	edelweiss	PROPN
easat-5378	48	3	applied	apply	VERB
easat-5378	48	4	science	science	NOUN
easat-5378	48	5	and	and	CCONJ
easat-5378	48	6	technology	technology	NOUN
easat-5378	48	7	issn	issn	PROPN
easat-5378	48	8	:	:	PUNCT
easat-5378	48	9	2576	2576	NUM
easat-5378	48	10	-	-	SYM
easat-5378	48	11	8484	8484	NUM
easat-5378	48	12	vol	vol	NOUN
easat-5378	48	13	.	.	PROPN
easat-5378	49	1	9	9	NUM
easat-5378	49	2	,	,	PUNCT
easat-5378	49	3	no	no	INTJ
easat-5378	49	4	.	.	NOUN
easat-5378	50	1	3	3	NUM
easat-5378	50	2	:	:	PUNCT
easat-5378	50	3	891	891	NUM
easat-5378	50	4	-	-	SYM
easat-5378	50	5	904	904	NUM
easat-5378	50	6	,	,	PUNCT
easat-5378	50	7	2025	2025	NUM
easat-5378	50	8	doi	doi	NOUN
easat-5378	50	9	:	:	PUNCT
easat-5378	50	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	50	11	©	©	PROPN
easat-5378	50	12	2025	2025	NUM
easat-5378	50	13	by	by	ADP
easat-5378	50	14	the	the	DET
easat-5378	50	15	author	author	NOUN
easat-5378	50	16	;	;	PUNCT
easat-5378	50	17	licensee	licensee	PROPN
easat-5378	50	18	learning	learn	VERB
easat-5378	50	19	gate	gate	NOUN
easat-5378	50	20	in	in	ADP
easat-5378	50	21	the	the	DET
easat-5378	50	22	next	next	ADJ
easat-5378	50	23	section	section	NOUN
easat-5378	50	24	,	,	PUNCT
easat-5378	50	25	we	we	PRON
easat-5378	50	26	present	present	VERB
easat-5378	50	27	some	some	DET
easat-5378	50	28	results	result	NOUN
easat-5378	50	29	of	of	ADP
easat-5378	50	30	the	the	DET
easat-5378	50	31	previous	previous	ADJ
easat-5378	50	32	studies	study	NOUN
easat-5378	50	33	as	as	ADP
easat-5378	50	34	preliminaries	preliminary	NOUN
easat-5378	50	35	.	.	PUNCT
easat-5378	51	1	we	we	PRON
easat-5378	51	2	then	then	ADV
easat-5378	51	3	extend	extend	VERB
easat-5378	51	4	the	the	DET
easat-5378	51	5	results	result	NOUN
easat-5378	51	6	of	of	ADP
easat-5378	51	7	previous	previous	ADJ
easat-5378	51	8	studies	study	NOUN
easat-5378	51	9	on	on	ADP
easat-5378	51	10	the	the	DET
easat-5378	51	11	generalizations	generalization	NOUN
easat-5378	51	12	of	of	ADP
easat-5378	51	13	factoriangular	factoriangular	ADJ
easat-5378	51	14	numbers	number	NOUN
easat-5378	51	15	to	to	PART
easat-5378	51	16	provide	provide	VERB
easat-5378	51	17	some	some	DET
easat-5378	51	18	explorations	exploration	NOUN
easat-5378	51	19	on	on	ADP
easat-5378	51	20	the	the	DET
easat-5378	51	21	further	further	ADJ
easat-5378	51	22	generalizations	generalization	NOUN
easat-5378	51	23	of	of	ADP
easat-5378	51	24	factoriangular	factoriangular	ADJ
easat-5378	51	25	numbers	number	NOUN
easat-5378	51	26	as	as	ADP
easat-5378	51	27	the	the	DET
easat-5378	51	28	main	main	ADJ
easat-5378	51	29	results	result	NOUN
easat-5378	51	30	.	.	PUNCT
easat-5378	52	1	3	3	X
easat-5378	52	2	.	.	NOUN
easat-5378	52	3	results	result	NOUN
easat-5378	52	4	and	and	CCONJ
easat-5378	52	5	discussion	discussion	NOUN
easat-5378	52	6	3.1	3.1	NUM
easat-5378	52	7	.	.	PUNCT
easat-5378	52	8	preliminaries	preliminary	NOUN
easat-5378	52	9	a	a	DET
easat-5378	52	10	number	number	NOUN
easat-5378	52	11	that	that	PRON
easat-5378	52	12	is	be	AUX
easat-5378	52	13	a	a	DET
easat-5378	52	14	sum	sum	NOUN
easat-5378	52	15	of	of	ADP
easat-5378	52	16	a	a	DET
easat-5378	52	17	factorial	factorial	NOUN
easat-5378	52	18	and	and	CCONJ
easat-5378	52	19	its	its	PRON
easat-5378	52	20	corresponding	corresponding	ADJ
easat-5378	52	21	triangular	triangular	NOUN
easat-5378	52	22	number	number	NOUN
easat-5378	52	23	is	be	AUX
easat-5378	52	24	referred	refer	VERB
easat-5378	52	25	to	to	ADP
easat-5378	52	26	as	as	ADP
easat-5378	52	27	factoriangular	factoriangular	ADJ
easat-5378	52	28	number	number	NOUN
easat-5378	52	29	[	[	X
easat-5378	52	30	10	10	NUM
easat-5378	52	31	]	]	PUNCT
easat-5378	52	32	.	.	PUNCT
easat-5378	53	1	the	the	DET
easat-5378	53	2	factoriangular	factoriangular	ADJ
easat-5378	53	3	number	number	NOUN
easat-5378	53	4	is	be	AUX
easat-5378	53	5	formally	formally	ADV
easat-5378	53	6	defined	define	VERB
easat-5378	53	7	as	as	SCONJ
easat-5378	53	8	follows	follow	VERB
easat-5378	53	9	:	:	PUNCT
easat-5378	53	10	definition	definition	NOUN
easat-5378	53	11	3.1	3.1	NUM
easat-5378	53	12	:	:	PUNCT
easat-5378	53	13	the	the	DET
easat-5378	53	14	nth	nth	ADJ
easat-5378	53	15	factoriangular	factoriangular	ADJ
easat-5378	53	16	number	number	NOUN
easat-5378	53	17	is	be	AUX
easat-5378	53	18	defined	define	VERB
easat-5378	53	19	by	by	ADP
easat-5378	53	20	the	the	DET
easat-5378	53	21	formula	formula	NOUN
easat-5378	53	22	𝐹𝑡𝑛	𝐹𝑡𝑛	PROPN
easat-5378	53	23	=	=	SYM
easat-5378	53	24	𝑛	𝑛	NOUN
easat-5378	53	25	!	!	PUNCT
easat-5378	54	1	+	+	CCONJ
easat-5378	55	1	𝑇𝑛	𝑇𝑛	ADV
easat-5378	55	2	where	where	SCONJ
easat-5378	55	3	𝑛	𝑛	X
easat-5378	55	4	!	!	PUNCT
easat-5378	55	5	=	=	SYM
easat-5378	55	6	1	1	NUM
easat-5378	55	7	⋅	⋅	NUM
easat-5378	55	8	2	2	NUM
easat-5378	55	9	⋅	⋅	X
easat-5378	55	10	3	3	NUM
easat-5378	55	11	⋅⋅⋅	⋅⋅⋅	X
easat-5378	55	12	𝑛	𝑛	PROPN
easat-5378	55	13	and	and	CCONJ
easat-5378	55	14	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	55	15	=	=	NOUN
easat-5378	55	16	1	1	NUM
easat-5378	55	17	+	+	NUM
easat-5378	55	18	2	2	NUM
easat-5378	55	19	+	+	SYM
easat-5378	55	20	3	3	NUM
easat-5378	55	21	+	+	NOUN
easat-5378	55	22	.	.	PUNCT
easat-5378	55	23	.	.	PUNCT
easat-5378	55	24	.	.	PUNCT
easat-5378	56	1	+	+	ADV
easat-5378	56	2	𝑛	𝑛	NOUN
easat-5378	56	3	=	=	SYM
easat-5378	56	4	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	56	5	+	+	CCONJ
easat-5378	56	6	1)/2	1)/2	NOUN
easat-5378	56	7	.	.	PUNCT
easat-5378	57	1	the	the	DET
easat-5378	57	2	first	first	ADJ
easat-5378	57	3	few	few	ADJ
easat-5378	57	4	factoriangular	factoriangular	ADJ
easat-5378	57	5	numbers	number	NOUN
easat-5378	57	6	are	be	AUX
easat-5378	57	7	given	give	VERB
easat-5378	57	8	in	in	ADP
easat-5378	57	9	the	the	DET
easat-5378	57	10	sequence	sequence	NOUN
easat-5378	57	11	{	{	PUNCT
easat-5378	57	12	2	2	NUM
easat-5378	57	13	,	,	PUNCT
easat-5378	57	14	5	5	NUM
easat-5378	57	15	,	,	PUNCT
easat-5378	57	16	12	12	NUM
easat-5378	57	17	,	,	PUNCT
easat-5378	57	18	34	34	NUM
easat-5378	57	19	,	,	PUNCT
easat-5378	57	20	135	135	NUM
easat-5378	57	21	,	,	PUNCT
easat-5378	57	22	741	741	NUM
easat-5378	57	23	,	,	PUNCT
easat-5378	57	24	5068	5068	NUM
easat-5378	57	25	,	,	PUNCT
easat-5378	57	26	40356	40356	NUM
easat-5378	57	27	,	,	PUNCT
easat-5378	57	28	362925	362925	NUM
easat-5378	57	29	,	,	PUNCT
easat-5378	57	30	3628855	3628855	NUM
easat-5378	57	31	,	,	PUNCT
easat-5378	57	32	…	…	PUNCT
easat-5378	57	33	}	}	PUNCT
easat-5378	57	34	.	.	PUNCT
easat-5378	58	1	this	this	DET
easat-5378	58	2	sequence	sequence	NOUN
easat-5378	58	3	,	,	PUNCT
easat-5378	58	4	with	with	ADP
easat-5378	58	5	2	2	NUM
easat-5378	58	6	(	(	PUNCT
easat-5378	58	7	i.e.	i.e.	X
easat-5378	58	8	,	,	PUNCT
easat-5378	58	9	1	1	NUM
easat-5378	58	10	!	!	PUNCT
easat-5378	59	1	+	+	CCONJ
easat-5378	59	2	t1	t1	NOUN
easat-5378	59	3	)	)	PUNCT
easat-5378	59	4	as	as	ADP
easat-5378	59	5	the	the	DET
easat-5378	59	6	first	first	ADJ
easat-5378	59	7	term	term	NOUN
easat-5378	59	8	,	,	PUNCT
easat-5378	59	9	appeared	appear	VERB
easat-5378	59	10	in	in	ADP
easat-5378	59	11	oeis	oeis	NOUN
easat-5378	59	12	as	as	ADP
easat-5378	59	13	a101292	a101292	NOUN
easat-5378	59	14	in	in	ADP
easat-5378	59	15	2004	2004	NUM
easat-5378	59	16	.	.	PUNCT
easat-5378	60	1	in	in	ADP
easat-5378	60	2	2016	2016	NUM
easat-5378	60	3	,	,	PUNCT
easat-5378	60	4	1	1	NUM
easat-5378	60	5	(	(	PUNCT
easat-5378	60	6	i.e.	i.e.	X
easat-5378	60	7	,	,	PUNCT
easat-5378	60	8	0	0	NUM
easat-5378	60	9	!	!	PUNCT
easat-5378	60	10	+	+	CCONJ
easat-5378	60	11	t0	t0	NOUN
easat-5378	60	12	)	)	PUNCT
easat-5378	60	13	was	be	AUX
easat-5378	60	14	appended	append	VERB
easat-5378	60	15	as	as	ADP
easat-5378	60	16	the	the	DET
easat-5378	60	17	first	first	ADJ
easat-5378	60	18	term	term	NOUN
easat-5378	60	19	[	[	X
easat-5378	60	20	13	13	NUM
easat-5378	60	21	]	]	PUNCT
easat-5378	60	22	.	.	PUNCT
easat-5378	61	1	we	we	PRON
easat-5378	61	2	call	call	VERB
easat-5378	61	3	the	the	DET
easat-5378	61	4	numbers	number	NOUN
easat-5378	61	5	in	in	ADP
easat-5378	61	6	this	this	DET
easat-5378	61	7	sequence	sequence	NOUN
easat-5378	61	8	n	n	CCONJ
easat-5378	61	9	-	-	PUNCT
easat-5378	61	10	factoriangular	factoriangular	ADJ
easat-5378	61	11	numbers	number	NOUN
easat-5378	61	12	.	.	PUNCT
easat-5378	62	1	the	the	DET
easat-5378	62	2	terms	term	NOUN
easat-5378	62	3	of	of	ADP
easat-5378	62	4	the	the	DET
easat-5378	62	5	sequence	sequence	NOUN
easat-5378	62	6	of	of	ADP
easat-5378	62	7	factoriangular	factoriangular	ADJ
easat-5378	62	8	numbers	number	NOUN
easat-5378	62	9	{	{	PUNCT
easat-5378	62	10	𝐹𝑡𝑛	𝐹𝑡𝑛	PROPN
easat-5378	62	11	}	}	PUNCT
easat-5378	62	12	for	for	ADP
easat-5378	62	13	natural	natural	ADJ
easat-5378	62	14	numbers	number	NOUN
easat-5378	62	15	𝑛	𝑛	PRON
easat-5378	62	16	≥	≥	NUM
easat-5378	62	17	1	1	NUM
easat-5378	62	18	are	be	AUX
easat-5378	62	19	of	of	ADP
easat-5378	62	20	the	the	DET
easat-5378	62	21	form	form	NOUN
easat-5378	62	22	𝐹𝑡𝑛	𝐹𝑡𝑛	PROPN
easat-5378	62	23	=	=	PUNCT
easat-5378	62	24	(	(	PUNCT
easat-5378	62	25	1	1	NUM
easat-5378	62	26	⋅	⋅	NUM
easat-5378	62	27	2	2	NUM
easat-5378	62	28	⋅	⋅	X
easat-5378	62	29	3	3	NUM
easat-5378	62	30	⋅⋅⋅	⋅⋅⋅	X
easat-5378	62	31	𝑛	𝑛	PROPN
easat-5378	62	32	)	)	PUNCT
easat-5378	62	33	+	+	CCONJ
easat-5378	62	34	(	(	PUNCT
easat-5378	62	35	1	1	NUM
easat-5378	62	36	+	+	NUM
easat-5378	62	37	2	2	NUM
easat-5378	62	38	+	+	SYM
easat-5378	62	39	3	3	NUM
easat-5378	62	40	+	+	NOUN
easat-5378	62	41	.	.	PUNCT
easat-5378	62	42	.	.	PUNCT
easat-5378	62	43	.	.	PUNCT
easat-5378	63	1	+	+	ADV
easat-5378	63	2	𝑛	𝑛	NOUN
easat-5378	63	3	)	)	PUNCT
easat-5378	63	4	.	.	PUNCT
easat-5378	64	1	there	there	PRON
easat-5378	64	2	are	be	VERB
easat-5378	64	3	several	several	ADJ
easat-5378	64	4	ways	way	NOUN
easat-5378	64	5	of	of	ADP
easat-5378	64	6	generalizing	generalize	VERB
easat-5378	64	7	this	this	DET
easat-5378	64	8	sequence	sequence	NOUN
easat-5378	64	9	.	.	PUNCT
easat-5378	65	1	a	a	DET
easat-5378	65	2	generalization	generalization	NOUN
easat-5378	65	3	of	of	ADP
easat-5378	65	4	the	the	DET
easat-5378	65	5	sequence	sequence	NOUN
easat-5378	65	6	of	of	ADP
easat-5378	65	7	n	n	CCONJ
easat-5378	65	8	-	-	PUNCT
easat-5378	65	9	factoriangular	factoriangular	ADJ
easat-5378	65	10	numbers	number	NOUN
easat-5378	65	11	[	[	X
easat-5378	65	12	19	19	NUM
easat-5378	65	13	]	]	PUNCT
easat-5378	65	14	is	be	AUX
easat-5378	65	15	the	the	DET
easat-5378	65	16	sequence	sequence	NOUN
easat-5378	65	17	{	{	PUNCT
easat-5378	65	18	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NOUN
easat-5378	65	19	}	}	PUNCT
easat-5378	65	20	for	for	ADP
easat-5378	65	21	natural	natural	ADJ
easat-5378	65	22	numbers	number	NOUN
easat-5378	65	23	𝑛	𝑛	PROPN
easat-5378	65	24	,	,	PUNCT
easat-5378	65	25	𝑘	𝑘	DET
easat-5378	65	26	≥	≥	NOUN
easat-5378	65	27	1	1	NUM
easat-5378	65	28	,	,	PUNCT
easat-5378	65	29	which	which	PRON
easat-5378	65	30	follow	follow	VERB
easat-5378	65	31	the	the	DET
easat-5378	65	32	form	form	NOUN
easat-5378	65	33	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NUM
easat-5378	65	34	=	=	PUNCT
easat-5378	65	35	(	(	PUNCT
easat-5378	65	36	1	1	NUM
easat-5378	65	37	⋅	⋅	NUM
easat-5378	65	38	2	2	NUM
easat-5378	65	39	⋅	⋅	X
easat-5378	65	40	3	3	NUM
easat-5378	65	41	⋅⋅⋅	⋅⋅⋅	X
easat-5378	65	42	𝑛	𝑛	PROPN
easat-5378	65	43	)	)	PUNCT
easat-5378	65	44	+	+	CCONJ
easat-5378	65	45	(	(	PUNCT
easat-5378	65	46	1	1	NUM
easat-5378	65	47	+	+	NUM
easat-5378	65	48	2	2	NUM
easat-5378	65	49	+	+	SYM
easat-5378	65	50	3	3	NUM
easat-5378	65	51	+	+	NOUN
easat-5378	65	52	.	.	PUNCT
easat-5378	65	53	.	.	PUNCT
easat-5378	65	54	.	.	PUNCT
easat-5378	66	1	+	+	ADV
easat-5378	66	2	𝑘	𝑘	X
easat-5378	66	3	)	)	PUNCT
easat-5378	66	4	we	we	PRON
easat-5378	66	5	call	call	VERB
easat-5378	66	6	the	the	DET
easat-5378	66	7	numbers	number	NOUN
easat-5378	66	8	in	in	ADP
easat-5378	66	9	this	this	DET
easat-5378	66	10	sequence	sequence	NOUN
easat-5378	66	11	(	(	PUNCT
easat-5378	66	12	𝑛	𝑛	ADJ
easat-5378	66	13	,	,	PUNCT
easat-5378	66	14	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	66	15	numbers	number	NOUN
easat-5378	66	16	and	and	CCONJ
easat-5378	66	17	define	define	VERB
easat-5378	66	18	as	as	SCONJ
easat-5378	66	19	follows	follow	VERB
easat-5378	66	20	:	:	PUNCT
easat-5378	66	21	definition	definition	NOUN
easat-5378	66	22	3.2	3.2	NUM
easat-5378	66	23	:	:	PUNCT
easat-5378	66	24	the	the	DET
easat-5378	66	25	(	(	PUNCT
easat-5378	66	26	𝑛	𝑛	ADJ
easat-5378	66	27	,	,	PUNCT
easat-5378	66	28	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	66	29	number	number	NOUN
easat-5378	66	30	is	be	AUX
easat-5378	66	31	defined	define	VERB
easat-5378	66	32	by	by	ADP
easat-5378	66	33	the	the	DET
easat-5378	66	34	formula	formula	NOUN
easat-5378	66	35	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NUM
easat-5378	66	36	=	=	SYM
easat-5378	66	37	𝑛	𝑛	NOUN
easat-5378	66	38	!	!	PUNCT
easat-5378	67	1	+	+	CCONJ
easat-5378	68	1	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	68	2	where	where	SCONJ
easat-5378	68	3	𝑛	𝑛	X
easat-5378	68	4	!	!	PUNCT
easat-5378	68	5	=	=	SYM
easat-5378	69	1	1	1	NUM
easat-5378	69	2	⋅	⋅	NUM
easat-5378	69	3	2	2	NUM
easat-5378	69	4	⋅	⋅	X
easat-5378	69	5	3	3	NUM
easat-5378	69	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	69	7	𝑛	𝑛	PROPN
easat-5378	69	8	and	and	CCONJ
easat-5378	69	9	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	69	10	=	=	SYM
easat-5378	69	11	1	1	NUM
easat-5378	69	12	+	+	NUM
easat-5378	69	13	2	2	NUM
easat-5378	69	14	+	+	SYM
easat-5378	69	15	3	3	NUM
easat-5378	69	16	+	+	NOUN
easat-5378	69	17	.	.	PUNCT
easat-5378	69	18	.	.	PUNCT
easat-5378	69	19	.	.	PUNCT
easat-5378	70	1	+	+	NOUN
easat-5378	70	2	𝑘	𝑘	X
easat-5378	70	3	=	=	X
easat-5378	70	4	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	70	5	+	+	CCONJ
easat-5378	70	6	1)/2	1)/2	NUM
easat-5378	70	7	for	for	ADP
easat-5378	70	8	natural	natural	ADJ
easat-5378	70	9	numbers	number	NOUN
easat-5378	70	10	𝑛	𝑛	PROPN
easat-5378	70	11	,	,	PUNCT
easat-5378	70	12	𝑘	𝑘	DET
easat-5378	70	13	≥	≥	NOUN
easat-5378	70	14	1	1	NUM
easat-5378	70	15	.	.	PUNCT
easat-5378	71	1	from	from	ADP
easat-5378	71	2	definitions	definition	NOUN
easat-5378	71	3	3.1	3.1	NUM
easat-5378	71	4	and	and	CCONJ
easat-5378	71	5	3.2	3.2	NUM
easat-5378	71	6	,	,	PUNCT
easat-5378	71	7	when	when	SCONJ
easat-5378	71	8	𝑛	𝑛	PROPN
easat-5378	71	9	=	=	SYM
easat-5378	71	10	𝑘	𝑘	PROPN
easat-5378	71	11	,	,	PUNCT
easat-5378	71	12	the	the	DET
easat-5378	71	13	(	(	PUNCT
easat-5378	71	14	𝑛	𝑛	ADJ
easat-5378	71	15	,	,	PUNCT
easat-5378	71	16	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	71	17	numbers	number	NOUN
easat-5378	71	18	are	be	AUX
easat-5378	71	19	the	the	DET
easat-5378	71	20	same	same	ADJ
easat-5378	71	21	as	as	ADP
easat-5378	71	22	the	the	DET
easat-5378	71	23	nfactoriangular	nfactoriangular	ADJ
easat-5378	71	24	numbers	number	NOUN
easat-5378	71	25	or	or	CCONJ
easat-5378	71	26	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NUM
easat-5378	71	27	=	=	PUNCT
easat-5378	71	28	𝐹𝑡𝑛.	𝐹𝑡𝑛.	VERB
easat-5378	71	29	the	the	DET
easat-5378	71	30	sequence	sequence	NOUN
easat-5378	71	31	of	of	ADP
easat-5378	71	32	the	the	DET
easat-5378	71	33	(	(	PUNCT
easat-5378	71	34	𝑛	𝑛	PROPN
easat-5378	71	35	,	,	PUNCT
easat-5378	71	36	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	71	37	numbers	number	NOUN
easat-5378	71	38	is	be	AUX
easat-5378	71	39	given	give	VERB
easat-5378	71	40	by	by	ADP
easat-5378	71	41	{	{	PUNCT
easat-5378	71	42	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NOUN
easat-5378	71	43	}	}	PUNCT
easat-5378	71	44	=	=	SYM
easat-5378	71	45	{	{	PUNCT
easat-5378	71	46	2	2	NUM
easat-5378	71	47	,	,	PUNCT
easat-5378	71	48	3	3	NUM
easat-5378	71	49	,	,	PUNCT
easat-5378	71	50	4	4	NUM
easat-5378	71	51	,	,	PUNCT
easat-5378	71	52	5	5	NUM
easat-5378	71	53	,	,	PUNCT
easat-5378	71	54	7	7	NUM
easat-5378	71	55	,	,	PUNCT
easat-5378	71	56	7	7	NUM
easat-5378	71	57	,	,	PUNCT
easat-5378	71	58	9	9	NUM
easat-5378	71	59	,	,	PUNCT
easat-5378	71	60	8	8	NUM
easat-5378	71	61	,	,	PUNCT
easat-5378	71	62	12	12	NUM
easat-5378	71	63	,	,	PUNCT
easat-5378	71	64	25	25	NUM
easat-5378	71	65	,	,	PUNCT
easat-5378	71	66	11	11	NUM
easat-5378	71	67	,	,	PUNCT
easat-5378	71	68	27	27	NUM
easat-5378	71	69	,	,	PUNCT
easat-5378	71	70	12	12	NUM
easat-5378	71	71	,	,	PUNCT
easat-5378	71	72	30	30	NUM
easat-5378	71	73	,	,	PUNCT
easat-5378	71	74	16	16	NUM
easat-5378	71	75	,	,	PUNCT
easat-5378	71	76	34	34	NUM
easat-5378	71	77	,	,	PUNCT
easat-5378	71	78	…	…	PUNCT
easat-5378	71	79	}	}	PUNCT
easat-5378	71	80	for	for	ADP
easat-5378	71	81	(	(	PUNCT
easat-5378	71	82	n	n	CCONJ
easat-5378	71	83	,	,	PUNCT
easat-5378	71	84	k	k	NOUN
easat-5378	71	85	)	)	PUNCT
easat-5378	71	86	=	=	SYM
easat-5378	71	87	(	(	PUNCT
easat-5378	71	88	1,1	1,1	NUM
easat-5378	71	89	)	)	PUNCT
easat-5378	71	90	,	,	PUNCT
easat-5378	71	91	(	(	PUNCT
easat-5378	71	92	2,1	2,1	NUM
easat-5378	71	93	)	)	PUNCT
easat-5378	71	94	,	,	PUNCT
easat-5378	71	95	(	(	PUNCT
easat-5378	71	96	1,2	1,2	NUM
easat-5378	71	97	)	)	PUNCT
easat-5378	71	98	,	,	PUNCT
easat-5378	71	99	(	(	PUNCT
easat-5378	71	100	2,2	2,2	NUM
easat-5378	71	101	)	)	PUNCT
easat-5378	71	102	,	,	PUNCT
easat-5378	71	103	(	(	PUNCT
easat-5378	71	104	3,1	3,1	NUM
easat-5378	71	105	)	)	PUNCT
easat-5378	71	106	,	,	PUNCT
easat-5378	71	107	(	(	PUNCT
easat-5378	71	108	1,3	1,3	NUM
easat-5378	71	109	)	)	PUNCT
easat-5378	71	110	,	,	PUNCT
easat-5378	71	111	(	(	PUNCT
easat-5378	71	112	3,2	3,2	NUM
easat-5378	71	113	)	)	PUNCT
easat-5378	71	114	,	,	PUNCT
easat-5378	71	115	(	(	PUNCT
easat-5378	71	116	2,3	2,3	NUM
easat-5378	71	117	)	)	PUNCT
easat-5378	71	118	,	,	PUNCT
easat-5378	71	119	(	(	PUNCT
easat-5378	71	120	3,3	3,3	NOUN
easat-5378	71	121	)	)	PUNCT
easat-5378	71	122	,	,	PUNCT
easat-5378	71	123	(	(	PUNCT
easat-5378	71	124	4,1	4,1	NOUN
easat-5378	71	125	)	)	PUNCT
easat-5378	71	126	,	,	PUNCT
easat-5378	71	127	(	(	PUNCT
easat-5378	71	128	1,4	1,4	NUM
easat-5378	71	129	)	)	PUNCT
easat-5378	71	130	,	,	PUNCT
easat-5378	71	131	(	(	PUNCT
easat-5378	71	132	4,2	4,2	NUM
easat-5378	71	133	)	)	PUNCT
easat-5378	71	134	,	,	PUNCT
easat-5378	71	135	(	(	PUNCT
easat-5378	71	136	2,4	2,4	NUM
easat-5378	71	137	)	)	PUNCT
easat-5378	71	138	,	,	PUNCT
easat-5378	71	139	(	(	PUNCT
easat-5378	71	140	4,3	4,3	NUM
easat-5378	71	141	)	)	PUNCT
easat-5378	71	142	,	,	PUNCT
easat-5378	71	143	(	(	PUNCT
easat-5378	71	144	3,4	3,4	NUM
easat-5378	71	145	)	)	PUNCT
easat-5378	71	146	,	,	PUNCT
easat-5378	71	147	(	(	PUNCT
easat-5378	71	148	4,4	4,4	NUM
easat-5378	71	149	)	)	PUNCT
easat-5378	71	150	,	,	PUNCT
easat-5378	71	151	…	…	PUNCT
easat-5378	71	152	notice	notice	VERB
easat-5378	71	153	that	that	SCONJ
easat-5378	71	154	in	in	ADP
easat-5378	71	155	this	this	DET
easat-5378	71	156	sequence	sequence	NOUN
easat-5378	71	157	of	of	ADP
easat-5378	71	158	(	(	PUNCT
easat-5378	71	159	𝑛	𝑛	ADJ
easat-5378	71	160	,	,	PUNCT
easat-5378	71	161	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	71	162	numbers	number	NOUN
easat-5378	71	163	,	,	PUNCT
easat-5378	71	164	the	the	DET
easat-5378	71	165	term	term	NOUN
easat-5378	71	166	in	in	ADP
easat-5378	71	167	the	the	DET
easat-5378	71	168	1st	1st	ADJ
easat-5378	71	169	,	,	PUNCT
easat-5378	71	170	4th	4th	ADJ
easat-5378	71	171	,	,	PUNCT
easat-5378	71	172	9th	9th	ADJ
easat-5378	71	173	,	,	PUNCT
easat-5378	71	174	16th	16th	ADJ
easat-5378	71	175	,	,	PUNCT
easat-5378	71	176	and	and	CCONJ
easat-5378	71	177	so	so	ADV
easat-5378	71	178	on	on	ADV
easat-5378	71	179	are	be	AUX
easat-5378	71	180	the	the	DET
easat-5378	71	181	n	n	CCONJ
easat-5378	71	182	-	-	PUNCT
easat-5378	71	183	factoriangular	factoriangular	ADJ
easat-5378	71	184	numbers	number	NOUN
easat-5378	71	185	.	.	PUNCT
easat-5378	72	1	these	these	DET
easat-5378	72	2	terms	term	NOUN
easat-5378	72	3	are	be	AUX
easat-5378	72	4	the	the	DET
easat-5378	72	5	entries	entry	NOUN
easat-5378	72	6	in	in	ADP
easat-5378	72	7	the	the	DET
easat-5378	72	8	main	main	ADJ
easat-5378	72	9	diagonal	diagonal	NOUN
easat-5378	72	10	from	from	ADP
easat-5378	72	11	the	the	DET
easat-5378	72	12	top	top	ADV
easat-5378	72	13	-	-	PUNCT
easat-5378	72	14	left	left	ADJ
easat-5378	72	15	to	to	ADP
easat-5378	72	16	the	the	DET
easat-5378	72	17	bottom	bottom	ADJ
easat-5378	72	18	-	-	PUNCT
easat-5378	72	19	right	right	NOUN
easat-5378	72	20	(	(	PUNCT
easat-5378	72	21	i.e	i.e	X
easat-5378	72	22	,	,	PUNCT
easat-5378	72	23	when	when	SCONJ
easat-5378	72	24	n	n	PROPN
easat-5378	72	25	=	=	SYM
easat-5378	72	26	k	k	NOUN
easat-5378	72	27	)	)	PUNCT
easat-5378	72	28	of	of	ADP
easat-5378	72	29	the	the	DET
easat-5378	72	30	following	follow	VERB
easat-5378	72	31	table	table	NOUN
easat-5378	72	32	:	:	PUNCT
easat-5378	72	33	table	table	NOUN
easat-5378	72	34	1	1	NUM
easat-5378	72	35	.	.	PUNCT
easat-5378	72	36	table	table	NOUN
easat-5378	72	37	of	of	ADP
easat-5378	72	38	(	(	PUNCT
easat-5378	72	39	n	n	CCONJ
easat-5378	72	40	,	,	PUNCT
easat-5378	72	41	k)-factoriangular	k)-factoriangular	ADJ
easat-5378	72	42	numbers	number	NOUN
easat-5378	72	43	.	.	PUNCT
easat-5378	73	1	𝒏	𝒏	PROPN
easat-5378	73	2	\	\	PROPN
easat-5378	73	3	𝒌	𝒌	PRON
easat-5378	73	4	1	1	NUM
easat-5378	73	5	2	2	NUM
easat-5378	73	6	3	3	NUM
easat-5378	73	7	4	4	NUM
easat-5378	73	8	5	5	NUM
easat-5378	73	9	6	6	NUM
easat-5378	73	10	7	7	NUM
easat-5378	73	11	𝒌	𝒌	ADP
easat-5378	73	12	1	1	NUM
easat-5378	73	13	2	2	NUM
easat-5378	73	14	4	4	NUM
easat-5378	73	15	7	7	NUM
easat-5378	73	16	11	11	NUM
easat-5378	73	17	16	16	NUM
easat-5378	73	18	22	22	NUM
easat-5378	73	19	29	29	NUM
easat-5378	73	20	1	1	NUM
easat-5378	73	21	+	+	CCONJ
easat-5378	73	22	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	73	23	2	2	NUM
easat-5378	73	24	3	3	NUM
easat-5378	73	25	5	5	NUM
easat-5378	73	26	8	8	NUM
easat-5378	73	27	12	12	NUM
easat-5378	73	28	17	17	NUM
easat-5378	73	29	23	23	NUM
easat-5378	73	30	30	30	NUM
easat-5378	73	31	2	2	NUM
easat-5378	73	32	+	+	CCONJ
easat-5378	73	33	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	73	34	3	3	NUM
easat-5378	73	35	7	7	NUM
easat-5378	73	36	9	9	NUM
easat-5378	73	37	12	12	NUM
easat-5378	73	38	16	16	NUM
easat-5378	73	39	21	21	NUM
easat-5378	73	40	27	27	NUM
easat-5378	73	41	34	34	NUM
easat-5378	73	42	6	6	NUM
easat-5378	74	1	+	+	CCONJ
easat-5378	75	1	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	75	2	4	4	NUM
easat-5378	75	3	25	25	NUM
easat-5378	75	4	27	27	NUM
easat-5378	75	5	30	30	NUM
easat-5378	75	6	34	34	NUM
easat-5378	75	7	39	39	NUM
easat-5378	75	8	45	45	NUM
easat-5378	75	9	52	52	NUM
easat-5378	75	10	24	24	NUM
easat-5378	75	11	+	+	CCONJ
easat-5378	76	1	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	76	2	5	5	NUM
easat-5378	76	3	121	121	NUM
easat-5378	76	4	123	123	NUM
easat-5378	76	5	126	126	NUM
easat-5378	76	6	130	130	NUM
easat-5378	76	7	135	135	NUM
easat-5378	76	8	141	141	NUM
easat-5378	76	9	148	148	NUM
easat-5378	76	10	120	120	NUM
easat-5378	76	11	+	+	CCONJ
easat-5378	77	1	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	77	2	6	6	NUM
easat-5378	77	3	721	721	NUM
easat-5378	77	4	723	723	NUM
easat-5378	77	5	726	726	NUM
easat-5378	77	6	730	730	NUM
easat-5378	77	7	735	735	NUM
easat-5378	77	8	741	741	NUM
easat-5378	77	9	748	748	NUM
easat-5378	77	10	720	720	NUM
easat-5378	77	11	+	+	CCONJ
easat-5378	78	1	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	78	2	7	7	NUM
easat-5378	78	3	5041	5041	NUM
easat-5378	78	4	5043	5043	NUM
easat-5378	78	5	5046	5046	NUM
easat-5378	78	6	5050	5050	NUM
easat-5378	78	7	5055	5055	NUM
easat-5378	78	8	5061	5061	NUM
easat-5378	78	9	5068	5068	NUM
easat-5378	78	10	5040	5040	NUM
easat-5378	79	1	+	+	CCONJ
easat-5378	79	2	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	79	3	𝑛	𝑛	DET
easat-5378	79	4	𝑛	𝑛	NOUN
easat-5378	79	5	!	!	PUNCT
easat-5378	80	1	+	+	CCONJ
easat-5378	80	2	1	1	NUM
easat-5378	80	3	𝑛	𝑛	NOUN
easat-5378	80	4	!	!	PUNCT
easat-5378	81	1	+	+	CCONJ
easat-5378	81	2	3	3	NUM
easat-5378	81	3	𝑛	𝑛	NOUN
easat-5378	81	4	!	!	PUNCT
easat-5378	82	1	+	+	CCONJ
easat-5378	82	2	6	6	NUM
easat-5378	82	3	𝑛	𝑛	NOUN
easat-5378	82	4	!	!	PUNCT
easat-5378	83	1	+	+	CCONJ
easat-5378	83	2	10	10	NUM
easat-5378	83	3	𝑛	𝑛	NOUN
easat-5378	83	4	!	!	PUNCT
easat-5378	84	1	+	+	CCONJ
easat-5378	84	2	15	15	NUM
easat-5378	84	3	𝑛	𝑛	NOUN
easat-5378	84	4	!	!	PUNCT
easat-5378	85	1	+	+	CCONJ
easat-5378	85	2	21	21	NUM
easat-5378	85	3	𝑛	𝑛	NOUN
easat-5378	85	4	!	!	PUNCT
easat-5378	86	1	+	+	CCONJ
easat-5378	86	2	28	28	NUM
easat-5378	86	3	𝑛	𝑛	NOUN
easat-5378	86	4	!	!	PUNCT
easat-5378	87	1	+	+	CCONJ
easat-5378	88	1	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	88	2	894	894	NUM
easat-5378	88	3	edelweiss	edelweiss	PROPN
easat-5378	88	4	applied	apply	VERB
easat-5378	88	5	science	science	NOUN
easat-5378	88	6	and	and	CCONJ
easat-5378	88	7	technology	technology	NOUN
easat-5378	88	8	issn	issn	PROPN
easat-5378	88	9	:	:	PUNCT
easat-5378	88	10	2576	2576	NUM
easat-5378	88	11	-	-	SYM
easat-5378	88	12	8484	8484	NUM
easat-5378	88	13	vol	vol	NOUN
easat-5378	88	14	.	.	PROPN
easat-5378	89	1	9	9	NUM
easat-5378	89	2	,	,	PUNCT
easat-5378	89	3	no	no	INTJ
easat-5378	89	4	.	.	NOUN
easat-5378	90	1	3	3	NUM
easat-5378	90	2	:	:	PUNCT
easat-5378	90	3	891	891	NUM
easat-5378	90	4	-	-	SYM
easat-5378	90	5	904	904	NUM
easat-5378	90	6	,	,	PUNCT
easat-5378	90	7	2025	2025	NUM
easat-5378	90	8	doi	doi	NOUN
easat-5378	90	9	:	:	PUNCT
easat-5378	90	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	90	11	©	©	PROPN
easat-5378	90	12	2025	2025	NUM
easat-5378	90	13	by	by	ADP
easat-5378	90	14	the	the	DET
easat-5378	90	15	author	author	NOUN
easat-5378	90	16	;	;	PUNCT
easat-5378	90	17	licensee	licensee	PROPN
easat-5378	90	18	learning	learning	NOUN
easat-5378	90	19	gate	gate	VERB
easat-5378	90	20	the	the	DET
easat-5378	90	21	n	n	CCONJ
easat-5378	90	22	-	-	PUNCT
easat-5378	90	23	factoriangular	factoriangular	ADJ
easat-5378	90	24	numbers	number	NOUN
easat-5378	90	25	were	be	AUX
easat-5378	90	26	also	also	ADV
easat-5378	90	27	generalized	generalize	VERB
easat-5378	90	28	into	into	ADP
easat-5378	90	29	multiple	multiple	ADJ
easat-5378	90	30	factoriangular	factoriangular	ADJ
easat-5378	90	31	numbers	number	NOUN
easat-5378	90	32	[	[	X
easat-5378	90	33	18	18	NUM
easat-5378	90	34	]	]	PUNCT
easat-5378	90	35	to	to	PART
easat-5378	90	36	have	have	VERB
easat-5378	90	37	𝐹𝑡(𝑛	𝐹𝑡(𝑛	NUM
easat-5378	90	38	,	,	PUNCT
easat-5378	90	39	𝑘	𝑘	NOUN
easat-5378	90	40	)	)	PUNCT
easat-5378	90	41	=	=	SYM
easat-5378	90	42	(	(	PUNCT
easat-5378	90	43	𝑛!)𝑘	𝑛!)𝑘	X
easat-5378	91	1	+	+	CCONJ
easat-5378	91	2	∑	∑	PROPN
easat-5378	91	3	𝑛𝑘	𝑛𝑘	X
easat-5378	91	4	wherein	wherein	NOUN
easat-5378	91	5	,	,	PUNCT
easat-5378	91	6	∑	∑	ADP
easat-5378	91	7	𝑛𝑘	𝑛𝑘	X
easat-5378	91	8	=	=	SYM
easat-5378	91	9	𝑇𝑛(𝑘	𝑇𝑛(𝑘	NOUN
easat-5378	91	10	)	)	PUNCT
easat-5378	91	11	.	.	PUNCT
easat-5378	92	1	this	this	PRON
easat-5378	92	2	is	be	AUX
easat-5378	92	3	similar	similar	ADJ
easat-5378	92	4	to	to	ADP
easat-5378	92	5	a	a	DET
easat-5378	92	6	generalization	generalization	NOUN
easat-5378	92	7	of	of	ADP
easat-5378	92	8	the	the	DET
easat-5378	92	9	sequence	sequence	NOUN
easat-5378	92	10	of	of	ADP
easat-5378	92	11	n	n	CCONJ
easat-5378	92	12	-	-	PUNCT
easat-5378	92	13	factoriangular	factoriangular	ADJ
easat-5378	92	14	numbers	number	NOUN
easat-5378	92	15	[	[	X
easat-5378	92	16	20	20	NUM
easat-5378	92	17	]	]	PUNCT
easat-5378	92	18	into	into	ADP
easat-5378	92	19	the	the	DET
easat-5378	92	20	sequence	sequence	NOUN
easat-5378	92	21	{	{	PUNCT
easat-5378	92	22	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	ADJ
easat-5378	92	23	)	)	PUNCT
easat-5378	92	24	}	}	PUNCT
easat-5378	92	25	for	for	ADP
easat-5378	92	26	natural	natural	ADJ
easat-5378	92	27	numbers	number	NOUN
easat-5378	92	28	𝑛	𝑛	PROPN
easat-5378	92	29	,	,	PUNCT
easat-5378	92	30	𝑚	𝑚	X
easat-5378	92	31	≥	≥	NUM
easat-5378	92	32	1	1	NUM
easat-5378	92	33	,	,	PUNCT
easat-5378	92	34	which	which	PRON
easat-5378	92	35	follow	follow	VERB
easat-5378	92	36	the	the	DET
easat-5378	92	37	form	form	NOUN
easat-5378	92	38	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	ADJ
easat-5378	92	39	)	)	PUNCT
easat-5378	92	40	=	=	SYM
easat-5378	93	1	(	(	PUNCT
easat-5378	93	2	1𝑚	1𝑚	X
easat-5378	93	3	⋅	⋅	PROPN
easat-5378	93	4	2𝑚	2𝑚	NOUN
easat-5378	93	5	⋅	⋅	PROPN
easat-5378	93	6	3𝑚	3𝑚	PROPN
easat-5378	93	7	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	93	8	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	93	9	)	)	PUNCT
easat-5378	93	10	+	+	CCONJ
easat-5378	93	11	(	(	PUNCT
easat-5378	93	12	1𝑚	1𝑚	NOUN
easat-5378	93	13	+	+	CCONJ
easat-5378	93	14	2𝑚	2𝑚	NOUN
easat-5378	93	15	+	+	CCONJ
easat-5378	93	16	3𝑚+	3𝑚+	NUM
easat-5378	93	17	.	.	PUNCT
easat-5378	93	18	.	.	PUNCT
easat-5378	93	19	.	.	PUNCT
easat-5378	94	1	+	+	ADV
easat-5378	94	2	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	94	3	)	)	PUNCT
easat-5378	94	4	we	we	PRON
easat-5378	94	5	call	call	VERB
easat-5378	94	6	the	the	DET
easat-5378	94	7	numbers	number	NOUN
easat-5378	94	8	in	in	ADP
easat-5378	94	9	this	this	DET
easat-5378	94	10	sequence	sequence	NOUN
easat-5378	94	11	as	as	ADP
easat-5378	94	12	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	94	13	numbers	number	NOUN
easat-5378	94	14	and	and	CCONJ
easat-5378	94	15	define	define	VERB
easat-5378	94	16	as	as	SCONJ
easat-5378	94	17	follows	follow	VERB
easat-5378	94	18	:	:	PUNCT
easat-5378	94	19	definition	definition	NOUN
easat-5378	94	20	3.3	3.3	NUM
easat-5378	94	21	:	:	PUNCT
easat-5378	94	22	the	the	DET
easat-5378	94	23	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	94	24	number	number	NOUN
easat-5378	94	25	is	be	AUX
easat-5378	94	26	defined	define	VERB
easat-5378	94	27	by	by	ADP
easat-5378	94	28	the	the	DET
easat-5378	94	29	formula	formula	NOUN
easat-5378	94	30	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	NOUN
easat-5378	94	31	)	)	PUNCT
easat-5378	94	32	=	=	SYM
easat-5378	95	1	(	(	PUNCT
easat-5378	95	2	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	95	3	+	+	CCONJ
easat-5378	95	4	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NOUN
easat-5378	95	5	)	)	PUNCT
easat-5378	95	6	where	where	SCONJ
easat-5378	95	7	(	(	PUNCT
easat-5378	95	8	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	95	9	=	=	SYM
easat-5378	95	10	1𝑚	1𝑚	NUM
easat-5378	95	11	⋅	⋅	PROPN
easat-5378	95	12	2𝑚	2𝑚	NOUN
easat-5378	95	13	⋅	⋅	PROPN
easat-5378	95	14	3𝑚	3𝑚	NUM
easat-5378	95	15	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	95	16	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	95	17	and	and	CCONJ
easat-5378	95	18	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NUM
easat-5378	95	19	)	)	PUNCT
easat-5378	95	20	=	=	SYM
easat-5378	95	21	1𝑚	1𝑚	NOUN
easat-5378	95	22	+	+	CCONJ
easat-5378	95	23	2𝑚	2𝑚	NOUN
easat-5378	95	24	+	+	CCONJ
easat-5378	95	25	3𝑚+	3𝑚+	NUM
easat-5378	95	26	.	.	PUNCT
easat-5378	95	27	.	.	PUNCT
easat-5378	95	28	.	.	PUNCT
easat-5378	96	1	+	+	ADV
easat-5378	96	2	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	96	3	for	for	ADP
easat-5378	96	4	natural	natural	ADJ
easat-5378	96	5	numbers	number	NOUN
easat-5378	96	6	,	,	PUNCT
easat-5378	96	7	1n	1n	NUM
easat-5378	96	8	m	m	NOUN
easat-5378	96	9			NUM
easat-5378	96	10	.	.	PUNCT
easat-5378	97	1	from	from	ADP
easat-5378	97	2	the	the	DET
easat-5378	97	3	definitions	definition	NOUN
easat-5378	97	4	3.1	3.1	NUM
easat-5378	97	5	and	and	CCONJ
easat-5378	97	6	3.3	3.3	NUM
easat-5378	97	7	,	,	PUNCT
easat-5378	97	8	when	when	SCONJ
easat-5378	97	9	𝑚	𝑚	PROPN
easat-5378	97	10	=	=	SYM
easat-5378	97	11	1	1	NUM
easat-5378	97	12	,	,	PUNCT
easat-5378	97	13	the	the	DET
easat-5378	97	14	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	97	15	numbers	number	NOUN
easat-5378	97	16	are	be	AUX
easat-5378	97	17	the	the	DET
easat-5378	97	18	same	same	ADJ
easat-5378	97	19	as	as	ADP
easat-5378	97	20	the	the	DET
easat-5378	97	21	n	n	CCONJ
easat-5378	97	22	-	-	PUNCT
easat-5378	97	23	factoriangular	factoriangular	ADJ
easat-5378	97	24	numbers	number	NOUN
easat-5378	97	25	or	or	CCONJ
easat-5378	97	26	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	ADJ
easat-5378	97	27	)	)	PUNCT
easat-5378	98	1	=	=	SYM
easat-5378	98	2	𝐹𝑡𝑛.	𝐹𝑡𝑛.	PROPN
easat-5378	98	3	here	here	ADV
easat-5378	98	4	,	,	PUNCT
easat-5378	98	5	𝑆1(𝑛	𝑆1(𝑛	PROPN
easat-5378	98	6	)	)	PUNCT
easat-5378	98	7	is	be	AUX
easat-5378	98	8	the	the	DET
easat-5378	98	9	same	same	ADJ
easat-5378	98	10	as	as	ADP
easat-5378	98	11	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	98	12	,	,	PUNCT
easat-5378	98	13	and	and	CCONJ
easat-5378	98	14	thus	thus	ADV
easat-5378	98	15	,	,	PUNCT
easat-5378	98	16	𝐹𝑡𝑛(1	𝐹𝑡𝑛(1	NUM
easat-5378	98	17	)	)	PUNCT
easat-5378	98	18	=	=	SYM
easat-5378	98	19	𝑛	𝑛	NOUN
easat-5378	98	20	!	!	PUNCT
easat-5378	99	1	+	+	PUNCT
easat-5378	99	2	𝑆1(𝑛	𝑆1(𝑛	X
easat-5378	99	3	)	)	PUNCT
easat-5378	99	4	=	=	PUNCT
easat-5378	99	5	(	(	PUNCT
easat-5378	99	6	1	1	NUM
easat-5378	99	7	⋅	⋅	NUM
easat-5378	99	8	2	2	NUM
easat-5378	99	9	⋅	⋅	X
easat-5378	99	10	3	3	NUM
easat-5378	99	11	⋅⋅⋅	⋅⋅⋅	X
easat-5378	99	12	𝑛	𝑛	PROPN
easat-5378	99	13	)	)	PUNCT
easat-5378	99	14	+	+	CCONJ
easat-5378	99	15	(	(	PUNCT
easat-5378	99	16	1	1	NUM
easat-5378	99	17	+	+	NUM
easat-5378	99	18	2	2	NUM
easat-5378	99	19	+	+	SYM
easat-5378	99	20	3	3	NUM
easat-5378	99	21	+	+	NOUN
easat-5378	99	22	.	.	PUNCT
easat-5378	99	23	.	.	PUNCT
easat-5378	99	24	.	.	PUNCT
easat-5378	100	1	+	+	PUNCT
easat-5378	100	2	𝑛	𝑛	X
easat-5378	100	3	)	)	PUNCT
easat-5378	100	4	=	=	SYM
easat-5378	100	5	𝑛	𝑛	NOUN
easat-5378	100	6	!	!	PUNCT
easat-5378	100	7	+	+	CCONJ
easat-5378	100	8	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	100	9	+	+	CCONJ
easat-5378	100	10	1	1	NUM
easat-5378	100	11	)	)	SYM
easat-5378	100	12	2	2	NUM
easat-5378	100	13	=	=	SYM
easat-5378	100	14	𝐹𝑡𝑛	𝐹𝑡𝑛	PROPN
easat-5378	100	15	for	for	ADP
easat-5378	100	16	the	the	DET
easat-5378	100	17	next	next	ADJ
easat-5378	100	18	few	few	ADJ
easat-5378	100	19	specific	specific	ADJ
easat-5378	100	20	cases	case	NOUN
easat-5378	100	21	of	of	ADP
easat-5378	100	22	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	100	23	numbers	number	NOUN
easat-5378	100	24	,	,	PUNCT
easat-5378	100	25	that	that	PRON
easat-5378	100	26	is	be	AUX
easat-5378	100	27	when	when	SCONJ
easat-5378	100	28	𝑚	𝑚	PROPN
easat-5378	100	29	=	=	SYM
easat-5378	100	30	2	2	NUM
easat-5378	100	31	,	,	PUNCT
easat-5378	100	32	3	3	NUM
easat-5378	100	33	,	,	PUNCT
easat-5378	100	34	4	4	NUM
easat-5378	100	35	,	,	PUNCT
easat-5378	100	36	5	5	NUM
easat-5378	100	37	,	,	PUNCT
easat-5378	100	38	6	6	NUM
easat-5378	100	39	,	,	PUNCT
easat-5378	100	40	7	7	NUM
easat-5378	100	41	,	,	PUNCT
easat-5378	100	42	8	8	NUM
easat-5378	100	43	,	,	PUNCT
easat-5378	100	44	9	9	NUM
easat-5378	100	45	,	,	PUNCT
easat-5378	100	46	we	we	PRON
easat-5378	100	47	have	have	VERB
easat-5378	100	48	𝐹𝑡𝑛(2	𝐹𝑡𝑛(2	NUM
easat-5378	100	49	)	)	PUNCT
easat-5378	101	1	=	=	SYM
easat-5378	101	2	(	(	PUNCT
easat-5378	101	3	𝑛!)2	𝑛!)2	PROPN
easat-5378	101	4	+	+	NUM
easat-5378	101	5	𝑆2(𝑛	𝑆2(𝑛	NOUN
easat-5378	101	6	)	)	PUNCT
easat-5378	101	7	=	=	PUNCT
easat-5378	101	8	(	(	PUNCT
easat-5378	101	9	12	12	NUM
easat-5378	101	10	⋅	⋅	NUM
easat-5378	101	11	22	22	NUM
easat-5378	101	12	⋅	⋅	PROPN
easat-5378	101	13	32	32	NUM
easat-5378	101	14	⋅⋅⋅	⋅⋅⋅	NOUN
easat-5378	101	15	𝑛2	𝑛2	NOUN
easat-5378	101	16	)	)	PUNCT
easat-5378	101	17	+	+	CCONJ
easat-5378	101	18	(	(	PUNCT
easat-5378	101	19	12	12	NUM
easat-5378	101	20	+	+	NUM
easat-5378	101	21	22	22	NUM
easat-5378	101	22	+	+	NUM
easat-5378	101	23	32	32	NUM
easat-5378	101	24	+	+	NOUN
easat-5378	101	25	.	.	PUNCT
easat-5378	101	26	.	.	PUNCT
easat-5378	101	27	.	.	PUNCT
easat-5378	102	1	+	+	NUM
easat-5378	102	2	𝑛2	𝑛2	NOUN
easat-5378	102	3	)	)	PUNCT
easat-5378	102	4	𝐹𝑡𝑛(3	𝐹𝑡𝑛(3	NUM
easat-5378	102	5	)	)	PUNCT
easat-5378	103	1	=	=	PRON
easat-5378	103	2	(	(	PUNCT
easat-5378	103	3	𝑛!)3	𝑛!)3	PROPN
easat-5378	103	4	+	+	SYM
easat-5378	103	5	𝑆3(𝑛	𝑆3(𝑛	NUM
easat-5378	103	6	)	)	PUNCT
easat-5378	103	7	=	=	PUNCT
easat-5378	104	1	(	(	PUNCT
easat-5378	104	2	13	13	NUM
easat-5378	104	3	⋅	⋅	PROPN
easat-5378	104	4	23	23	NUM
easat-5378	104	5	⋅	⋅	PROPN
easat-5378	104	6	33	33	NUM
easat-5378	104	7	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	104	8	𝑛3	𝑛3	PROPN
easat-5378	104	9	)	)	PUNCT
easat-5378	105	1	+	+	CCONJ
easat-5378	105	2	(	(	PUNCT
easat-5378	105	3	13	13	NUM
easat-5378	105	4	+	+	NUM
easat-5378	105	5	23	23	NUM
easat-5378	105	6	+	+	SYM
easat-5378	105	7	33	33	NUM
easat-5378	105	8	+	+	NOUN
easat-5378	105	9	.	.	PUNCT
easat-5378	105	10	.	.	PUNCT
easat-5378	105	11	.	.	PUNCT
easat-5378	106	1	+	+	ADV
easat-5378	106	2	𝑛3	𝑛3	NOUN
easat-5378	106	3	)	)	PUNCT
easat-5378	106	4	𝐹𝑡𝑛(4	𝐹𝑡𝑛(4	NUM
easat-5378	106	5	)	)	PUNCT
easat-5378	107	1	=	=	PRON
easat-5378	107	2	(	(	PUNCT
easat-5378	107	3	𝑛!)4	𝑛!)4	NOUN
easat-5378	107	4	+	+	CCONJ
easat-5378	107	5	𝑆4(𝑛	𝑆4(𝑛	NUM
easat-5378	107	6	)	)	PUNCT
easat-5378	107	7	=	=	PUNCT
easat-5378	107	8	(	(	PUNCT
easat-5378	107	9	14	14	NUM
easat-5378	107	10	⋅	⋅	NUM
easat-5378	107	11	24	24	NUM
easat-5378	107	12	⋅	⋅	PROPN
easat-5378	107	13	34	34	NUM
easat-5378	107	14	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	107	15	𝑛4	𝑛4	NOUN
easat-5378	107	16	)	)	PUNCT
easat-5378	107	17	+	+	CCONJ
easat-5378	108	1	(	(	PUNCT
easat-5378	108	2	14	14	NUM
easat-5378	108	3	+	+	NUM
easat-5378	108	4	24	24	NUM
easat-5378	108	5	+	+	NUM
easat-5378	108	6	34	34	NUM
easat-5378	108	7	+	+	NOUN
easat-5378	108	8	.	.	PUNCT
easat-5378	108	9	.	.	PUNCT
easat-5378	108	10	.	.	PUNCT
easat-5378	109	1	+	+	ADJ
easat-5378	109	2	𝑛4	𝑛4	NOUN
easat-5378	109	3	)	)	PUNCT
easat-5378	109	4	𝐹𝑡𝑛(5	𝐹𝑡𝑛(5	NUM
easat-5378	109	5	)	)	PUNCT
easat-5378	109	6	=	=	PRON
easat-5378	109	7	(	(	PUNCT
easat-5378	109	8	𝑛!)5	𝑛!)5	X
easat-5378	109	9	+	+	NUM
easat-5378	109	10	𝑆5(𝑛	𝑆5(𝑛	NOUN
easat-5378	109	11	)	)	PUNCT
easat-5378	110	1	=	=	PUNCT
easat-5378	110	2	(	(	PUNCT
easat-5378	110	3	15	15	NUM
easat-5378	110	4	⋅	⋅	NUM
easat-5378	110	5	25	25	NUM
easat-5378	110	6	⋅	⋅	PROPN
easat-5378	110	7	35	35	NUM
easat-5378	110	8	⋅⋅⋅	⋅⋅⋅	NOUN
easat-5378	110	9	𝑛5	𝑛5	NOUN
easat-5378	110	10	)	)	PUNCT
easat-5378	110	11	+	+	CCONJ
easat-5378	111	1	(	(	PUNCT
easat-5378	111	2	15	15	NUM
easat-5378	111	3	+	+	NUM
easat-5378	111	4	25	25	NUM
easat-5378	111	5	+	+	NUM
easat-5378	111	6	35	35	NUM
easat-5378	111	7	+	+	NOUN
easat-5378	111	8	.	.	PUNCT
easat-5378	111	9	.	.	PUNCT
easat-5378	111	10	.	.	PUNCT
easat-5378	112	1	+	+	ADJ
easat-5378	112	2	𝑛5	𝑛5	NOUN
easat-5378	112	3	)	)	PUNCT
easat-5378	112	4	𝐹𝑡𝑛(6	𝐹𝑡𝑛(6	NUM
easat-5378	112	5	)	)	PUNCT
easat-5378	112	6	=	=	NOUN
easat-5378	112	7	(	(	PUNCT
easat-5378	112	8	𝑛!)6	𝑛!)6	PROPN
easat-5378	112	9	+	+	PROPN
easat-5378	112	10	𝑆6(𝑛	𝑆6(𝑛	ADJ
easat-5378	112	11	)	)	PUNCT
easat-5378	112	12	=	=	PUNCT
easat-5378	113	1	(	(	PUNCT
easat-5378	113	2	16	16	NUM
easat-5378	113	3	⋅	⋅	PROPN
easat-5378	113	4	26	26	NUM
easat-5378	113	5	⋅	⋅	PROPN
easat-5378	113	6	36	36	NUM
easat-5378	113	7	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	113	8	𝑛6	𝑛6	PROPN
easat-5378	113	9	)	)	PUNCT
easat-5378	114	1	+	+	CCONJ
easat-5378	114	2	(	(	PUNCT
easat-5378	114	3	16	16	NUM
easat-5378	114	4	+	+	NUM
easat-5378	114	5	26	26	NUM
easat-5378	114	6	+	+	SYM
easat-5378	114	7	36	36	NUM
easat-5378	114	8	+	+	NOUN
easat-5378	114	9	.	.	PUNCT
easat-5378	114	10	.	.	PUNCT
easat-5378	114	11	.	.	PUNCT
easat-5378	115	1	+	+	NOUN
easat-5378	115	2	𝑛6	𝑛6	PROPN
easat-5378	115	3	)	)	PUNCT
easat-5378	115	4	𝐹𝑡𝑛(7	𝐹𝑡𝑛(7	NUM
easat-5378	115	5	)	)	PUNCT
easat-5378	116	1	=	=	PRON
easat-5378	116	2	(	(	PUNCT
easat-5378	116	3	𝑛!)7	𝑛!)7	PROPN
easat-5378	116	4	+	+	CCONJ
easat-5378	116	5	𝑆7(𝑛	𝑆7(𝑛	PROPN
easat-5378	116	6	)	)	PUNCT
easat-5378	116	7	=	=	PUNCT
easat-5378	116	8	(	(	PUNCT
easat-5378	116	9	17	17	NUM
easat-5378	116	10	⋅	⋅	NUM
easat-5378	116	11	27	27	NUM
easat-5378	116	12	⋅	⋅	PROPN
easat-5378	116	13	37	37	NUM
easat-5378	116	14	⋅⋅⋅	⋅⋅⋅	X
easat-5378	116	15	𝑛7	𝑛7	NOUN
easat-5378	116	16	)	)	PUNCT
easat-5378	116	17	+	+	CCONJ
easat-5378	116	18	(	(	PUNCT
easat-5378	116	19	17	17	NUM
easat-5378	116	20	+	+	NUM
easat-5378	116	21	27	27	NUM
easat-5378	116	22	+	+	CCONJ
easat-5378	116	23	37	37	NUM
easat-5378	116	24	+	+	NOUN
easat-5378	116	25	.	.	PUNCT
easat-5378	116	26	.	.	PUNCT
easat-5378	116	27	.	.	PUNCT
easat-5378	117	1	+	+	NOUN
easat-5378	117	2	𝑛7	𝑛7	NOUN
easat-5378	117	3	)	)	PUNCT
easat-5378	117	4	𝐹𝑡𝑛(8	𝐹𝑡𝑛(8	NUM
easat-5378	117	5	)	)	PUNCT
easat-5378	117	6	=	=	SYM
easat-5378	117	7	(	(	PUNCT
easat-5378	117	8	𝑛!)8	𝑛!)8	PROPN
easat-5378	117	9	+	+	PROPN
easat-5378	117	10	𝑆8(𝑛	𝑆8(𝑛	PROPN
easat-5378	117	11	)	)	PUNCT
easat-5378	118	1	=	=	PUNCT
easat-5378	118	2	(	(	PUNCT
easat-5378	118	3	18	18	NUM
easat-5378	118	4	⋅	⋅	NUM
easat-5378	118	5	28	28	NUM
easat-5378	118	6	⋅	⋅	PROPN
easat-5378	118	7	38	38	NUM
easat-5378	118	8	⋅⋅⋅	⋅⋅⋅	NOUN
easat-5378	118	9	𝑛8	𝑛8	NOUN
easat-5378	118	10	)	)	PUNCT
easat-5378	118	11	+	+	CCONJ
easat-5378	118	12	(	(	PUNCT
easat-5378	118	13	15	15	NUM
easat-5378	118	14	+	+	NUM
easat-5378	118	15	28	28	NUM
easat-5378	118	16	+	+	CCONJ
easat-5378	118	17	38	38	NUM
easat-5378	118	18	+	+	NOUN
easat-5378	118	19	.	.	PUNCT
easat-5378	118	20	.	.	PUNCT
easat-5378	118	21	.	.	PUNCT
easat-5378	119	1	+	+	PUNCT
easat-5378	119	2	𝑛8	𝑛8	NOUN
easat-5378	119	3	)	)	PUNCT
easat-5378	119	4	𝐹𝑡𝑛(9	𝐹𝑡𝑛(9	NUM
easat-5378	119	5	)	)	PUNCT
easat-5378	120	1	=	=	NOUN
easat-5378	120	2	(	(	PUNCT
easat-5378	120	3	𝑛!)9	𝑛!)9	ADP
easat-5378	120	4	+	+	NUM
easat-5378	120	5	𝑆9(𝑛	𝑆9(𝑛	NUM
easat-5378	120	6	)	)	PUNCT
easat-5378	120	7	=	=	NOUN
easat-5378	120	8	(	(	PUNCT
easat-5378	120	9	19	19	NUM
easat-5378	120	10	⋅	⋅	NUM
easat-5378	120	11	29	29	NUM
easat-5378	120	12	⋅	⋅	PROPN
easat-5378	120	13	39	39	NUM
easat-5378	120	14	⋅⋅⋅	⋅⋅⋅	X
easat-5378	120	15	𝑛9	𝑛9	PROPN
easat-5378	120	16	)	)	PUNCT
easat-5378	120	17	+	+	CCONJ
easat-5378	121	1	(	(	PUNCT
easat-5378	121	2	19	19	NUM
easat-5378	121	3	+	+	NUM
easat-5378	121	4	29	29	NUM
easat-5378	121	5	+	+	CCONJ
easat-5378	121	6	39	39	NUM
easat-5378	121	7	+	+	NOUN
easat-5378	121	8	.	.	PUNCT
easat-5378	121	9	.	.	PUNCT
easat-5378	121	10	.	.	PUNCT
easat-5378	122	1	+	+	VERB
easat-5378	122	2	𝑛9	𝑛9	NOUN
easat-5378	122	3	)	)	PUNCT
easat-5378	122	4	theorem	theorem	VERB
easat-5378	122	5	3.4	3.4	NUM
easat-5378	122	6	:	:	PUNCT
easat-5378	122	7	for	for	ADP
easat-5378	122	8	natural	natural	ADJ
easat-5378	122	9	number	number	NOUN
easat-5378	122	10	1n	1n	NUM
easat-5378	122	11			NUM
easat-5378	122	12	,	,	PUNCT
easat-5378	122	13	the	the	DET
easat-5378	122	14	𝑛(2)-	𝑛(2)-	NOUN
easat-5378	122	15	,	,	PUNCT
easat-5378	122	16	𝑛(3)-	𝑛(3)-	NOUN
easat-5378	122	17	,	,	PUNCT
easat-5378	122	18	𝑛(4)-	𝑛(4)-	NOUN
easat-5378	122	19	,	,	PUNCT
easat-5378	122	20	𝑛(5)-	𝑛(5)-	PROPN
easat-5378	122	21	,	,	PUNCT
easat-5378	122	22	𝑛(6)-	𝑛(6)-	NOUN
easat-5378	122	23	,	,	PUNCT
easat-5378	122	24	𝑛(7)-	𝑛(7)-	PROPN
easat-5378	122	25	,	,	PUNCT
easat-5378	122	26	𝑛(8)-	𝑛(8)-	ADP
easat-5378	122	27	,	,	PUNCT
easat-5378	122	28	and	and	CCONJ
easat-5378	122	29	𝑛(9)factoriangular	𝑛(9)factoriangular	ADJ
easat-5378	122	30	numbers	number	NOUN
easat-5378	122	31	are	be	AUX
easat-5378	122	32	,	,	PUNCT
easat-5378	122	33	respectively	respectively	ADV
easat-5378	122	34	,	,	PUNCT
easat-5378	122	35	given	give	VERB
easat-5378	122	36	by	by	ADP
easat-5378	122	37	the	the	DET
easat-5378	122	38	formulas	formula	NOUN
easat-5378	122	39	𝐹𝑡𝑛(2	𝐹𝑡𝑛(2	NUM
easat-5378	122	40	)	)	PUNCT
easat-5378	123	1	=	=	SYM
easat-5378	123	2	(	(	PUNCT
easat-5378	123	3	𝑛!)2	𝑛!)2	PROPN
easat-5378	123	4	+	+	NUM
easat-5378	123	5	1	1	NUM
easat-5378	123	6	3	3	NUM
easat-5378	123	7	(	(	PUNCT
easat-5378	123	8	2𝑛	2𝑛	PROPN
easat-5378	123	9	+	+	PROPN
easat-5378	123	10	1)𝑇𝑛	1)𝑇𝑛	PROPN
easat-5378	123	11	𝐹𝑡𝑛(3	𝐹𝑡𝑛(3	NUM
easat-5378	123	12	)	)	PUNCT
easat-5378	123	13	=	=	PRON
easat-5378	123	14	(	(	PUNCT
easat-5378	123	15	𝑛!)3	𝑛!)3	PROPN
easat-5378	123	16	+	+	CCONJ
easat-5378	123	17	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	123	18	2	2	NUM
easat-5378	123	19	𝐹𝑡𝑛(4	𝐹𝑡𝑛(4	NUM
easat-5378	123	20	)	)	PUNCT
easat-5378	124	1	=	=	SYM
easat-5378	124	2	(	(	PUNCT
easat-5378	124	3	𝑛!)4	𝑛!)4	NOUN
easat-5378	124	4	+	+	CCONJ
easat-5378	124	5	1	1	NUM
easat-5378	124	6	15	15	NUM
easat-5378	124	7	(	(	PUNCT
easat-5378	124	8	6𝑛3	6𝑛3	NUM
easat-5378	124	9	+	+	CCONJ
easat-5378	124	10	9𝑛2	9𝑛2	NUM
easat-5378	124	11	+	+	CCONJ
easat-5378	124	12	𝑛	𝑛	DET
easat-5378	124	13	−	−	PROPN
easat-5378	124	14	1)𝑇𝑛	1)𝑇𝑛	PROPN
easat-5378	124	15	𝐹𝑡𝑛(5	𝐹𝑡𝑛(5	NUM
easat-5378	124	16	)	)	PUNCT
easat-5378	124	17	=	=	SYM
easat-5378	124	18	(	(	PUNCT
easat-5378	124	19	𝑛!)5	𝑛!)5	X
easat-5378	124	20	+	+	CCONJ
easat-5378	124	21	1	1	NUM
easat-5378	124	22	3	3	NUM
easat-5378	124	23	(	(	PUNCT
easat-5378	124	24	2𝑛2	2𝑛2	NUM
easat-5378	124	25	+	+	CCONJ
easat-5378	124	26	2𝑛	2𝑛	NUM
easat-5378	124	27	−	−	PROPN
easat-5378	125	1	1)𝑇𝑛	1)𝑇𝑛	PROPN
easat-5378	125	2	2	2	NUM
easat-5378	125	3	𝐹𝑡𝑛(6	𝐹𝑡𝑛(6	NUM
easat-5378	125	4	)	)	PUNCT
easat-5378	126	1	=	=	NOUN
easat-5378	126	2	(	(	PUNCT
easat-5378	126	3	𝑛!)6	𝑛!)6	PROPN
easat-5378	126	4	+	+	CCONJ
easat-5378	126	5	1	1	NUM
easat-5378	126	6	21	21	NUM
easat-5378	126	7	(	(	PUNCT
easat-5378	126	8	6𝑛5	6𝑛5	NUM
easat-5378	126	9	+	+	CCONJ
easat-5378	126	10	15𝑛4	15𝑛4	NUM
easat-5378	126	11	+	+	NUM
easat-5378	126	12	6𝑛3	6𝑛3	NUM
easat-5378	127	1	−	−	NUM
easat-5378	127	2	6𝑛2	6𝑛2	NUM
easat-5378	127	3	−	−	NOUN
easat-5378	127	4	𝑛	𝑛	PROPN
easat-5378	127	5	+	+	PROPN
easat-5378	127	6	1)𝑇𝑛	1)𝑇𝑛	PROPN
easat-5378	127	7	𝐹𝑡𝑛(7	𝐹𝑡𝑛(7	NUM
easat-5378	127	8	)	)	PUNCT
easat-5378	128	1	=	=	PUNCT
easat-5378	128	2	(	(	PUNCT
easat-5378	128	3	𝑛!)7	𝑛!)7	PROPN
easat-5378	128	4	+	+	CCONJ
easat-5378	128	5	1	1	NUM
easat-5378	128	6	6	6	NUM
easat-5378	128	7	(	(	PUNCT
easat-5378	128	8	3𝑛4	3𝑛4	NUM
easat-5378	128	9	+	+	NUM
easat-5378	128	10	6𝑛3	6𝑛3	NUM
easat-5378	128	11	−	−	NOUN
easat-5378	128	12	𝑛2	𝑛2	NOUN
easat-5378	128	13	−	−	PROPN
easat-5378	128	14	4𝑛	4𝑛	NOUN
easat-5378	128	15	+	+	X
easat-5378	128	16	2)𝑇𝑛	2)𝑇𝑛	PROPN
easat-5378	128	17	2	2	NUM
easat-5378	128	18	𝐹𝑡𝑛(8	𝐹𝑡𝑛(8	NUM
easat-5378	128	19	)	)	PUNCT
easat-5378	128	20	=	=	SYM
easat-5378	129	1	(	(	PUNCT
easat-5378	129	2	𝑛!)8	𝑛!)8	PROPN
easat-5378	129	3	+	+	PROPN
easat-5378	129	4	1	1	NUM
easat-5378	129	5	45	45	NUM
easat-5378	129	6	(	(	PUNCT
easat-5378	129	7	10𝑛7	10𝑛7	NUM
easat-5378	129	8	+	+	SYM
easat-5378	129	9	35𝑛6	35𝑛6	NUM
easat-5378	129	10	+	+	NUM
easat-5378	129	11	25𝑛5	25𝑛5	NUM
easat-5378	129	12	−	−	NOUN
easat-5378	130	1	25𝑛4	25𝑛4	NUM
easat-5378	130	2	−	−	NOUN
easat-5378	130	3	17𝑛3	17𝑛3	NUM
easat-5378	131	1	+	+	NUM
easat-5378	132	1	17𝑛2	17𝑛2	NUM
easat-5378	132	2	+	+	NUM
easat-5378	132	3	3𝑛	3𝑛	NUM
easat-5378	132	4	−	−	PROPN
easat-5378	132	5	3)𝑇𝑛	3)𝑇𝑛	PROPN
easat-5378	132	6	𝐹𝑡𝑛(9	𝐹𝑡𝑛(9	NUM
easat-5378	132	7	)	)	PUNCT
easat-5378	133	1	=	=	NOUN
easat-5378	133	2	(	(	PUNCT
easat-5378	133	3	𝑛!)9	𝑛!)9	ADP
easat-5378	133	4	+	+	NOUN
easat-5378	133	5	1	1	NUM
easat-5378	133	6	5	5	NUM
easat-5378	133	7	(	(	PUNCT
easat-5378	133	8	2𝑛6	2𝑛6	NUM
easat-5378	133	9	+	+	CCONJ
easat-5378	133	10	6𝑛5	6𝑛5	NUM
easat-5378	133	11	+	+	NUM
easat-5378	133	12	𝑛4	𝑛4	NOUN
easat-5378	133	13	−	−	NOUN
easat-5378	133	14	8𝑛3	8𝑛3	NUM
easat-5378	134	1	+	+	CCONJ
easat-5378	134	2	𝑛2	𝑛2	NOUN
easat-5378	134	3	+	+	CCONJ
easat-5378	134	4	6𝑛	6𝑛	NUM
easat-5378	134	5	−	−	NOUN
easat-5378	135	1	3)𝑇𝑛	3)𝑇𝑛	PROPN
easat-5378	135	2	2	2	NUM
easat-5378	135	3	where	where	SCONJ
easat-5378	135	4	𝑛	𝑛	X
easat-5378	135	5	!	!	PUNCT
easat-5378	135	6	=	=	SYM
easat-5378	136	1	1	1	NUM
easat-5378	136	2	⋅	⋅	NUM
easat-5378	136	3	2	2	NUM
easat-5378	136	4	⋅	⋅	X
easat-5378	136	5	3	3	NUM
easat-5378	136	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	136	7	𝑛	𝑛	PROPN
easat-5378	136	8	and	and	CCONJ
easat-5378	136	9	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	136	10	=	=	NOUN
easat-5378	136	11	1	1	NUM
easat-5378	136	12	+	+	NUM
easat-5378	136	13	2	2	NUM
easat-5378	136	14	+	+	SYM
easat-5378	136	15	3	3	NUM
easat-5378	136	16	+	+	NOUN
easat-5378	136	17	.	.	PUNCT
easat-5378	136	18	.	.	PUNCT
easat-5378	136	19	.	.	PUNCT
easat-5378	137	1	+	+	ADV
easat-5378	137	2	𝑛	𝑛	NOUN
easat-5378	137	3	=	=	SYM
easat-5378	137	4	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	137	5	+	+	CCONJ
easat-5378	137	6	1)/2	1)/2	NOUN
easat-5378	137	7	.	.	PUNCT
easat-5378	138	1	a	a	DET
easat-5378	138	2	recent	recent	ADJ
easat-5378	138	3	paper	paper	NOUN
easat-5378	138	4	[	[	X
easat-5378	138	5	20	20	NUM
easat-5378	138	6	]	]	PUNCT
easat-5378	138	7	gives	give	VERB
easat-5378	138	8	the	the	DET
easat-5378	138	9	proof	proof	NOUN
easat-5378	138	10	of	of	ADP
easat-5378	138	11	theorem	theorem	ADJ
easat-5378	138	12	3.4	3.4	NUM
easat-5378	138	13	.	.	PUNCT
easat-5378	139	1	the	the	DET
easat-5378	139	2	proof	proof	NOUN
easat-5378	139	3	has	have	AUX
easat-5378	139	4	verified	verify	VERB
easat-5378	139	5	the	the	DET
easat-5378	139	6	following	following	NOUN
easat-5378	139	7	:	:	PUNCT
easat-5378	139	8	(	(	PUNCT
easat-5378	139	9	𝑛	𝑛	PROPN
easat-5378	139	10	+	+	X
easat-5378	139	11	1)2	1)2	NUM
easat-5378	139	12	−	−	NOUN
easat-5378	139	13	1	1	NUM
easat-5378	139	14	=	=	SYM
easat-5378	139	15	2𝑆1	2𝑆1	NUM
easat-5378	139	16	+	+	CCONJ
easat-5378	139	17	𝑛	𝑛	PROPN
easat-5378	139	18	(	(	PUNCT
easat-5378	139	19	𝑛	𝑛	PROPN
easat-5378	139	20	+	+	SYM
easat-5378	140	1	1)3	1)3	NUM
easat-5378	140	2	−	−	PROPN
easat-5378	140	3	1	1	NUM
easat-5378	140	4	=	=	SYM
easat-5378	140	5	3𝑆2	3𝑆2	NUM
easat-5378	140	6	+	+	CCONJ
easat-5378	140	7	3𝑆1	3𝑆1	NUM
easat-5378	140	8	+	+	CCONJ
easat-5378	140	9	𝑛	𝑛	DET
easat-5378	140	10	895	895	NUM
easat-5378	140	11	edelweiss	edelweiss	NOUN
easat-5378	140	12	applied	apply	VERB
easat-5378	140	13	science	science	NOUN
easat-5378	140	14	and	and	CCONJ
easat-5378	140	15	technology	technology	NOUN
easat-5378	140	16	issn	issn	PROPN
easat-5378	140	17	:	:	PUNCT
easat-5378	140	18	2576	2576	NUM
easat-5378	140	19	-	-	SYM
easat-5378	140	20	8484	8484	NUM
easat-5378	140	21	vol	vol	NOUN
easat-5378	140	22	.	.	PROPN
easat-5378	141	1	9	9	NUM
easat-5378	141	2	,	,	PUNCT
easat-5378	141	3	no	no	INTJ
easat-5378	141	4	.	.	NOUN
easat-5378	142	1	3	3	NUM
easat-5378	142	2	:	:	PUNCT
easat-5378	142	3	891	891	NUM
easat-5378	142	4	-	-	SYM
easat-5378	142	5	904	904	NUM
easat-5378	142	6	,	,	PUNCT
easat-5378	142	7	2025	2025	NUM
easat-5378	142	8	doi	doi	NOUN
easat-5378	142	9	:	:	PUNCT
easat-5378	142	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	142	11	©	©	PROPN
easat-5378	142	12	2025	2025	NUM
easat-5378	142	13	by	by	ADP
easat-5378	142	14	the	the	DET
easat-5378	142	15	author	author	NOUN
easat-5378	142	16	;	;	PUNCT
easat-5378	142	17	licensee	licensee	PROPN
easat-5378	142	18	learning	learning	NOUN
easat-5378	142	19	gate	gate	NOUN
easat-5378	142	20	(	(	PUNCT
easat-5378	142	21	𝑛	𝑛	PROPN
easat-5378	142	22	+	+	NOUN
easat-5378	142	23	1)4	1)4	NUM
easat-5378	142	24	−	−	NOUN
easat-5378	142	25	1	1	NUM
easat-5378	142	26	=	=	SYM
easat-5378	142	27	4𝑆3	4𝑆3	NUM
easat-5378	143	1	+	+	CCONJ
easat-5378	143	2	6𝑆2	6𝑆2	NUM
easat-5378	143	3	+	+	NUM
easat-5378	143	4	4𝑆1	4𝑆1	NUM
easat-5378	144	1	+	+	CCONJ
easat-5378	145	1	𝑛	𝑛	PROPN
easat-5378	145	2	(	(	PUNCT
easat-5378	145	3	𝑛	𝑛	PROPN
easat-5378	145	4	+	+	NUM
easat-5378	145	5	1)5	1)5	NUM
easat-5378	145	6	−	−	NOUN
easat-5378	145	7	1	1	NUM
easat-5378	145	8	=	=	SYM
easat-5378	145	9	5𝑆4	5𝑆4	NUM
easat-5378	145	10	+	+	NUM
easat-5378	145	11	10𝑆3	10𝑆3	NUM
easat-5378	145	12	+	+	NUM
easat-5378	145	13	10𝑆2	10𝑆2	NUM
easat-5378	145	14	+	+	SYM
easat-5378	145	15	5𝑆1	5𝑆1	NUM
easat-5378	145	16	+	+	CCONJ
easat-5378	145	17	𝑛	𝑛	ADJ
easat-5378	145	18	(	(	PUNCT
easat-5378	145	19	𝑛	𝑛	PROPN
easat-5378	145	20	+	+	PROPN
easat-5378	145	21	1)6	1)6	NUM
easat-5378	145	22	−	−	NOUN
easat-5378	145	23	1	1	NUM
easat-5378	145	24	=	=	SYM
easat-5378	145	25	6𝑆5	6𝑆5	NUM
easat-5378	146	1	+	+	NUM
easat-5378	146	2	15𝑆4	15𝑆4	NUM
easat-5378	146	3	+	+	CCONJ
easat-5378	146	4	20𝑆3	20𝑆3	NUM
easat-5378	146	5	+	+	CCONJ
easat-5378	146	6	15𝑆2	15𝑆2	NUM
easat-5378	146	7	+	+	SYM
easat-5378	146	8	6𝑆1	6𝑆1	NUM
easat-5378	146	9	+	+	CCONJ
easat-5378	146	10	𝑛	𝑛	PROPN
easat-5378	146	11	(	(	PUNCT
easat-5378	146	12	𝑛	𝑛	PROPN
easat-5378	146	13	+	+	NOUN
easat-5378	147	1	1)7	1)7	NUM
easat-5378	147	2	−	−	NOUN
easat-5378	147	3	1	1	NUM
easat-5378	147	4	=	=	SYM
easat-5378	147	5	7𝑆6	7𝑆6	PROPN
easat-5378	148	1	+	+	CCONJ
easat-5378	149	1	21𝑆5	21𝑆5	NUM
easat-5378	149	2	+	+	CCONJ
easat-5378	149	3	35𝑆4	35𝑆4	NUM
easat-5378	149	4	+	+	CCONJ
easat-5378	149	5	35𝑆3	35𝑆3	NUM
easat-5378	149	6	+	+	SYM
easat-5378	149	7	21𝑆2	21𝑆2	NUM
easat-5378	149	8	+	+	CCONJ
easat-5378	149	9	7𝑆1	7𝑆1	NUM
easat-5378	150	1	+	+	CCONJ
easat-5378	150	2	𝑛	𝑛	ADJ
easat-5378	150	3	(	(	PUNCT
easat-5378	150	4	𝑛	𝑛	PROPN
easat-5378	150	5	+	+	SYM
easat-5378	150	6	1)8	1)8	NUM
easat-5378	150	7	−	−	PROPN
easat-5378	150	8	1	1	NUM
easat-5378	150	9	=	=	SYM
easat-5378	150	10	8𝑆7	8𝑆7	NUM
easat-5378	151	1	+	+	CCONJ
easat-5378	151	2	28𝑆6	28𝑆6	NUM
easat-5378	151	3	+	+	SYM
easat-5378	151	4	56𝑆5	56𝑆5	NUM
easat-5378	151	5	+	+	CCONJ
easat-5378	151	6	70𝑆4	70𝑆4	NUM
easat-5378	151	7	+	+	CCONJ
easat-5378	151	8	56𝑆3	56𝑆3	NUM
easat-5378	151	9	+	+	CCONJ
easat-5378	151	10	28𝑆2	28𝑆2	NUM
easat-5378	151	11	+	+	NUM
easat-5378	151	12	8𝑆1	8𝑆1	NUM
easat-5378	152	1	+	+	CCONJ
easat-5378	152	2	𝑛	𝑛	ADJ
easat-5378	152	3	(	(	PUNCT
easat-5378	152	4	𝑛	𝑛	PROPN
easat-5378	152	5	+	+	NUM
easat-5378	152	6	1)9	1)9	NUM
easat-5378	152	7	−	−	NOUN
easat-5378	152	8	1	1	NUM
easat-5378	152	9	=	=	SYM
easat-5378	152	10	9𝑆8	9𝑆8	NOUN
easat-5378	152	11	+	+	CCONJ
easat-5378	152	12	36𝑆7	36𝑆7	PROPN
easat-5378	153	1	+	+	CCONJ
easat-5378	153	2	84𝑆6	84𝑆6	PROPN
easat-5378	153	3	+	+	CCONJ
easat-5378	153	4	126𝑆5	126𝑆5	NUM
easat-5378	153	5	+	+	CCONJ
easat-5378	153	6	126𝑆4	126𝑆4	NUM
easat-5378	154	1	+	+	NUM
easat-5378	154	2	84𝑆3	84𝑆3	NUM
easat-5378	155	1	+	+	NUM
easat-5378	155	2	36𝑆2	36𝑆2	NUM
easat-5378	155	3	+	+	SYM
easat-5378	155	4	9𝑆1	9𝑆1	NUM
easat-5378	155	5	+	+	CCONJ
easat-5378	155	6	𝑛	𝑛	PROPN
easat-5378	155	7	(	(	PUNCT
easat-5378	155	8	𝑛	𝑛	PROPN
easat-5378	155	9	+	+	X
easat-5378	155	10	1)10	1)10	ADJ
easat-5378	155	11	−	−	NOUN
easat-5378	155	12	1	1	NUM
easat-5378	155	13	=	=	SYM
easat-5378	155	14	10𝑆9	10𝑆9	NUM
easat-5378	156	1	+	+	CCONJ
easat-5378	156	2	45𝑆8	45𝑆8	NUM
easat-5378	156	3	+	+	NOUN
easat-5378	156	4	120𝑆7	120𝑆7	NUM
easat-5378	156	5	+	+	NUM
easat-5378	156	6	210𝑆6	210𝑆6	NUM
easat-5378	156	7	+	+	CCONJ
easat-5378	156	8	252𝑆5	252𝑆5	NUM
easat-5378	156	9	+	+	NUM
easat-5378	156	10	210𝑆4	210𝑆4	NUM
easat-5378	156	11	+	+	CCONJ
easat-5378	156	12	120𝑆3	120𝑆3	NUM
easat-5378	156	13	+	+	SYM
easat-5378	156	14	45𝑆2	45𝑆2	NUM
easat-5378	156	15	+	+	CCONJ
easat-5378	156	16	10𝑆1	10𝑆1	NUM
easat-5378	156	17	+	+	NUM
easat-5378	156	18	𝑛	𝑛	NOUN
easat-5378	156	19	for	for	ADP
easat-5378	156	20	ease	ease	NOUN
easat-5378	156	21	of	of	ADP
easat-5378	156	22	writing	writing	NOUN
easat-5378	156	23	of	of	ADP
easat-5378	156	24	the	the	DET
easat-5378	156	25	above	above	NOUN
easat-5378	156	26	and	and	CCONJ
easat-5378	156	27	to	to	PART
easat-5378	156	28	avoid	avoid	VERB
easat-5378	156	29	confusion	confusion	NOUN
easat-5378	156	30	as	as	ADP
easat-5378	156	31	to	to	ADP
easat-5378	156	32	whether	whether	SCONJ
easat-5378	156	33	a	a	DET
easat-5378	156	34	function	function	NOUN
easat-5378	156	35	or	or	CCONJ
easat-5378	156	36	a	a	DET
easat-5378	156	37	multiplication	multiplication	NOUN
easat-5378	156	38	,	,	PUNCT
easat-5378	156	39	the	the	DET
easat-5378	156	40	sum	sum	NOUN
easat-5378	156	41	of	of	ADP
easat-5378	156	42	powers	power	NOUN
easat-5378	156	43	of	of	ADP
easat-5378	156	44	natural	natural	ADJ
easat-5378	156	45	numbers	number	NOUN
easat-5378	156	46	n	n	PRON
easat-5378	156	47	has	have	AUX
easat-5378	156	48	been	be	AUX
easat-5378	156	49	written	write	VERB
easat-5378	156	50	simply	simply	ADV
easat-5378	156	51	as	as	SCONJ
easat-5378	156	52	𝑆𝑚	𝑆𝑚	PROPN
easat-5378	156	53	instead	instead	ADV
easat-5378	156	54	of	of	ADP
easat-5378	156	55	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NUM
easat-5378	156	56	)	)	PUNCT
easat-5378	156	57	(	(	PUNCT
easat-5378	156	58	e.g.	e.g.	ADV
easat-5378	156	59	,	,	PUNCT
easat-5378	156	60	𝑆1	𝑆1	NOUN
easat-5378	156	61	instead	instead	ADV
easat-5378	156	62	of	of	ADP
easat-5378	156	63	𝑆1(𝑛	𝑆1(𝑛	PROPN
easat-5378	156	64	)	)	PUNCT
easat-5378	156	65	)	)	PUNCT
easat-5378	156	66	.	.	PUNCT
easat-5378	157	1	the	the	DET
easat-5378	157	2	sequences	sequence	NOUN
easat-5378	157	3	of	of	ADP
easat-5378	157	4	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	157	5	numbers	number	NOUN
easat-5378	157	6	,	,	PUNCT
easat-5378	157	7	for	for	ADP
easat-5378	157	8	some	some	DET
easat-5378	157	9	specific	specific	ADJ
easat-5378	157	10	𝑚	𝑚	X
easat-5378	157	11	≥	≥	NUM
easat-5378	157	12	1	1	NUM
easat-5378	157	13	,	,	PUNCT
easat-5378	157	14	are	be	AUX
easat-5378	157	15	given	give	VERB
easat-5378	157	16	as	as	SCONJ
easat-5378	157	17	follows	follow	VERB
easat-5378	157	18	:	:	PUNCT
easat-5378	157	19	{	{	PUNCT
easat-5378	157	20	𝐹𝑡𝑛(1	𝐹𝑡𝑛(1	NUM
easat-5378	157	21	)	)	PUNCT
easat-5378	157	22	}	}	PUNCT
easat-5378	157	23	=	=	SYM
easat-5378	157	24	{	{	PUNCT
easat-5378	157	25	2	2	NUM
easat-5378	157	26	,	,	PUNCT
easat-5378	157	27	5	5	NUM
easat-5378	157	28	,	,	PUNCT
easat-5378	157	29	12	12	NUM
easat-5378	157	30	,	,	PUNCT
easat-5378	157	31	34	34	NUM
easat-5378	157	32	,	,	PUNCT
easat-5378	157	33	135	135	NUM
easat-5378	157	34	,	,	PUNCT
easat-5378	157	35	.	.	PUNCT
easat-5378	157	36	.	.	PUNCT
easat-5378	157	37	.	.	PUNCT
easat-5378	158	1	}	}	PUNCT
easat-5378	159	1	for	for	ADP
easat-5378	159	2	𝑚	𝑚	NOUN
easat-5378	159	3	=	=	SYM
easat-5378	159	4	1	1	NUM
easat-5378	159	5	{	{	PUNCT
easat-5378	159	6	𝐹𝑡𝑛(2	𝐹𝑡𝑛(2	NUM
easat-5378	159	7	)	)	PUNCT
easat-5378	159	8	}	}	PUNCT
easat-5378	159	9	=	=	SYM
easat-5378	159	10	{	{	PUNCT
easat-5378	159	11	2	2	NUM
easat-5378	159	12	,	,	PUNCT
easat-5378	159	13	9	9	NUM
easat-5378	159	14	,	,	PUNCT
easat-5378	159	15	50	50	NUM
easat-5378	159	16	,	,	PUNCT
easat-5378	159	17	606	606	NUM
easat-5378	159	18	,	,	PUNCT
easat-5378	159	19	14455	14455	NUM
easat-5378	159	20	,	,	PUNCT
easat-5378	159	21	.	.	PUNCT
easat-5378	159	22	.	.	PUNCT
easat-5378	159	23	.	.	PUNCT
easat-5378	160	1	}	}	PUNCT
easat-5378	160	2	for	for	ADP
easat-5378	160	3	𝑚	𝑚	NOUN
easat-5378	160	4	=	=	SYM
easat-5378	160	5	2	2	NUM
easat-5378	160	6	{	{	PUNCT
easat-5378	160	7	𝐹𝑡𝑛(3	𝐹𝑡𝑛(3	NUM
easat-5378	160	8	)	)	PUNCT
easat-5378	160	9	}	}	PUNCT
easat-5378	161	1	=	=	SYM
easat-5378	161	2	{	{	PUNCT
easat-5378	161	3	2	2	NUM
easat-5378	161	4	,	,	PUNCT
easat-5378	161	5	17	17	NUM
easat-5378	161	6	,	,	PUNCT
easat-5378	161	7	252	252	NUM
easat-5378	161	8	,	,	PUNCT
easat-5378	161	9	13924	13924	NUM
easat-5378	161	10	,	,	PUNCT
easat-5378	161	11	1728225	1728225	NUM
easat-5378	161	12	,	,	PUNCT
easat-5378	161	13	.	.	PUNCT
easat-5378	161	14	.	.	PUNCT
easat-5378	161	15	.	.	PUNCT
easat-5378	162	1	}	}	PUNCT
easat-5378	163	1	for	for	ADP
easat-5378	163	2	𝑚	𝑚	NOUN
easat-5378	163	3	=	=	SYM
easat-5378	163	4	3	3	NUM
easat-5378	163	5	{	{	PUNCT
easat-5378	163	6	𝐹𝑡𝑛(4	𝐹𝑡𝑛(4	NUM
easat-5378	163	7	)	)	PUNCT
easat-5378	163	8	}	}	PUNCT
easat-5378	163	9	=	=	SYM
easat-5378	163	10	{	{	PUNCT
easat-5378	163	11	2	2	NUM
easat-5378	163	12	,	,	PUNCT
easat-5378	163	13	33	33	NUM
easat-5378	163	14	,	,	PUNCT
easat-5378	163	15	1394	1394	NUM
easat-5378	163	16	,	,	PUNCT
easat-5378	163	17	332130	332130	NUM
easat-5378	163	18	,	,	PUNCT
easat-5378	163	19	207360979	207360979	NUM
easat-5378	163	20	,	,	PUNCT
easat-5378	163	21	.	.	PUNCT
easat-5378	163	22	.	.	PUNCT
easat-5378	163	23	.	.	PUNCT
easat-5378	164	1	}	}	PUNCT
easat-5378	165	1	for	for	ADP
easat-5378	165	2	𝑚	𝑚	NOUN
easat-5378	165	3	=	=	SYM
easat-5378	165	4	4	4	NUM
easat-5378	165	5	{	{	PUNCT
easat-5378	165	6	𝐹𝑡𝑛(5	𝐹𝑡𝑛(5	NOUN
easat-5378	165	7	)	)	PUNCT
easat-5378	165	8	}	}	PUNCT
easat-5378	165	9	=	=	SYM
easat-5378	165	10	{	{	PUNCT
easat-5378	165	11	2	2	NUM
easat-5378	165	12	,	,	PUNCT
easat-5378	165	13	65	65	NUM
easat-5378	165	14	,	,	PUNCT
easat-5378	165	15	8052	8052	NUM
easat-5378	165	16	,	,	PUNCT
easat-5378	165	17	7963924	7963924	NUM
easat-5378	165	18	,	,	PUNCT
easat-5378	165	19	24883204425	24883204425	NUM
easat-5378	165	20	,	,	PUNCT
easat-5378	165	21	.	.	PUNCT
easat-5378	165	22	.	.	PUNCT
easat-5378	165	23	.	.	PUNCT
easat-5378	166	1	}	}	PUNCT
easat-5378	166	2	for	for	ADP
easat-5378	166	3	𝑚	𝑚	NOUN
easat-5378	166	4	=	=	SYM
easat-5378	166	5	5	5	NUM
easat-5378	166	6	{	{	PUNCT
easat-5378	166	7	𝐹𝑡𝑛(6	𝐹𝑡𝑛(6	NUM
easat-5378	166	8	)	)	PUNCT
easat-5378	166	9	}	}	PUNCT
easat-5378	167	1	=	=	SYM
easat-5378	167	2	{	{	PUNCT
easat-5378	167	3	2	2	NUM
easat-5378	167	4	,	,	PUNCT
easat-5378	167	5	129	129	NUM
easat-5378	167	6	,	,	PUNCT
easat-5378	167	7	47450	47450	NUM
easat-5378	167	8	,	,	PUNCT
easat-5378	167	9	191107866	191107866	NUM
easat-5378	167	10	,	,	PUNCT
easat-5378	167	11	2985984020515	2985984020515	NUM
easat-5378	167	12	,	,	PUNCT
easat-5378	167	13	.	.	PUNCT
easat-5378	167	14	.	.	PUNCT
easat-5378	167	15	.	.	PUNCT
easat-5378	168	1	}	}	PUNCT
easat-5378	169	1	for	for	ADP
easat-5378	169	2	𝑚	𝑚	NOUN
easat-5378	169	3	=	=	SYM
easat-5378	169	4	6	6	NUM
easat-5378	169	5	{	{	PUNCT
easat-5378	169	6	𝐹𝑡𝑛(7	𝐹𝑡𝑛(7	NUM
easat-5378	169	7	)	)	PUNCT
easat-5378	169	8	}	}	PUNCT
easat-5378	169	9	=	=	SYM
easat-5378	169	10	{	{	PUNCT
easat-5378	169	11	2	2	NUM
easat-5378	169	12	,	,	PUNCT
easat-5378	169	13	257	257	NUM
easat-5378	169	14	,	,	PUNCT
easat-5378	169	15	282252	282252	NUM
easat-5378	169	16	,	,	PUNCT
easat-5378	169	17	4586490124	4586490124	NUM
easat-5378	169	18	,	,	PUNCT
easat-5378	169	19	358318080096825	358318080096825	NUM
easat-5378	169	20	,	,	PUNCT
easat-5378	169	21	.	.	PUNCT
easat-5378	169	22	.	.	PUNCT
easat-5378	169	23	.	.	PUNCT
easat-5378	169	24	}	}	PUNCT
easat-5378	170	1	for	for	ADP
easat-5378	170	2	𝑚	𝑚	NOUN
easat-5378	170	3	=	=	SYM
easat-5378	170	4	7	7	NUM
easat-5378	170	5	{	{	PUNCT
easat-5378	170	6	𝐹𝑡𝑛(8	𝐹𝑡𝑛(8	NUM
easat-5378	170	7	)	)	PUNCT
easat-5378	170	8	}	}	PUNCT
easat-5378	170	9	=	=	SYM
easat-5378	170	10	{	{	PUNCT
easat-5378	170	11	2	2	NUM
easat-5378	170	12	,	,	PUNCT
easat-5378	170	13	513	513	NUM
easat-5378	170	14	,	,	PUNCT
easat-5378	170	15	1686434	1686434	NUM
easat-5378	170	16	,	,	PUNCT
easat-5378	170	17	11062023330	11062023330	NUM
easat-5378	170	18	,	,	PUNCT
easat-5378	170	19	1334357900462979	1334357900462979	NUM
easat-5378	170	20	,	,	PUNCT
easat-5378	170	21	.	.	PUNCT
easat-5378	170	22	.	.	PUNCT
easat-5378	170	23	.	.	PUNCT
easat-5378	171	1	}	}	PUNCT
easat-5378	172	1	for	for	ADP
easat-5378	172	2	𝑚	𝑚	NOUN
easat-5378	172	3	=	=	SYM
easat-5378	172	4	8	8	NUM
easat-5378	172	5	{	{	PUNCT
easat-5378	172	6	𝐹𝑡𝑛(9	𝐹𝑡𝑛(9	NUM
easat-5378	172	7	)	)	PUNCT
easat-5378	172	8	}	}	PUNCT
easat-5378	172	9	=	=	SYM
easat-5378	172	10	{	{	PUNCT
easat-5378	172	11	2	2	NUM
easat-5378	172	12	,	,	PUNCT
easat-5378	172	13	1025	1025	NUM
easat-5378	172	14	,	,	PUNCT
easat-5378	172	15	10097892	10097892	NUM
easat-5378	172	16	,	,	PUNCT
easat-5378	172	17	286607822564	286607822564	NUM
easat-5378	172	18	,	,	PUNCT
easat-5378	172	19	559780352002235465	559780352002235465	NUM
easat-5378	172	20	,	,	PUNCT
easat-5378	172	21	.	.	PUNCT
easat-5378	172	22	.	.	PUNCT
easat-5378	172	23	.	.	PUNCT
easat-5378	173	1	}	}	PUNCT
easat-5378	173	2	for	for	ADP
easat-5378	173	3	𝑚	𝑚	NOUN
easat-5378	173	4	=	=	SYM
easat-5378	173	5	9	9	NUM
easat-5378	173	6	we	we	PRON
easat-5378	173	7	present	present	VERB
easat-5378	173	8	the	the	DET
easat-5378	173	9	next	next	ADJ
easat-5378	173	10	theorem	theorem	NOUN
easat-5378	173	11	for	for	ADP
easat-5378	173	12	the	the	DET
easat-5378	173	13	general	general	ADJ
easat-5378	173	14	case	case	NOUN
easat-5378	173	15	of	of	ADP
easat-5378	173	16	the	the	DET
easat-5378	173	17	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	173	18	numbers	number	NOUN
easat-5378	173	19	.	.	PUNCT
easat-5378	174	1	theorem	theorem	VERB
easat-5378	174	2	3.5	3.5	NUM
easat-5378	174	3	:	:	PUNCT
easat-5378	174	4	for	for	ADP
easat-5378	174	5	natural	natural	ADJ
easat-5378	174	6	numbers	number	NOUN
easat-5378	174	7	𝑛	𝑛	PROPN
easat-5378	174	8	,	,	PUNCT
easat-5378	174	9	𝑚	𝑚	X
easat-5378	174	10	≥	≥	NUM
easat-5378	174	11	1	1	NUM
easat-5378	174	12	,	,	PUNCT
easat-5378	174	13	the	the	DET
easat-5378	174	14	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	174	15	numbers	number	NOUN
easat-5378	174	16	can	can	AUX
easat-5378	174	17	be	be	AUX
easat-5378	174	18	determined	determine	VERB
easat-5378	174	19	by	by	ADP
easat-5378	174	20	the	the	DET
easat-5378	174	21	formula	formula	NOUN
easat-5378	174	22	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	NOUN
easat-5378	174	23	)	)	PUNCT
easat-5378	174	24	=	=	SYM
easat-5378	175	1	(	(	PUNCT
easat-5378	175	2	𝑛!)𝑚	𝑛!)𝑚	VERB
easat-5378	175	3	+	+	CCONJ
easat-5378	175	4	1	1	NUM
easat-5378	175	5	𝑚	𝑚	X
easat-5378	175	6	+	+	NOUN
easat-5378	175	7	1	1	NUM
easat-5378	176	1	[	[	X
easat-5378	176	2	(	(	PUNCT
easat-5378	176	3	𝑛	𝑛	PROPN
easat-5378	176	4	+	+	NUM
easat-5378	176	5	1)[(𝑛	1)[(𝑛	NUM
easat-5378	176	6	+	+	CCONJ
easat-5378	176	7	1)𝑚	1)𝑚	NUM
easat-5378	177	1	−	−	NOUN
easat-5378	177	2	1	1	NUM
easat-5378	177	3	]	]	PUNCT
easat-5378	177	4	−	−	PUNCT
easat-5378	177	5	∑	∑	INTJ
easat-5378	177	6	(	(	PUNCT
easat-5378	177	7	𝑚	𝑚	PROPN
easat-5378	177	8	+	+	PROPN
easat-5378	177	9	1	1	NUM
easat-5378	177	10	𝑖	𝑖	X
easat-5378	177	11	)	)	PUNCT
easat-5378	177	12	𝑚−1	𝑚−1	PROPN
easat-5378	177	13	𝑖=1	𝑖=1	PROPN
easat-5378	177	14	𝑆𝑖(𝑛	𝑆𝑖(𝑛	NOUN
easat-5378	177	15	)	)	PUNCT
easat-5378	177	16	]	]	PUNCT
easat-5378	177	17	where	where	SCONJ
easat-5378	177	18	𝑛	𝑛	X
easat-5378	177	19	!	!	PUNCT
easat-5378	177	20	=	=	SYM
easat-5378	178	1	1	1	NUM
easat-5378	178	2	⋅	⋅	NUM
easat-5378	178	3	2	2	NUM
easat-5378	178	4	⋅	⋅	X
easat-5378	178	5	3	3	NUM
easat-5378	178	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	178	7	𝑛	𝑛	PROPN
easat-5378	178	8	and	and	CCONJ
easat-5378	178	9	𝑆𝑖(𝑛	𝑆𝑖(𝑛	PRON
easat-5378	178	10	)	)	PUNCT
easat-5378	178	11	=	=	SYM
easat-5378	178	12	1𝑖	1𝑖	NOUN
easat-5378	178	13	+	+	CCONJ
easat-5378	178	14	2𝑖	2𝑖	NOUN
easat-5378	178	15	+	+	CCONJ
easat-5378	178	16	3𝑖+	3𝑖+	NUM
easat-5378	178	17	.	.	PUNCT
easat-5378	178	18	.	.	PUNCT
easat-5378	178	19	.	.	PUNCT
easat-5378	179	1	+	+	X
easat-5378	179	2	𝑛𝑖.	𝑛𝑖.	VERB
easat-5378	179	3	a	a	DET
easat-5378	179	4	recent	recent	ADJ
easat-5378	179	5	paper	paper	NOUN
easat-5378	180	1	[	[	X
easat-5378	180	2	18	18	NUM
easat-5378	180	3	]	]	PUNCT
easat-5378	180	4	gives	give	VERB
easat-5378	180	5	the	the	DET
easat-5378	180	6	proof	proof	NOUN
easat-5378	180	7	of	of	ADP
easat-5378	180	8	theorem	theorem	ADJ
easat-5378	180	9	3.5	3.5	NUM
easat-5378	180	10	.	.	PUNCT
easat-5378	181	1	the	the	DET
easat-5378	181	2	proof	proof	NOUN
easat-5378	181	3	has	have	AUX
easat-5378	181	4	verified	verify	VERB
easat-5378	181	5	the	the	DET
easat-5378	181	6	following	following	NOUN
easat-5378	181	7	:	:	PUNCT
easat-5378	181	8	(	(	PUNCT
easat-5378	181	9	𝑛	𝑛	PROPN
easat-5378	181	10	+	+	NUM
easat-5378	181	11	1)𝑚+1	1)𝑚+1	NUM
easat-5378	181	12	−	−	NOUN
easat-5378	181	13	1	1	NUM
easat-5378	181	14	=	=	SYM
easat-5378	181	15	(	(	PUNCT
easat-5378	181	16	𝑚	𝑚	PROPN
easat-5378	181	17	+	+	PROPN
easat-5378	181	18	1	1	NUM
easat-5378	181	19	𝑚	𝑚	NOUN
easat-5378	181	20	)	)	PUNCT
easat-5378	182	1	𝑆𝑚	𝑆𝑚	NOUN
easat-5378	183	1	+	+	CCONJ
easat-5378	183	2	(	(	PUNCT
easat-5378	183	3	𝑚	𝑚	PROPN
easat-5378	183	4	+	+	PROPN
easat-5378	183	5	1	1	NUM
easat-5378	183	6	𝑚	𝑚	ADP
easat-5378	183	7	−	−	PROPN
easat-5378	183	8	1	1	NUM
easat-5378	183	9	)	)	PUNCT
easat-5378	183	10	𝑆𝑚−1	𝑆𝑚−1	PROPN
easat-5378	184	1	+	+	CCONJ
easat-5378	184	2	(	(	PUNCT
easat-5378	184	3	𝑚	𝑚	PROPN
easat-5378	184	4	+	+	PROPN
easat-5378	184	5	1	1	NUM
easat-5378	184	6	𝑚	𝑚	ADP
easat-5378	184	7	−	−	PROPN
easat-5378	184	8	2	2	NUM
easat-5378	184	9	)	)	PUNCT
easat-5378	184	10	𝑆𝑚−2	𝑆𝑚−2	PROPN
easat-5378	184	11	+	+	PROPN
easat-5378	184	12	.	.	PUNCT
easat-5378	184	13	.	.	PUNCT
easat-5378	184	14	.	.	PUNCT
easat-5378	185	1	+	+	CCONJ
easat-5378	185	2	(	(	PUNCT
easat-5378	185	3	𝑚	𝑚	X
easat-5378	185	4	+	+	NOUN
easat-5378	185	5	1	1	NUM
easat-5378	185	6	1	1	NUM
easat-5378	185	7	)	)	PUNCT
easat-5378	185	8	𝑆1	𝑆1	NOUN
easat-5378	185	9	+	+	CCONJ
easat-5378	185	10	𝑛	𝑛	PROPN
easat-5378	185	11	wherein	wherein	NOUN
easat-5378	185	12	,	,	PUNCT
easat-5378	185	13	the	the	DET
easat-5378	185	14	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NOUN
easat-5378	185	15	)	)	PUNCT
easat-5378	185	16	is	be	AUX
easat-5378	185	17	simply	simply	ADV
easat-5378	185	18	written	write	VERB
easat-5378	185	19	again	again	ADV
easat-5378	185	20	as	as	SCONJ
easat-5378	185	21	𝑆𝑚.	𝑆𝑚.	PROPN
easat-5378	185	22	we	we	PRON
easat-5378	185	23	present	present	VERB
easat-5378	185	24	the	the	DET
easat-5378	185	25	next	next	ADJ
easat-5378	185	26	two	two	NUM
easat-5378	185	27	theorems	theorem	NOUN
easat-5378	185	28	for	for	ADP
easat-5378	185	29	the	the	DET
easat-5378	185	30	even	even	ADJ
easat-5378	185	31	and	and	CCONJ
easat-5378	185	32	odd	odd	ADJ
easat-5378	185	33	m	m	VERB
easat-5378	185	34	in	in	ADP
easat-5378	185	35	the	the	DET
easat-5378	185	36	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	185	37	numbers	number	NOUN
easat-5378	185	38	.	.	PUNCT
easat-5378	186	1	theorem	theorem	VERB
easat-5378	186	2	3.6	3.6	NUM
easat-5378	186	3	:	:	PUNCT
easat-5378	186	4	the	the	DET
easat-5378	186	5	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	186	6	number	number	NOUN
easat-5378	186	7	for	for	ADP
easat-5378	186	8	even	even	ADV
easat-5378	186	9	𝑚	𝑚	NOUN
easat-5378	186	10	=	=	SYM
easat-5378	186	11	2𝑘	2𝑘	NUM
easat-5378	186	12	is	be	AUX
easat-5378	186	13	given	give	VERB
easat-5378	186	14	by	by	ADP
easat-5378	186	15	the	the	DET
easat-5378	186	16	formula	formula	NOUN
easat-5378	186	17	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	NOUN
easat-5378	186	18	)	)	PUNCT
easat-5378	186	19	=	=	SYM
easat-5378	186	20	𝐹𝑡𝑛(2𝑘	𝐹𝑡𝑛(2𝑘	NOUN
easat-5378	186	21	)	)	PUNCT
easat-5378	186	22	=	=	SYM
easat-5378	187	1	(	(	PUNCT
easat-5378	187	2	𝑛!)2𝑘	𝑛!)2𝑘	X
easat-5378	187	3	+	+	NUM
easat-5378	187	4	2𝑛	2𝑛	NUM
easat-5378	187	5	+	+	CCONJ
easat-5378	187	6	1	1	NUM
easat-5378	187	7	2𝑘	2𝑘	NUM
easat-5378	187	8	+	+	CCONJ
easat-5378	187	9	1	1	NUM
easat-5378	187	10	[	[	X
easat-5378	187	11	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	187	12	+	+	CCONJ
easat-5378	187	13	𝑃(𝑛2𝑘−3)]𝑇𝑛	𝑃(𝑛2𝑘−3)]𝑇𝑛	ADJ
easat-5378	187	14	where	where	SCONJ
easat-5378	187	15	𝑛	𝑛	X
easat-5378	187	16	!	!	PUNCT
easat-5378	187	17	=	=	SYM
easat-5378	188	1	1	1	NUM
easat-5378	188	2	⋅	⋅	NUM
easat-5378	188	3	2	2	NUM
easat-5378	188	4	⋅	⋅	X
easat-5378	188	5	3	3	NUM
easat-5378	188	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	188	7	𝑛	𝑛	PROPN
easat-5378	188	8	,	,	PUNCT
easat-5378	188	9	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	188	10	=	=	NOUN
easat-5378	188	11	1	1	NUM
easat-5378	188	12	+	+	NUM
easat-5378	188	13	2	2	NUM
easat-5378	188	14	+	+	SYM
easat-5378	188	15	3	3	NUM
easat-5378	188	16	+	+	NOUN
easat-5378	188	17	.	.	PUNCT
easat-5378	188	18	.	.	PUNCT
easat-5378	188	19	.	.	PUNCT
easat-5378	189	1	+	+	ADV
easat-5378	189	2	𝑛	𝑛	NOUN
easat-5378	189	3	=	=	SYM
easat-5378	189	4	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	189	5	+	+	CCONJ
easat-5378	189	6	1)/2	1)/2	NUM
easat-5378	189	7	,	,	PUNCT
easat-5378	189	8	and	and	CCONJ
easat-5378	189	9	𝑃(𝑛2𝑘−3	𝑃(𝑛2𝑘−3	NOUN
easat-5378	189	10	)	)	PUNCT
easat-5378	189	11	is	be	AUX
easat-5378	189	12	a	a	DET
easat-5378	189	13	polynomial	polynomial	NOUN
easat-5378	189	14	in	in	ADP
easat-5378	189	15	𝑛	𝑛	PRON
easat-5378	189	16	of	of	ADP
easat-5378	189	17	degree	degree	NOUN
easat-5378	189	18	2𝑘	2𝑘	NUM
easat-5378	189	19	−	−	NOUN
easat-5378	189	20	3	3	NUM
easat-5378	189	21	,	,	PUNCT
easat-5378	189	22	for	for	ADP
easat-5378	189	23	natural	natural	ADJ
easat-5378	189	24	numbers	number	NOUN
easat-5378	189	25	𝑛	𝑛	PROPN
easat-5378	189	26	,	,	PUNCT
easat-5378	189	27	𝑘	𝑘	DET
easat-5378	189	28	≥	≥	NOUN
easat-5378	189	29	1	1	NUM
easat-5378	189	30	.	.	PUNCT
easat-5378	190	1	a	a	DET
easat-5378	190	2	recent	recent	ADJ
easat-5378	190	3	paper	paper	NOUN
easat-5378	190	4	[	[	X
easat-5378	190	5	20	20	NUM
easat-5378	190	6	]	]	PUNCT
easat-5378	190	7	gives	give	VERB
easat-5378	190	8	the	the	DET
easat-5378	190	9	proof	proof	NOUN
easat-5378	190	10	of	of	ADP
easat-5378	190	11	theorem	theorem	ADJ
easat-5378	190	12	3.6	3.6	NUM
easat-5378	190	13	.	.	PUNCT
easat-5378	191	1	the	the	DET
easat-5378	191	2	proof	proof	NOUN
easat-5378	191	3	has	have	AUX
easat-5378	191	4	verified	verify	VERB
easat-5378	191	5	the	the	DET
easat-5378	191	6	following	following	NOUN
easat-5378	191	7	:	:	PUNCT
easat-5378	191	8	𝑆2(1	𝑆2(1	NOUN
easat-5378	191	9	)	)	PUNCT
easat-5378	191	10	=	=	SYM
easat-5378	192	1	𝑆2	𝑆2	NOUN
easat-5378	192	2	=	=	SYM
easat-5378	192	3	2𝑛	2𝑛	PROPN
easat-5378	193	1	+	+	CCONJ
easat-5378	193	2	1	1	NUM
easat-5378	193	3	3	3	NUM
easat-5378	193	4	𝑇𝑛	𝑇𝑛	ADV
easat-5378	193	5	𝑆2(2	𝑆2(2	ADJ
easat-5378	193	6	)	)	PUNCT
easat-5378	194	1	=	=	SYM
easat-5378	194	2	𝑆4	𝑆4	PROPN
easat-5378	194	3	=	=	SYM
easat-5378	194	4	2𝑛	2𝑛	PROPN
easat-5378	195	1	+	+	CCONJ
easat-5378	195	2	1	1	NUM
easat-5378	195	3	5	5	NUM
easat-5378	195	4	[	[	X
easat-5378	195	5	𝑛2	𝑛2	NOUN
easat-5378	195	6	+	+	CCONJ
easat-5378	195	7	(	(	PUNCT
easat-5378	195	8	𝑛	𝑛	PRON
easat-5378	195	9	−	−	NUM
easat-5378	195	10	1	1	NUM
easat-5378	195	11	3	3	NUM
easat-5378	195	12	)	)	PUNCT
easat-5378	195	13	]	]	PUNCT
easat-5378	195	14	𝑇𝑛	𝑇𝑛	NOUN
easat-5378	195	15	𝑆2(3	𝑆2(3	NOUN
easat-5378	195	16	)	)	PUNCT
easat-5378	195	17	=	=	PUNCT
easat-5378	195	18	𝑆6	𝑆6	NOUN
easat-5378	195	19	=	=	PUNCT
easat-5378	195	20	2𝑛	2𝑛	PROPN
easat-5378	196	1	+	+	CCONJ
easat-5378	196	2	1	1	NUM
easat-5378	196	3	7	7	NUM
easat-5378	196	4	[	[	X
easat-5378	196	5	𝑛4	𝑛4	NOUN
easat-5378	196	6	+	+	CCONJ
easat-5378	196	7	(	(	PUNCT
easat-5378	196	8	2𝑛3	2𝑛3	NUM
easat-5378	196	9	−	−	NOUN
easat-5378	196	10	𝑛	𝑛	PROPN
easat-5378	197	1	+	+	NOUN
easat-5378	197	2	1	1	NUM
easat-5378	197	3	3	3	NUM
easat-5378	197	4	)	)	PUNCT
easat-5378	198	1	]	]	PUNCT
easat-5378	198	2	𝑇𝑛	𝑇𝑛	NOUN
easat-5378	198	3	𝑆2(4	𝑆2(4	NOUN
easat-5378	198	4	)	)	PUNCT
easat-5378	198	5	=	=	SYM
easat-5378	198	6	𝑆8	𝑆8	PROPN
easat-5378	198	7	=	=	PROPN
easat-5378	198	8	2𝑛	2𝑛	PROPN
easat-5378	198	9	+	+	CCONJ
easat-5378	198	10	1	1	NUM
easat-5378	198	11	9	9	NUM
easat-5378	198	12	[	[	X
easat-5378	198	13	𝑛6	𝑛6	PROPN
easat-5378	198	14	+	+	CCONJ
easat-5378	198	15	(	(	PUNCT
easat-5378	198	16	3𝑛5	3𝑛5	NUM
easat-5378	198	17	+	+	NUM
easat-5378	198	18	𝑛4	𝑛4	NOUN
easat-5378	198	19	−	−	PROPN
easat-5378	198	20	3𝑛3	3𝑛3	NUM
easat-5378	198	21	−	−	NOUN
easat-5378	198	22	1	1	NUM
easat-5378	198	23	5	5	NUM
easat-5378	198	24	𝑛2	𝑛2	NOUN
easat-5378	198	25	+	+	CCONJ
easat-5378	198	26	9	9	NUM
easat-5378	198	27	5	5	NUM
easat-5378	198	28	𝑛	𝑛	DET
easat-5378	198	29	−	−	NUM
easat-5378	198	30	3	3	NUM
easat-5378	198	31	5	5	NUM
easat-5378	198	32	)	)	PUNCT
easat-5378	198	33	]	]	PUNCT
easat-5378	198	34	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	198	35	896	896	NUM
easat-5378	198	36	edelweiss	edelweiss	PROPN
easat-5378	198	37	applied	apply	VERB
easat-5378	198	38	science	science	NOUN
easat-5378	198	39	and	and	CCONJ
easat-5378	198	40	technology	technology	NOUN
easat-5378	198	41	issn	issn	PROPN
easat-5378	198	42	:	:	PUNCT
easat-5378	198	43	2576	2576	NUM
easat-5378	198	44	-	-	SYM
easat-5378	198	45	8484	8484	NUM
easat-5378	198	46	vol	vol	NOUN
easat-5378	198	47	.	.	PROPN
easat-5378	199	1	9	9	NUM
easat-5378	199	2	,	,	PUNCT
easat-5378	199	3	no	no	INTJ
easat-5378	199	4	.	.	NOUN
easat-5378	200	1	3	3	NUM
easat-5378	200	2	:	:	PUNCT
easat-5378	200	3	891	891	NUM
easat-5378	200	4	-	-	SYM
easat-5378	200	5	904	904	NUM
easat-5378	200	6	,	,	PUNCT
easat-5378	200	7	2025	2025	NUM
easat-5378	200	8	doi	doi	NOUN
easat-5378	200	9	:	:	PUNCT
easat-5378	200	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	200	11	©	©	PROPN
easat-5378	200	12	2025	2025	NUM
easat-5378	200	13	by	by	ADP
easat-5378	200	14	the	the	DET
easat-5378	200	15	author	author	NOUN
easat-5378	200	16	;	;	PUNCT
easat-5378	200	17	licensee	licensee	PROPN
easat-5378	200	18	learning	learning	NOUN
easat-5378	200	19	gate	gate	NOUN
easat-5378	200	20	𝑆2𝑘	𝑆2𝑘	PROPN
easat-5378	200	21	=	=	SYM
easat-5378	200	22	2𝑛	2𝑛	PROPN
easat-5378	201	1	+	+	CCONJ
easat-5378	201	2	1	1	NUM
easat-5378	201	3	2𝑘	2𝑘	NUM
easat-5378	201	4	+	+	CCONJ
easat-5378	201	5	1	1	NUM
easat-5378	201	6	[	[	X
easat-5378	201	7	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	201	8	+	+	CCONJ
easat-5378	201	9	𝑃(𝑛2𝑘−3)]𝑇𝑛	𝑃(𝑛2𝑘−3)]𝑇𝑛	ADJ
easat-5378	201	10	wherein	wherein	NOUN
easat-5378	201	11	,	,	PUNCT
easat-5378	201	12	the	the	DET
easat-5378	201	13	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NOUN
easat-5378	201	14	)	)	PUNCT
easat-5378	201	15	is	be	AUX
easat-5378	201	16	simply	simply	ADV
easat-5378	201	17	written	write	VERB
easat-5378	201	18	again	again	ADV
easat-5378	201	19	as	as	SCONJ
easat-5378	201	20	𝑆𝑚	𝑆𝑚	PROPN
easat-5378	201	21	(	(	PUNCT
easat-5378	201	22	e.g.	e.g.	ADV
easat-5378	201	23	,	,	PUNCT
easat-5378	201	24	𝑆2	𝑆2	PROPN
easat-5378	201	25	means	mean	VERB
easat-5378	201	26	𝑆2(𝑛	𝑆2(𝑛	NOUN
easat-5378	201	27	)	)	PUNCT
easat-5378	201	28	)	)	PUNCT
easat-5378	201	29	.	.	PUNCT
easat-5378	202	1	theorem	theorem	VERB
easat-5378	202	2	3.7	3.7	NUM
easat-5378	202	3	:	:	PUNCT
easat-5378	202	4	the	the	DET
easat-5378	202	5	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	202	6	number	number	NOUN
easat-5378	202	7	for	for	ADP
easat-5378	202	8	odd	odd	ADJ
easat-5378	202	9	𝑚	𝑚	NOUN
easat-5378	202	10	=	=	SYM
easat-5378	202	11	2𝑘	2𝑘	NOUN
easat-5378	203	1	+	+	CCONJ
easat-5378	203	2	1	1	NUM
easat-5378	203	3	is	be	AUX
easat-5378	203	4	given	give	VERB
easat-5378	203	5	by	by	ADP
easat-5378	203	6	the	the	DET
easat-5378	203	7	formula	formula	NOUN
easat-5378	203	8	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	NOUN
easat-5378	203	9	)	)	PUNCT
easat-5378	203	10	=	=	SYM
easat-5378	203	11	𝐹𝑡𝑛(2𝑘+1	𝐹𝑡𝑛(2𝑘+1	NOUN
easat-5378	203	12	)	)	PUNCT
easat-5378	203	13	=	=	NOUN
easat-5378	203	14	(	(	PUNCT
easat-5378	203	15	𝑛!)2𝑘+1	𝑛!)2𝑘+1	X
easat-5378	203	16	+	+	CCONJ
easat-5378	203	17	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	203	18	+	+	CCONJ
easat-5378	203	19	1	1	NUM
easat-5378	203	20	)	)	PUNCT
easat-5378	203	21	𝑘	𝑘	NOUN
easat-5378	204	1	+	+	ADJ
easat-5378	204	2	1	1	NUM
easat-5378	205	1	[	[	X
easat-5378	205	2	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	205	3	+	+	CCONJ
easat-5378	205	4	𝑃(𝑛2𝑘−3)]𝑇𝑛	𝑃(𝑛2𝑘−3)]𝑇𝑛	ADJ
easat-5378	205	5	where	where	SCONJ
easat-5378	205	6	𝑛	𝑛	X
easat-5378	205	7	!	!	PUNCT
easat-5378	205	8	=	=	SYM
easat-5378	205	9	1	1	NUM
easat-5378	205	10	⋅	⋅	NUM
easat-5378	205	11	2	2	NUM
easat-5378	205	12	⋅	⋅	X
easat-5378	205	13	3	3	NUM
easat-5378	205	14	⋅⋅⋅	⋅⋅⋅	X
easat-5378	205	15	𝑛	𝑛	PROPN
easat-5378	205	16	,	,	PUNCT
easat-5378	205	17	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	205	18	=	=	NOUN
easat-5378	205	19	1	1	NUM
easat-5378	205	20	+	+	NUM
easat-5378	205	21	2	2	NUM
easat-5378	205	22	+	+	SYM
easat-5378	205	23	3	3	NUM
easat-5378	205	24	+	+	NOUN
easat-5378	205	25	.	.	PUNCT
easat-5378	205	26	.	.	PUNCT
easat-5378	205	27	.	.	PUNCT
easat-5378	206	1	+	+	ADV
easat-5378	206	2	𝑛	𝑛	NOUN
easat-5378	206	3	=	=	SYM
easat-5378	206	4	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	206	5	+	+	CCONJ
easat-5378	206	6	1)/2	1)/2	NUM
easat-5378	206	7	,	,	PUNCT
easat-5378	206	8	and	and	CCONJ
easat-5378	206	9	𝑃(𝑛2𝑘−3	𝑃(𝑛2𝑘−3	NOUN
easat-5378	206	10	)	)	PUNCT
easat-5378	206	11	is	be	AUX
easat-5378	206	12	a	a	DET
easat-5378	206	13	polynomial	polynomial	NOUN
easat-5378	206	14	in	in	ADP
easat-5378	206	15	𝑛	𝑛	PRON
easat-5378	206	16	of	of	ADP
easat-5378	206	17	degree	degree	NOUN
easat-5378	206	18	2𝑘	2𝑘	NUM
easat-5378	206	19	−	−	NOUN
easat-5378	206	20	3	3	NUM
easat-5378	206	21	,	,	PUNCT
easat-5378	206	22	for	for	ADP
easat-5378	206	23	natural	natural	ADJ
easat-5378	206	24	numbers	number	NOUN
easat-5378	206	25	𝑛	𝑛	PROPN
easat-5378	206	26	,	,	PUNCT
easat-5378	206	27	𝑘	𝑘	DET
easat-5378	206	28	≥	≥	NOUN
easat-5378	206	29	1	1	NUM
easat-5378	206	30	.	.	PUNCT
easat-5378	207	1	a	a	DET
easat-5378	207	2	recent	recent	ADJ
easat-5378	207	3	paper	paper	NOUN
easat-5378	207	4	[	[	X
easat-5378	207	5	20	20	NUM
easat-5378	207	6	]	]	PUNCT
easat-5378	207	7	gives	give	VERB
easat-5378	207	8	the	the	DET
easat-5378	207	9	proof	proof	NOUN
easat-5378	207	10	of	of	ADP
easat-5378	207	11	theorem	theorem	ADJ
easat-5378	207	12	3.7	3.7	NUM
easat-5378	207	13	.	.	PUNCT
easat-5378	208	1	the	the	DET
easat-5378	208	2	proof	proof	NOUN
easat-5378	208	3	has	have	AUX
easat-5378	208	4	verified	verify	VERB
easat-5378	208	5	the	the	DET
easat-5378	208	6	following	follow	VERB
easat-5378	208	7	:	:	PUNCT
easat-5378	208	8	𝑆2(1)+1	𝑆2(1)+1	NOUN
easat-5378	208	9	=	=	SYM
easat-5378	208	10	𝑆3	𝑆3	PROPN
easat-5378	208	11	=	=	PUNCT
easat-5378	208	12	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	208	13	+	+	CCONJ
easat-5378	208	14	1	1	NUM
easat-5378	208	15	)	)	SYM
easat-5378	208	16	2	2	NUM
easat-5378	208	17	𝑇𝑛	𝑇𝑛	NOUN
easat-5378	208	18	𝑆2(2)+1	𝑆2(2)+1	NOUN
easat-5378	208	19	=	=	SYM
easat-5378	208	20	𝑆5	𝑆5	PROPN
easat-5378	208	21	=	=	PUNCT
easat-5378	208	22	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	208	23	+	+	CCONJ
easat-5378	208	24	1	1	NUM
easat-5378	208	25	)	)	PUNCT
easat-5378	208	26	3	3	NUM
easat-5378	208	27	[	[	X
easat-5378	208	28	𝑛2	𝑛2	NOUN
easat-5378	208	29	+	+	CCONJ
easat-5378	208	30	(	(	PUNCT
easat-5378	208	31	𝑛	𝑛	PRON
easat-5378	208	32	−	−	NUM
easat-5378	208	33	1	1	NUM
easat-5378	208	34	2	2	NUM
easat-5378	208	35	)	)	PUNCT
easat-5378	208	36	]	]	PUNCT
easat-5378	208	37	𝑇𝑛	𝑇𝑛	NOUN
easat-5378	208	38	𝑆2(3)+1	𝑆2(3)+1	NOUN
easat-5378	208	39	=	=	SYM
easat-5378	208	40	𝑆7	𝑆7	NOUN
easat-5378	208	41	=	=	PUNCT
easat-5378	208	42	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	208	43	+	+	CCONJ
easat-5378	208	44	1	1	NUM
easat-5378	208	45	)	)	PUNCT
easat-5378	208	46	4	4	NUM
easat-5378	208	47	[	[	X
easat-5378	208	48	𝑛4	𝑛4	NOUN
easat-5378	208	49	+	+	CCONJ
easat-5378	208	50	(	(	PUNCT
easat-5378	208	51	2𝑛3	2𝑛3	NUM
easat-5378	208	52	−	−	NUM
easat-5378	208	53	1	1	NUM
easat-5378	208	54	3	3	NUM
easat-5378	208	55	𝑛2	𝑛2	NOUN
easat-5378	208	56	−	−	PROPN
easat-5378	208	57	4	4	NUM
easat-5378	208	58	3	3	NUM
easat-5378	208	59	𝑛	𝑛	NOUN
easat-5378	208	60	+	+	CCONJ
easat-5378	208	61	2	2	NUM
easat-5378	208	62	3	3	NUM
easat-5378	208	63	)	)	PUNCT
easat-5378	208	64	]	]	PUNCT
easat-5378	209	1	𝑇𝑛	𝑇𝑛	NOUN
easat-5378	209	2	𝑆2(4)+1	𝑆2(4)+1	NOUN
easat-5378	209	3	=	=	SYM
easat-5378	209	4	𝑆9	𝑆9	NOUN
easat-5378	209	5	=	=	PUNCT
easat-5378	209	6	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	209	7	+	+	CCONJ
easat-5378	209	8	1	1	NUM
easat-5378	209	9	)	)	PUNCT
easat-5378	209	10	5	5	NUM
easat-5378	210	1	[	[	X
easat-5378	210	2	𝑛6	𝑛6	NOUN
easat-5378	210	3	+	+	CCONJ
easat-5378	210	4	(	(	PUNCT
easat-5378	210	5	3𝑛5	3𝑛5	NUM
easat-5378	210	6	+	+	CCONJ
easat-5378	210	7	1	1	NUM
easat-5378	210	8	2	2	NUM
easat-5378	210	9	𝑛4	𝑛4	NOUN
easat-5378	210	10	−	−	NOUN
easat-5378	210	11	4𝑛3	4𝑛3	NUM
easat-5378	211	1	+	+	CCONJ
easat-5378	211	2	1	1	NUM
easat-5378	211	3	2	2	NUM
easat-5378	211	4	𝑛2	𝑛2	NOUN
easat-5378	211	5	+	+	NOUN
easat-5378	211	6	3𝑛	3𝑛	NUM
easat-5378	211	7	−	−	NOUN
easat-5378	211	8	3	3	NUM
easat-5378	211	9	2	2	NUM
easat-5378	211	10	)	)	PUNCT
easat-5378	211	11	]	]	PUNCT
easat-5378	211	12	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	211	13	𝑆2𝑘+1	𝑆2𝑘+1	PROPN
easat-5378	211	14	=	=	PUNCT
easat-5378	211	15	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	211	16	+	+	CCONJ
easat-5378	211	17	1	1	X
easat-5378	211	18	)	)	PUNCT
easat-5378	211	19	𝑘	𝑘	NOUN
easat-5378	212	1	+	+	ADJ
easat-5378	212	2	1	1	NUM
easat-5378	212	3	[	[	X
easat-5378	212	4	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	212	5	+	+	CCONJ
easat-5378	212	6	𝑃(𝑛2𝑘−3)]𝑇𝑛	𝑃(𝑛2𝑘−3)]𝑇𝑛	ADJ
easat-5378	212	7	wherein	wherein	NOUN
easat-5378	212	8	,	,	PUNCT
easat-5378	212	9	the	the	DET
easat-5378	212	10	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NOUN
easat-5378	212	11	)	)	PUNCT
easat-5378	212	12	is	be	AUX
easat-5378	212	13	simply	simply	ADV
easat-5378	212	14	written	write	VERB
easat-5378	212	15	again	again	ADV
easat-5378	212	16	as	as	ADP
easat-5378	212	17	𝑆𝑚	𝑆𝑚	PROPN
easat-5378	212	18	(	(	PUNCT
easat-5378	212	19	e.g.	e.g.	ADV
easat-5378	212	20	,	,	PUNCT
easat-5378	212	21	𝑆3	𝑆3	PROPN
easat-5378	212	22	means	mean	VERB
easat-5378	212	23	𝑆3(𝑛	𝑆3(𝑛	PROPN
easat-5378	212	24	)	)	PUNCT
easat-5378	212	25	)	)	PUNCT
easat-5378	212	26	.	.	PUNCT
easat-5378	213	1	we	we	PRON
easat-5378	213	2	also	also	ADV
easat-5378	213	3	present	present	VERB
easat-5378	213	4	the	the	DET
easat-5378	213	5	next	next	ADJ
easat-5378	213	6	two	two	NUM
easat-5378	213	7	theorems	theorem	NOUN
easat-5378	213	8	for	for	ADP
easat-5378	213	9	the	the	DET
easat-5378	213	10	even	even	ADJ
easat-5378	213	11	and	and	CCONJ
easat-5378	213	12	odd	odd	ADJ
easat-5378	213	13	m	m	VERB
easat-5378	213	14	in	in	ADP
easat-5378	213	15	the	the	DET
easat-5378	213	16	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	213	17	numbers	number	NOUN
easat-5378	213	18	that	that	PRON
easat-5378	213	19	involves	involve	VERB
easat-5378	213	20	representation	representation	NOUN
easat-5378	213	21	in	in	ADP
easat-5378	213	22	the	the	DET
easat-5378	213	23	sum	sum	NOUN
easat-5378	213	24	of	of	ADP
easat-5378	213	25	powers	power	NOUN
easat-5378	213	26	instead	instead	ADV
easat-5378	213	27	of	of	ADP
easat-5378	213	28	the	the	DET
easat-5378	213	29	triangular	triangular	NOUN
easat-5378	213	30	numbers	number	NOUN
easat-5378	213	31	.	.	PUNCT
easat-5378	214	1	theorem	theorem	VERB
easat-5378	214	2	3.8	3.8	NUM
easat-5378	214	3	:	:	PUNCT
easat-5378	214	4	the	the	DET
easat-5378	214	5	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	214	6	number	number	NOUN
easat-5378	214	7	for	for	ADP
easat-5378	214	8	even	even	ADV
easat-5378	214	9	𝑚	𝑚	NOUN
easat-5378	214	10	=	=	SYM
easat-5378	214	11	2𝑘	2𝑘	NUM
easat-5378	214	12	is	be	AUX
easat-5378	214	13	given	give	VERB
easat-5378	214	14	by	by	ADP
easat-5378	214	15	the	the	DET
easat-5378	214	16	formula	formula	NOUN
easat-5378	214	17	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	NOUN
easat-5378	214	18	)	)	PUNCT
easat-5378	214	19	=	=	SYM
easat-5378	214	20	𝐹𝑡𝑛(2𝑘	𝐹𝑡𝑛(2𝑘	NOUN
easat-5378	214	21	)	)	PUNCT
easat-5378	214	22	=	=	SYM
easat-5378	215	1	(	(	PUNCT
easat-5378	215	2	𝑛!)2𝑘	𝑛!)2𝑘	X
easat-5378	215	3	+	+	NUM
easat-5378	215	4	3	3	NUM
easat-5378	215	5	2𝑘	2𝑘	NUM
easat-5378	216	1	+	+	CCONJ
easat-5378	216	2	1	1	NUM
easat-5378	216	3	[	[	X
easat-5378	216	4	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	216	5	+	+	NUM
easat-5378	216	6	𝑃(𝑛2𝑘−3)]𝑆2(𝑛	𝑃(𝑛2𝑘−3)]𝑆2(𝑛	NOUN
easat-5378	216	7	)	)	PUNCT
easat-5378	216	8	where	where	SCONJ
easat-5378	216	9	𝑛	𝑛	X
easat-5378	216	10	!	!	PUNCT
easat-5378	216	11	=	=	SYM
easat-5378	217	1	1	1	NUM
easat-5378	217	2	⋅	⋅	NUM
easat-5378	217	3	2	2	NUM
easat-5378	217	4	⋅	⋅	X
easat-5378	217	5	3	3	NUM
easat-5378	217	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	217	7	𝑛	𝑛	PROPN
easat-5378	217	8	,	,	PUNCT
easat-5378	217	9	𝑆2(𝑛	𝑆2(𝑛	NOUN
easat-5378	217	10	)	)	PUNCT
easat-5378	217	11	=	=	PUNCT
easat-5378	217	12	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	217	13	+	+	CCONJ
easat-5378	217	14	1)(2𝑛	1)(2𝑛	NUM
easat-5378	217	15	+	+	SYM
easat-5378	217	16	1)/6	1)/6	NUM
easat-5378	217	17	,	,	PUNCT
easat-5378	217	18	𝑆3(𝑛	𝑆3(𝑛	NUM
easat-5378	217	19	)	)	PUNCT
easat-5378	217	20	=	=	SYM
easat-5378	217	21	𝑛2(𝑛	𝑛2(𝑛	PUNCT
easat-5378	217	22	+	+	X
easat-5378	217	23	1)2/4	1)2/4	NUM
easat-5378	217	24	,	,	PUNCT
easat-5378	217	25	and	and	CCONJ
easat-5378	217	26	𝑃(𝑛2𝑘−3	𝑃(𝑛2𝑘−3	NOUN
easat-5378	217	27	)	)	PUNCT
easat-5378	217	28	is	be	AUX
easat-5378	217	29	a	a	DET
easat-5378	217	30	polynomial	polynomial	NOUN
easat-5378	217	31	in	in	ADP
easat-5378	217	32	𝑛	𝑛	PRON
easat-5378	217	33	of	of	ADP
easat-5378	217	34	degree	degree	NOUN
easat-5378	217	35	2𝑘	2𝑘	NUM
easat-5378	217	36	−	−	NOUN
easat-5378	217	37	3	3	NUM
easat-5378	217	38	,	,	PUNCT
easat-5378	217	39	for	for	ADP
easat-5378	217	40	natural	natural	ADJ
easat-5378	217	41	numbers	number	NOUN
easat-5378	217	42	𝑛	𝑛	PRON
easat-5378	217	43	≥	≥	NOUN
easat-5378	217	44	1and	1and	NUM
easat-5378	217	45	𝑘	𝑘	X
easat-5378	217	46	>	>	X
easat-5378	217	47	1	1	NUM
easat-5378	217	48	.	.	PUNCT
easat-5378	218	1	a	a	DET
easat-5378	218	2	recent	recent	ADJ
easat-5378	218	3	paper	paper	NOUN
easat-5378	218	4	[	[	X
easat-5378	218	5	20	20	NUM
easat-5378	218	6	]	]	PUNCT
easat-5378	218	7	gives	give	VERB
easat-5378	218	8	the	the	DET
easat-5378	218	9	proof	proof	NOUN
easat-5378	218	10	of	of	ADP
easat-5378	218	11	theorem	theorem	ADJ
easat-5378	218	12	3.8	3.8	NUM
easat-5378	218	13	.	.	PUNCT
easat-5378	219	1	the	the	DET
easat-5378	219	2	proof	proof	NOUN
easat-5378	219	3	has	have	AUX
easat-5378	219	4	verified	verify	VERB
easat-5378	219	5	the	the	DET
easat-5378	219	6	following	following	NOUN
easat-5378	219	7	:	:	PUNCT
easat-5378	219	8	𝑆2(2	𝑆2(2	X
easat-5378	219	9	)	)	PUNCT
easat-5378	219	10	=	=	SYM
easat-5378	220	1	𝑆4	𝑆4	PROPN
easat-5378	220	2	=	=	SYM
easat-5378	220	3	3	3	NUM
easat-5378	220	4	5	5	NUM
easat-5378	220	5	[	[	X
easat-5378	220	6	𝑛2	𝑛2	NOUN
easat-5378	220	7	+	+	CCONJ
easat-5378	220	8	(	(	PUNCT
easat-5378	220	9	𝑛	𝑛	PRON
easat-5378	220	10	−	−	NUM
easat-5378	220	11	1	1	NUM
easat-5378	220	12	3	3	NUM
easat-5378	220	13	)	)	PUNCT
easat-5378	220	14	]	]	PUNCT
easat-5378	220	15	𝑆2	𝑆2	PROPN
easat-5378	220	16	𝑆2(3	𝑆2(3	NOUN
easat-5378	220	17	)	)	PUNCT
easat-5378	220	18	=	=	PUNCT
easat-5378	220	19	𝑆6	𝑆6	NOUN
easat-5378	220	20	=	=	NOUN
easat-5378	220	21	3	3	NUM
easat-5378	220	22	7	7	NUM
easat-5378	220	23	[	[	X
easat-5378	220	24	𝑛4	𝑛4	NOUN
easat-5378	220	25	+	+	CCONJ
easat-5378	220	26	(	(	PUNCT
easat-5378	220	27	2𝑛3	2𝑛3	NUM
easat-5378	220	28	−	−	NOUN
easat-5378	220	29	𝑛	𝑛	PROPN
easat-5378	221	1	+	+	NOUN
easat-5378	221	2	1	1	NUM
easat-5378	221	3	3	3	NUM
easat-5378	221	4	)	)	PUNCT
easat-5378	221	5	]	]	PUNCT
easat-5378	221	6	𝑆2	𝑆2	PROPN
easat-5378	221	7	𝑆2(4	𝑆2(4	PROPN
easat-5378	221	8	)	)	PUNCT
easat-5378	222	1	=	=	SYM
easat-5378	222	2	𝑆8	𝑆8	NOUN
easat-5378	222	3	=	=	NOUN
easat-5378	222	4	3	3	NUM
easat-5378	222	5	9	9	NUM
easat-5378	222	6	[	[	X
easat-5378	222	7	𝑛6	𝑛6	PROPN
easat-5378	222	8	+	+	CCONJ
easat-5378	222	9	(	(	PUNCT
easat-5378	222	10	3𝑛5	3𝑛5	NUM
easat-5378	222	11	+	+	NUM
easat-5378	222	12	𝑛4	𝑛4	NOUN
easat-5378	222	13	−	−	PROPN
easat-5378	222	14	3𝑛3	3𝑛3	NUM
easat-5378	222	15	−	−	NOUN
easat-5378	222	16	1	1	NUM
easat-5378	222	17	5	5	NUM
easat-5378	222	18	𝑛2	𝑛2	NOUN
easat-5378	222	19	+	+	CCONJ
easat-5378	222	20	9	9	NUM
easat-5378	222	21	5	5	NUM
easat-5378	222	22	𝑛	𝑛	DET
easat-5378	222	23	−	−	NUM
easat-5378	222	24	3	3	NUM
easat-5378	222	25	5	5	NUM
easat-5378	222	26	)	)	PUNCT
easat-5378	222	27	]	]	PUNCT
easat-5378	222	28	𝑆2	𝑆2	PROPN
easat-5378	222	29	𝑆2𝑘	𝑆2𝑘	PROPN
easat-5378	222	30	=	=	PUNCT
easat-5378	222	31	3	3	NUM
easat-5378	222	32	2𝑘	2𝑘	NUM
easat-5378	223	1	+	+	CCONJ
easat-5378	223	2	1	1	NUM
easat-5378	223	3	[	[	X
easat-5378	223	4	𝑛2𝑘−2	𝑛2𝑘−2	NUM
easat-5378	223	5	+	+	CCONJ
easat-5378	223	6	𝑃(𝑛2𝑘−3)]𝑆2	𝑃(𝑛2𝑘−3)]𝑆2	NOUN
easat-5378	223	7	where	where	SCONJ
easat-5378	223	8	in	in	ADV
easat-5378	223	9	,	,	PUNCT
easat-5378	223	10	the	the	DET
easat-5378	223	11	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NOUN
easat-5378	223	12	)	)	PUNCT
easat-5378	223	13	is	be	AUX
easat-5378	223	14	simply	simply	ADV
easat-5378	223	15	written	write	VERB
easat-5378	223	16	again	again	ADV
easat-5378	223	17	as	as	SCONJ
easat-5378	223	18	𝑆𝑚	𝑆𝑚	PROPN
easat-5378	223	19	(	(	PUNCT
easat-5378	223	20	e.g.	e.g.	ADV
easat-5378	223	21	,	,	PUNCT
easat-5378	223	22	𝑆4	𝑆4	PROPN
easat-5378	223	23	means	mean	VERB
easat-5378	223	24	𝑆4(𝑛	𝑆4(𝑛	PROPN
easat-5378	223	25	)	)	PUNCT
easat-5378	223	26	)	)	PUNCT
easat-5378	223	27	.	.	PUNCT
easat-5378	224	1	theorem	theorem	VERB
easat-5378	224	2	3.9	3.9	NUM
easat-5378	224	3	:	:	PUNCT
easat-5378	224	4	the	the	DET
easat-5378	224	5	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	224	6	number	number	NOUN
easat-5378	224	7	for	for	ADP
easat-5378	224	8	odd	odd	ADJ
easat-5378	224	9	𝑚	𝑚	NOUN
easat-5378	224	10	=	=	SYM
easat-5378	224	11	2𝑘	2𝑘	NOUN
easat-5378	225	1	+	+	CCONJ
easat-5378	225	2	1	1	NUM
easat-5378	225	3	is	be	AUX
easat-5378	225	4	given	give	VERB
easat-5378	225	5	by	by	ADP
easat-5378	225	6	the	the	DET
easat-5378	225	7	formula	formula	NOUN
easat-5378	225	8	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	NOUN
easat-5378	225	9	)	)	PUNCT
easat-5378	225	10	=	=	SYM
easat-5378	225	11	𝐹𝑡𝑛(2𝑘+1	𝐹𝑡𝑛(2𝑘+1	NOUN
easat-5378	225	12	)	)	PUNCT
easat-5378	225	13	=	=	NOUN
easat-5378	225	14	(	(	PUNCT
easat-5378	225	15	𝑛!)2𝑘+1	𝑛!)2𝑘+1	NUM
easat-5378	225	16	+	+	SYM
easat-5378	225	17	2	2	NUM
easat-5378	225	18	𝑘	𝑘	NOUN
easat-5378	225	19	+	+	NOUN
easat-5378	225	20	1	1	NUM
easat-5378	225	21	[	[	X
easat-5378	225	22	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	225	23	+	+	NUM
easat-5378	225	24	𝑃(𝑛2𝑘−3)]𝑆3(𝑛	𝑃(𝑛2𝑘−3)]𝑆3(𝑛	NOUN
easat-5378	225	25	)	)	PUNCT
easat-5378	225	26	where	where	SCONJ
easat-5378	225	27	𝑛	𝑛	X
easat-5378	225	28	!	!	PUNCT
easat-5378	225	29	=	=	SYM
easat-5378	226	1	1	1	NUM
easat-5378	226	2	⋅	⋅	NUM
easat-5378	226	3	2	2	NUM
easat-5378	226	4	⋅	⋅	X
easat-5378	226	5	3	3	NUM
easat-5378	226	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	226	7	𝑛	𝑛	PROPN
easat-5378	226	8	,	,	PUNCT
easat-5378	226	9	𝑆2(𝑛	𝑆2(𝑛	NOUN
easat-5378	226	10	)	)	PUNCT
easat-5378	226	11	=	=	PUNCT
easat-5378	226	12	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	226	13	+	+	CCONJ
easat-5378	226	14	1)(2𝑛	1)(2𝑛	NUM
easat-5378	226	15	+	+	SYM
easat-5378	226	16	1)/6	1)/6	NUM
easat-5378	226	17	,	,	PUNCT
easat-5378	226	18	𝑆3(𝑛	𝑆3(𝑛	NUM
easat-5378	226	19	)	)	PUNCT
easat-5378	226	20	=	=	SYM
easat-5378	226	21	𝑛2(𝑛	𝑛2(𝑛	PUNCT
easat-5378	226	22	+	+	X
easat-5378	226	23	1)2/4	1)2/4	NUM
easat-5378	226	24	,	,	PUNCT
easat-5378	226	25	and	and	CCONJ
easat-5378	226	26	𝑃(𝑛2𝑘−3	𝑃(𝑛2𝑘−3	NOUN
easat-5378	226	27	)	)	PUNCT
easat-5378	226	28	is	be	AUX
easat-5378	226	29	a	a	DET
easat-5378	226	30	polynomial	polynomial	NOUN
easat-5378	226	31	in	in	ADP
easat-5378	226	32	𝑛	𝑛	PRON
easat-5378	226	33	of	of	ADP
easat-5378	226	34	degree	degree	NOUN
easat-5378	226	35	2𝑘	2𝑘	NUM
easat-5378	226	36	−	−	NOUN
easat-5378	226	37	3	3	NUM
easat-5378	226	38	,	,	PUNCT
easat-5378	226	39	for	for	ADP
easat-5378	226	40	natural	natural	ADJ
easat-5378	226	41	numbers	number	NOUN
easat-5378	226	42	𝑛	𝑛	PRON
easat-5378	226	43	≥	≥	NOUN
easat-5378	226	44	1and	1and	NUM
easat-5378	226	45	𝑘	𝑘	X
easat-5378	226	46	>	>	X
easat-5378	226	47	1	1	NUM
easat-5378	226	48	.	.	PUNCT
easat-5378	227	1	a	a	DET
easat-5378	227	2	recent	recent	ADJ
easat-5378	227	3	paper	paper	NOUN
easat-5378	227	4	[	[	X
easat-5378	227	5	20	20	NUM
easat-5378	227	6	]	]	PUNCT
easat-5378	227	7	gives	give	VERB
easat-5378	227	8	the	the	DET
easat-5378	227	9	proof	proof	NOUN
easat-5378	227	10	of	of	ADP
easat-5378	227	11	theorem	theorem	ADJ
easat-5378	227	12	3.9	3.9	NUM
easat-5378	227	13	.	.	PUNCT
easat-5378	228	1	the	the	DET
easat-5378	228	2	proof	proof	NOUN
easat-5378	228	3	has	have	AUX
easat-5378	228	4	verified	verify	VERB
easat-5378	228	5	the	the	DET
easat-5378	228	6	following	following	NOUN
easat-5378	228	7	:	:	PUNCT
easat-5378	228	8	𝑆2(2)+1	𝑆2(2)+1	NOUN
easat-5378	228	9	=	=	SYM
easat-5378	228	10	𝑆5	𝑆5	PROPN
easat-5378	228	11	=	=	SYM
easat-5378	228	12	2	2	NUM
easat-5378	228	13	3	3	NUM
easat-5378	228	14	[	[	X
easat-5378	228	15	𝑛2	𝑛2	NOUN
easat-5378	228	16	+	+	CCONJ
easat-5378	228	17	(	(	PUNCT
easat-5378	228	18	𝑛	𝑛	PRON
easat-5378	228	19	−	−	NUM
easat-5378	228	20	1	1	NUM
easat-5378	228	21	2	2	NUM
easat-5378	228	22	)	)	PUNCT
easat-5378	228	23	]	]	PUNCT
easat-5378	228	24	𝑆3	𝑆3	PROPN
easat-5378	228	25	𝑆2(3)+1	𝑆2(3)+1	NOUN
easat-5378	228	26	=	=	SYM
easat-5378	228	27	𝑆7	𝑆7	NOUN
easat-5378	228	28	=	=	SYM
easat-5378	228	29	2	2	NUM
easat-5378	228	30	4	4	NUM
easat-5378	228	31	[	[	X
easat-5378	228	32	𝑛4	𝑛4	NOUN
easat-5378	228	33	+	+	CCONJ
easat-5378	228	34	(	(	PUNCT
easat-5378	228	35	2𝑛3	2𝑛3	NUM
easat-5378	228	36	−	−	NUM
easat-5378	228	37	1	1	NUM
easat-5378	228	38	3	3	NUM
easat-5378	228	39	𝑛2	𝑛2	NOUN
easat-5378	228	40	−	−	PROPN
easat-5378	228	41	4	4	NUM
easat-5378	228	42	3	3	NUM
easat-5378	228	43	𝑛	𝑛	NOUN
easat-5378	228	44	+	+	CCONJ
easat-5378	228	45	2	2	NUM
easat-5378	228	46	3	3	NUM
easat-5378	228	47	)	)	PUNCT
easat-5378	228	48	]	]	PUNCT
easat-5378	228	49	𝑆3	𝑆3	PROPN
easat-5378	228	50	897	897	PROPN
easat-5378	228	51	edelweiss	edelweiss	PROPN
easat-5378	228	52	applied	apply	VERB
easat-5378	228	53	science	science	NOUN
easat-5378	228	54	and	and	CCONJ
easat-5378	228	55	technology	technology	NOUN
easat-5378	228	56	issn	issn	PROPN
easat-5378	228	57	:	:	PUNCT
easat-5378	228	58	2576	2576	NUM
easat-5378	228	59	-	-	SYM
easat-5378	228	60	8484	8484	NUM
easat-5378	228	61	vol	vol	NOUN
easat-5378	228	62	.	.	PROPN
easat-5378	229	1	9	9	NUM
easat-5378	229	2	,	,	PUNCT
easat-5378	229	3	no	no	INTJ
easat-5378	229	4	.	.	NOUN
easat-5378	230	1	3	3	NUM
easat-5378	230	2	:	:	PUNCT
easat-5378	230	3	891	891	NUM
easat-5378	230	4	-	-	SYM
easat-5378	230	5	904	904	NUM
easat-5378	230	6	,	,	PUNCT
easat-5378	230	7	2025	2025	NUM
easat-5378	230	8	doi	doi	NOUN
easat-5378	230	9	:	:	PUNCT
easat-5378	230	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	230	11	©	©	PROPN
easat-5378	230	12	2025	2025	NUM
easat-5378	230	13	by	by	ADP
easat-5378	230	14	the	the	DET
easat-5378	230	15	author	author	NOUN
easat-5378	230	16	;	;	PUNCT
easat-5378	230	17	licensee	licensee	PROPN
easat-5378	230	18	learning	learn	VERB
easat-5378	230	19	gate	gate	NOUN
easat-5378	230	20	𝑆2(4)+1	𝑆2(4)+1	NOUN
easat-5378	230	21	=	=	PUNCT
easat-5378	230	22	𝑆8	𝑆8	NOUN
easat-5378	230	23	=	=	NOUN
easat-5378	230	24	2	2	NUM
easat-5378	230	25	5	5	NUM
easat-5378	230	26	[	[	X
easat-5378	230	27	𝑛6	𝑛6	NOUN
easat-5378	230	28	+	+	CCONJ
easat-5378	230	29	(	(	PUNCT
easat-5378	230	30	3𝑛5	3𝑛5	NUM
easat-5378	230	31	+	+	CCONJ
easat-5378	230	32	1	1	NUM
easat-5378	230	33	2	2	NUM
easat-5378	230	34	𝑛4	𝑛4	NOUN
easat-5378	230	35	−	−	NOUN
easat-5378	230	36	4𝑛3	4𝑛3	NUM
easat-5378	231	1	+	+	CCONJ
easat-5378	231	2	1	1	NUM
easat-5378	231	3	2	2	NUM
easat-5378	231	4	𝑛2	𝑛2	NOUN
easat-5378	231	5	+	+	NOUN
easat-5378	231	6	3𝑛	3𝑛	NUM
easat-5378	231	7	−	−	NOUN
easat-5378	231	8	3	3	NUM
easat-5378	231	9	2	2	NUM
easat-5378	231	10	)	)	PUNCT
easat-5378	231	11	]	]	PUNCT
easat-5378	231	12	𝑆3	𝑆3	PROPN
easat-5378	231	13	𝑆2𝑘+1	𝑆2𝑘+1	PROPN
easat-5378	231	14	=	=	SYM
easat-5378	232	1	2	2	X
easat-5378	232	2	𝑘	𝑘	NOUN
easat-5378	232	3	+	+	NOUN
easat-5378	232	4	1	1	NUM
easat-5378	233	1	[	[	X
easat-5378	233	2	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	233	3	+	+	X
easat-5378	233	4	𝑃(𝑛2𝑘−3)]𝑆3	𝑃(𝑛2𝑘−3)]𝑆3	NOUN
easat-5378	233	5	where	where	SCONJ
easat-5378	233	6	in	in	ADP
easat-5378	233	7	,	,	PUNCT
easat-5378	233	8	the	the	DET
easat-5378	233	9	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NOUN
easat-5378	233	10	)	)	PUNCT
easat-5378	233	11	is	be	AUX
easat-5378	233	12	simply	simply	ADV
easat-5378	233	13	written	write	VERB
easat-5378	233	14	again	again	ADV
easat-5378	233	15	as	as	SCONJ
easat-5378	233	16	𝑆𝑚	𝑆𝑚	PROPN
easat-5378	233	17	(	(	PUNCT
easat-5378	233	18	e.g.	e.g.	ADV
easat-5378	233	19	,	,	PUNCT
easat-5378	233	20	𝑆5	𝑆5	PROPN
easat-5378	233	21	means	mean	VERB
easat-5378	233	22	𝑆5(𝑛	𝑆5(𝑛	NOUN
easat-5378	233	23	)	)	PUNCT
easat-5378	233	24	)	)	PUNCT
easat-5378	233	25	.	.	PUNCT
easat-5378	234	1	the	the	DET
easat-5378	234	2	{	{	PUNCT
easat-5378	234	3	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NOUN
easat-5378	234	4	}	}	PUNCT
easat-5378	234	5	and	and	CCONJ
easat-5378	234	6	{	{	PUNCT
easat-5378	234	7	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	ADJ
easat-5378	234	8	)	)	PUNCT
easat-5378	234	9	}	}	PUNCT
easat-5378	234	10	can	can	AUX
easat-5378	234	11	be	be	AUX
easat-5378	234	12	combined	combine	VERB
easat-5378	234	13	to	to	PART
easat-5378	234	14	produce	produce	VERB
easat-5378	234	15	another	another	DET
easat-5378	234	16	generalization	generalization	NOUN
easat-5378	234	17	of	of	ADP
easat-5378	234	18	the	the	DET
easat-5378	234	19	sequence	sequence	NOUN
easat-5378	234	20	of	of	ADP
easat-5378	234	21	nfactoriangular	nfactoriangular	ADJ
easat-5378	234	22	numbers	number	NOUN
easat-5378	234	23	,	,	PUNCT
easat-5378	234	24	which	which	PRON
easat-5378	234	25	is	be	AUX
easat-5378	234	26	the	the	DET
easat-5378	234	27	sequence	sequence	NOUN
easat-5378	234	28	}	}	PUNCT
easat-5378	234	29	for	for	ADP
easat-5378	234	30	natural	natural	ADJ
easat-5378	234	31	numbers	number	NOUN
easat-5378	234	32	𝑛	𝑛	PROPN
easat-5378	234	33	,	,	PUNCT
easat-5378	234	34	𝑘	𝑘	PROPN
easat-5378	234	35	,	,	PUNCT
easat-5378	234	36	𝑚	𝑚	X
easat-5378	234	37	≥	≥	NUM
easat-5378	234	38	1	1	NUM
easat-5378	234	39	that	that	PRON
easat-5378	234	40	follow	follow	VERB
easat-5378	234	41	the	the	DET
easat-5378	234	42	form	form	NOUN
easat-5378	234	43	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	234	44	)	)	PUNCT
easat-5378	235	1	=	=	SYM
easat-5378	235	2	(	(	PUNCT
easat-5378	235	3	1𝑚	1𝑚	X
easat-5378	235	4	⋅	⋅	PROPN
easat-5378	235	5	2𝑚	2𝑚	NOUN
easat-5378	235	6	⋅	⋅	PROPN
easat-5378	235	7	3𝑚	3𝑚	PROPN
easat-5378	235	8	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	235	9	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	235	10	)	)	PUNCT
easat-5378	235	11	+	+	CCONJ
easat-5378	235	12	(	(	PUNCT
easat-5378	235	13	1𝑚	1𝑚	NOUN
easat-5378	235	14	+	+	CCONJ
easat-5378	235	15	2𝑚	2𝑚	NOUN
easat-5378	235	16	+	+	CCONJ
easat-5378	235	17	3𝑚+	3𝑚+	NUM
easat-5378	235	18	.	.	PUNCT
easat-5378	235	19	.	.	PUNCT
easat-5378	235	20	.	.	PUNCT
easat-5378	236	1	+	+	ADV
easat-5378	236	2	𝑘𝑚	𝑘𝑚	NOUN
easat-5378	236	3	)	)	PUNCT
easat-5378	236	4	we	we	PRON
easat-5378	236	5	call	call	VERB
easat-5378	236	6	the	the	DET
easat-5378	236	7	numbers	number	NOUN
easat-5378	236	8	in	in	ADP
easat-5378	236	9	this	this	DET
easat-5378	236	10	sequence	sequence	NOUN
easat-5378	236	11	as	as	ADP
easat-5378	236	12	(	(	PUNCT
easat-5378	236	13	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	236	14	)	)	PUNCT
easat-5378	236	15	,	,	PUNCT
easat-5378	236	16	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	236	17	numbers	number	NOUN
easat-5378	236	18	and	and	CCONJ
easat-5378	236	19	define	define	VERB
easat-5378	236	20	as	as	SCONJ
easat-5378	236	21	follows	follow	VERB
easat-5378	236	22	:	:	PUNCT
easat-5378	236	23	definition	definition	NOUN
easat-5378	236	24	3.10	3.10	NUM
easat-5378	236	25	:	:	PUNCT
easat-5378	236	26	the	the	DET
easat-5378	236	27	(	(	PUNCT
easat-5378	236	28	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	236	29	)	)	PUNCT
easat-5378	236	30	,	,	PUNCT
easat-5378	236	31	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	236	32	number	number	NOUN
easat-5378	236	33	is	be	AUX
easat-5378	236	34	defined	define	VERB
easat-5378	236	35	by	by	ADP
easat-5378	236	36	the	the	DET
easat-5378	236	37	formula	formula	NOUN
easat-5378	236	38	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	236	39	)	)	PUNCT
easat-5378	237	1	=	=	PUNCT
easat-5378	237	2	(	(	PUNCT
easat-5378	237	3	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	237	4	+	+	CCONJ
easat-5378	237	5	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	237	6	)	)	PUNCT
easat-5378	237	7	where	where	SCONJ
easat-5378	237	8	(	(	PUNCT
easat-5378	237	9	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	237	10	=	=	SYM
easat-5378	237	11	1𝑚	1𝑚	NUM
easat-5378	237	12	⋅	⋅	PROPN
easat-5378	237	13	2𝑚	2𝑚	NOUN
easat-5378	237	14	⋅	⋅	PROPN
easat-5378	237	15	3𝑚	3𝑚	NUM
easat-5378	237	16	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	237	17	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	237	18	and	and	CCONJ
easat-5378	237	19	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	237	20	)	)	PUNCT
easat-5378	237	21	=	=	SYM
easat-5378	237	22	1𝑚	1𝑚	NOUN
easat-5378	237	23	+	+	CCONJ
easat-5378	237	24	2𝑚	2𝑚	NOUN
easat-5378	237	25	+	+	CCONJ
easat-5378	237	26	3𝑚+	3𝑚+	NUM
easat-5378	237	27	.	.	PUNCT
easat-5378	237	28	.	.	PUNCT
easat-5378	237	29	.	.	PUNCT
easat-5378	238	1	+	+	ADV
easat-5378	238	2	𝑘𝑚	𝑘𝑚	NOUN
easat-5378	238	3	for	for	ADP
easat-5378	238	4	natural	natural	ADJ
easat-5378	238	5	numbers	number	NOUN
easat-5378	238	6	𝑛	𝑛	PROPN
easat-5378	238	7	,	,	PUNCT
easat-5378	238	8	𝑘	𝑘	PROPN
easat-5378	238	9	,	,	PUNCT
easat-5378	238	10	𝑚	𝑚	X
easat-5378	238	11	≥	≥	NUM
easat-5378	238	12	1	1	NUM
easat-5378	238	13	.	.	PUNCT
easat-5378	238	14	from	from	ADP
easat-5378	238	15	the	the	DET
easat-5378	238	16	above	above	ADJ
easat-5378	238	17	definitions	definition	NOUN
easat-5378	238	18	,	,	PUNCT
easat-5378	238	19	when	when	SCONJ
easat-5378	238	20	𝑚	𝑚	PROPN
easat-5378	238	21	=	=	SYM
easat-5378	238	22	1	1	NUM
easat-5378	238	23	,	,	PUNCT
easat-5378	238	24	the	the	DET
easat-5378	238	25	(	(	PUNCT
easat-5378	238	26	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	238	27	)	)	PUNCT
easat-5378	238	28	,	,	PUNCT
easat-5378	238	29	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	238	30	numbers	number	NOUN
easat-5378	238	31	are	be	AUX
easat-5378	238	32	the	the	DET
easat-5378	238	33	same	same	ADJ
easat-5378	238	34	as	as	ADP
easat-5378	238	35	the	the	DET
easat-5378	238	36	(	(	PUNCT
easat-5378	238	37	𝑛	𝑛	ADJ
easat-5378	238	38	,	,	PUNCT
easat-5378	238	39	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	238	40	numbers	number	NOUN
easat-5378	238	41	or	or	CCONJ
easat-5378	238	42	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	238	43	)	)	PUNCT
easat-5378	238	44	=	=	SYM
easat-5378	238	45	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NUM
easat-5378	238	46	;	;	PUNCT
easat-5378	238	47	when	when	SCONJ
easat-5378	238	48	𝑛	𝑛	PROPN
easat-5378	238	49	=	=	SYM
easat-5378	238	50	𝑘	𝑘	PROPN
easat-5378	238	51	,	,	PUNCT
easat-5378	238	52	the	the	DET
easat-5378	238	53	(	(	PUNCT
easat-5378	238	54	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	238	55	)	)	PUNCT
easat-5378	238	56	,	,	PUNCT
easat-5378	238	57	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	238	58	numbers	number	NOUN
easat-5378	238	59	are	be	AUX
easat-5378	238	60	the	the	DET
easat-5378	238	61	same	same	ADJ
easat-5378	238	62	as	as	ADP
easat-5378	238	63	the	the	DET
easat-5378	238	64	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	238	65	numbers	number	NOUN
easat-5378	238	66	or	or	CCONJ
easat-5378	238	67	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	238	68	)	)	PUNCT
easat-5378	238	69	=	=	SYM
easat-5378	238	70	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	ADJ
easat-5378	238	71	)	)	PUNCT
easat-5378	238	72	;	;	PUNCT
easat-5378	238	73	and	and	CCONJ
easat-5378	238	74	𝑚	𝑚	X
easat-5378	238	75	=	=	SYM
easat-5378	238	76	1	1	NUM
easat-5378	238	77	and	and	CCONJ
easat-5378	238	78	𝑛	𝑛	PROPN
easat-5378	238	79	=	=	SYM
easat-5378	238	80	𝑘	𝑘	PROPN
easat-5378	238	81	,	,	PUNCT
easat-5378	238	82	the	the	DET
easat-5378	238	83	(	(	PUNCT
easat-5378	238	84	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	238	85	)	)	PUNCT
easat-5378	238	86	,	,	PUNCT
easat-5378	238	87	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	238	88	numbers	number	NOUN
easat-5378	238	89	are	be	AUX
easat-5378	238	90	the	the	DET
easat-5378	238	91	same	same	ADJ
easat-5378	238	92	as	as	ADP
easat-5378	238	93	the	the	DET
easat-5378	238	94	n	n	CCONJ
easat-5378	238	95	-	-	PUNCT
easat-5378	238	96	factoriangular	factoriangular	ADJ
easat-5378	238	97	numbers	number	NOUN
easat-5378	238	98	or	or	CCONJ
easat-5378	238	99	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	238	100	)	)	PUNCT
easat-5378	239	1	=	=	PUNCT
easat-5378	239	2	𝐹𝑡𝑛.	𝐹𝑡𝑛.	VERB
easat-5378	239	3	the	the	DET
easat-5378	239	4	sequence	sequence	NOUN
easat-5378	239	5	of	of	ADP
easat-5378	239	6	(	(	PUNCT
easat-5378	239	7	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	239	8	)	)	PUNCT
easat-5378	239	9	,	,	PUNCT
easat-5378	239	10	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	239	11	numbers	number	NOUN
easat-5378	239	12	can	can	AUX
easat-5378	239	13	be	be	AUX
easat-5378	239	14	further	far	ADV
easat-5378	239	15	generalized	generalize	VERB
easat-5378	239	16	into	into	ADP
easat-5378	239	17	the	the	DET
easat-5378	239	18	sequence	sequence	NOUN
easat-5378	239	19	{	{	PUNCT
easat-5378	239	20	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	239	21	)	)	PUNCT
easat-5378	239	22	}	}	PUNCT
easat-5378	239	23	for	for	ADP
easat-5378	239	24	natural	natural	ADJ
easat-5378	239	25	numbers	number	NOUN
easat-5378	239	26	𝑛	𝑛	PROPN
easat-5378	239	27	,	,	PUNCT
easat-5378	239	28	𝑘	𝑘	PROPN
easat-5378	239	29	,	,	PUNCT
easat-5378	239	30	𝑎	𝑎	NOUN
easat-5378	239	31	,	,	PUNCT
easat-5378	239	32	𝑏	𝑏	PRON
easat-5378	239	33	≥	≥	NOUN
easat-5378	239	34	1	1	NUM
easat-5378	239	35	,	,	PUNCT
easat-5378	239	36	which	which	PRON
easat-5378	239	37	follow	follow	VERB
easat-5378	239	38	the	the	DET
easat-5378	239	39	form	form	NOUN
easat-5378	239	40	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	239	41	)	)	PUNCT
easat-5378	239	42	=	=	PRON
easat-5378	240	1	(	(	PUNCT
easat-5378	240	2	1𝑎	1𝑎	NUM
easat-5378	240	3	⋅	⋅	NUM
easat-5378	240	4	2𝑎	2𝑎	NUM
easat-5378	240	5	⋅	⋅	PROPN
easat-5378	240	6	3𝑎	3𝑎	PROPN
easat-5378	240	7	⋅⋅⋅	⋅⋅⋅	X
easat-5378	240	8	𝑛𝑎	𝑛𝑎	PROPN
easat-5378	240	9	)	)	PUNCT
easat-5378	240	10	+	+	CCONJ
easat-5378	240	11	(	(	PUNCT
easat-5378	240	12	1𝑏	1𝑏	NOUN
easat-5378	240	13	+	+	CCONJ
easat-5378	240	14	2𝑏	2𝑏	NUM
easat-5378	240	15	+	+	X
easat-5378	240	16	3𝑏+	3𝑏+	NUM
easat-5378	240	17	.	.	PUNCT
easat-5378	240	18	.	.	PUNCT
easat-5378	240	19	.	.	PUNCT
easat-5378	241	1	+	+	ADV
easat-5378	241	2	𝑘𝑏	𝑘𝑏	PROPN
easat-5378	241	3	)	)	PUNCT
easat-5378	241	4	we	we	PRON
easat-5378	241	5	call	call	VERB
easat-5378	241	6	the	the	DET
easat-5378	241	7	numbers	number	NOUN
easat-5378	241	8	in	in	ADP
easat-5378	241	9	this	this	DET
easat-5378	241	10	sequence	sequence	NOUN
easat-5378	241	11	as	as	ADP
easat-5378	241	12	(	(	PUNCT
easat-5378	241	13	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	241	14	)	)	PUNCT
easat-5378	241	15	,	,	PUNCT
easat-5378	241	16	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	241	17	numbers	number	NOUN
easat-5378	241	18	and	and	CCONJ
easat-5378	241	19	define	define	VERB
easat-5378	241	20	as	as	SCONJ
easat-5378	241	21	follows	follow	VERB
easat-5378	241	22	:	:	PUNCT
easat-5378	241	23	definition	definition	NOUN
easat-5378	241	24	3.11	3.11	NUM
easat-5378	241	25	:	:	PUNCT
easat-5378	241	26	the	the	DET
easat-5378	241	27	(	(	PUNCT
easat-5378	241	28	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	241	29	)	)	PUNCT
easat-5378	241	30	,	,	PUNCT
easat-5378	241	31	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	241	32	number	number	NOUN
easat-5378	241	33	is	be	AUX
easat-5378	241	34	defined	define	VERB
easat-5378	241	35	by	by	ADP
easat-5378	241	36	the	the	DET
easat-5378	241	37	formula	formula	NOUN
easat-5378	241	38	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	241	39	)	)	PUNCT
easat-5378	241	40	=	=	PUNCT
easat-5378	241	41	(	(	PUNCT
easat-5378	241	42	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	241	43	+	+	CCONJ
easat-5378	241	44	𝑆𝑏(𝑘	𝑆𝑏(𝑘	NOUN
easat-5378	241	45	)	)	PUNCT
easat-5378	241	46	where	where	SCONJ
easat-5378	241	47	(	(	PUNCT
easat-5378	241	48	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	241	49	=	=	SYM
easat-5378	241	50	1𝑎	1𝑎	NUM
easat-5378	241	51	⋅	⋅	NUM
easat-5378	241	52	2𝑎	2𝑎	NUM
easat-5378	241	53	⋅	⋅	PROPN
easat-5378	241	54	3𝑎	3𝑎	NUM
easat-5378	241	55	⋅⋅⋅	⋅⋅⋅	X
easat-5378	241	56	𝑛𝑎	𝑛𝑎	PROPN
easat-5378	241	57	and	and	CCONJ
easat-5378	241	58	𝑆𝑏(𝑘	𝑆𝑏(𝑘	NOUN
easat-5378	241	59	)	)	PUNCT
easat-5378	242	1	=	=	SYM
easat-5378	242	2	1𝑏	1𝑏	NOUN
easat-5378	242	3	+	+	CCONJ
easat-5378	242	4	2𝑏	2𝑏	NUM
easat-5378	242	5	+	+	X
easat-5378	242	6	3𝑏+	3𝑏+	NUM
easat-5378	242	7	.	.	PUNCT
easat-5378	242	8	.	.	PUNCT
easat-5378	242	9	.	.	PUNCT
easat-5378	243	1	+	+	ADJ
easat-5378	243	2	𝑘𝑏	𝑘𝑏	PROPN
easat-5378	243	3	for	for	ADP
easat-5378	243	4	natural	natural	ADJ
easat-5378	243	5	numbers	number	NOUN
easat-5378	243	6	𝑛	𝑛	PROPN
easat-5378	243	7	,	,	PUNCT
easat-5378	243	8	𝑘	𝑘	PROPN
easat-5378	243	9	,	,	PUNCT
easat-5378	243	10	𝑎	𝑎	NOUN
easat-5378	243	11	,	,	PUNCT
easat-5378	243	12	𝑏	𝑏	PRON
easat-5378	243	13	≥	≥	NOUN
easat-5378	243	14	1	1	NUM
easat-5378	243	15	.	.	PUNCT
easat-5378	243	16	from	from	ADP
easat-5378	243	17	the	the	DET
easat-5378	243	18	above	above	ADJ
easat-5378	243	19	definitions	definition	NOUN
easat-5378	243	20	,	,	PUNCT
easat-5378	243	21	when	when	SCONJ
easat-5378	243	22	𝑎	𝑎	X
easat-5378	243	23	=	=	SYM
easat-5378	243	24	𝑏	𝑏	NOUN
easat-5378	243	25	=	=	SYM
easat-5378	243	26	𝑚	𝑚	PROPN
easat-5378	243	27	,	,	PUNCT
easat-5378	243	28	the	the	DET
easat-5378	243	29	(	(	PUNCT
easat-5378	243	30	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	243	31	)	)	PUNCT
easat-5378	243	32	,	,	PUNCT
easat-5378	243	33	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	243	34	numbers	number	NOUN
easat-5378	243	35	are	be	AUX
easat-5378	243	36	the	the	DET
easat-5378	243	37	same	same	ADJ
easat-5378	243	38	as	as	ADP
easat-5378	243	39	the	the	DET
easat-5378	243	40	(	(	PUNCT
easat-5378	243	41	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	243	42	)	)	PUNCT
easat-5378	243	43	,	,	PUNCT
easat-5378	243	44	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	243	45	numbers	number	NOUN
easat-5378	243	46	or	or	CCONJ
easat-5378	243	47	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	243	48	)	)	PUNCT
easat-5378	244	1	=	=	SYM
easat-5378	244	2	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	244	3	)	)	PUNCT
easat-5378	244	4	;	;	PUNCT
easat-5378	244	5	when	when	SCONJ
easat-5378	244	6	𝑎	𝑎	X
easat-5378	244	7	=	=	SYM
easat-5378	244	8	𝑏	𝑏	NOUN
easat-5378	244	9	=	=	SYM
easat-5378	244	10	1	1	NUM
easat-5378	244	11	,	,	PUNCT
easat-5378	244	12	the	the	DET
easat-5378	244	13	(	(	PUNCT
easat-5378	244	14	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	244	15	)	)	PUNCT
easat-5378	244	16	,	,	PUNCT
easat-5378	244	17	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	244	18	numbers	number	NOUN
easat-5378	244	19	are	be	AUX
easat-5378	244	20	the	the	DET
easat-5378	244	21	same	same	ADJ
easat-5378	244	22	as	as	ADP
easat-5378	244	23	the	the	DET
easat-5378	244	24	(	(	PUNCT
easat-5378	244	25	𝑛	𝑛	ADJ
easat-5378	244	26	,	,	PUNCT
easat-5378	244	27	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	244	28	numbers	number	NOUN
easat-5378	244	29	or	or	CCONJ
easat-5378	244	30	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	244	31	)	)	PUNCT
easat-5378	244	32	=	=	SYM
easat-5378	244	33	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NOUN
easat-5378	244	34	;	;	PUNCT
easat-5378	244	35	when	when	SCONJ
easat-5378	244	36	𝑛	𝑛	PROPN
easat-5378	244	37	=	=	SYM
easat-5378	244	38	𝑘	𝑘	PROPN
easat-5378	244	39	and	and	CCONJ
easat-5378	244	40	𝑎	𝑎	X
easat-5378	244	41	=	=	SYM
easat-5378	244	42	𝑏	𝑏	NOUN
easat-5378	244	43	=	=	SYM
easat-5378	244	44	𝑚	𝑚	PROPN
easat-5378	244	45	,	,	PUNCT
easat-5378	244	46	the	the	DET
easat-5378	244	47	(	(	PUNCT
easat-5378	244	48	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	244	49	)	)	PUNCT
easat-5378	244	50	,	,	PUNCT
easat-5378	244	51	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	244	52	numbers	number	NOUN
easat-5378	244	53	are	be	AUX
easat-5378	244	54	the	the	DET
easat-5378	244	55	same	same	ADJ
easat-5378	244	56	as	as	ADP
easat-5378	244	57	the	the	DET
easat-5378	244	58	𝑛(𝑚)factoriangular	𝑛(𝑚)factoriangular	ADJ
easat-5378	244	59	numbers	number	NOUN
easat-5378	244	60	or	or	CCONJ
easat-5378	244	61	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NUM
easat-5378	244	62	)	)	PUNCT
easat-5378	244	63	=	=	PRON
easat-5378	244	64	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	ADJ
easat-5378	244	65	)	)	PUNCT
easat-5378	244	66	;	;	PUNCT
easat-5378	244	67	and	and	CCONJ
easat-5378	244	68	when	when	SCONJ
easat-5378	244	69	𝑛	𝑛	PROPN
easat-5378	244	70	=	=	SYM
easat-5378	244	71	𝑘	𝑘	PROPN
easat-5378	244	72	and	and	CCONJ
easat-5378	244	73	𝑎	𝑎	NOUN
easat-5378	244	74	=	=	SYM
easat-5378	244	75	𝑏	𝑏	NOUN
easat-5378	244	76	=	=	SYM
easat-5378	244	77	1	1	NUM
easat-5378	244	78	,	,	PUNCT
easat-5378	244	79	the	the	DET
easat-5378	244	80	(	(	PUNCT
easat-5378	244	81	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	244	82	)	)	PUNCT
easat-5378	244	83	,	,	PUNCT
easat-5378	244	84	𝑘(𝑏))factoriangular	𝑘(𝑏))factoriangular	PROPN
easat-5378	244	85	numbers	number	NOUN
easat-5378	244	86	are	be	AUX
easat-5378	244	87	the	the	DET
easat-5378	244	88	same	same	ADJ
easat-5378	244	89	as	as	ADP
easat-5378	244	90	the	the	DET
easat-5378	244	91	n	n	CCONJ
easat-5378	244	92	-	-	PUNCT
easat-5378	244	93	factoriangular	factoriangular	ADJ
easat-5378	244	94	numbers	number	NOUN
easat-5378	244	95	or	or	CCONJ
easat-5378	244	96	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	244	97	)	)	PUNCT
easat-5378	245	1	=	=	PUNCT
easat-5378	245	2	𝐹𝑡𝑛.	𝐹𝑡𝑛.	VERB
easat-5378	245	3	the	the	DET
easat-5378	245	4	above	above	ADJ
easat-5378	245	5	definitions	definition	NOUN
easat-5378	245	6	were	be	AUX
easat-5378	245	7	taken	take	VERB
easat-5378	245	8	from	from	ADP
easat-5378	245	9	previous	previous	ADJ
easat-5378	245	10	study	study	NOUN
easat-5378	245	11	castillo	castillo	PROPN
easat-5378	246	1	[	[	X
easat-5378	246	2	20	20	NUM
easat-5378	246	3	]	]	PUNCT
easat-5378	246	4	.	.	PUNCT
easat-5378	247	1	however	however	ADV
easat-5378	247	2	,	,	PUNCT
easat-5378	247	3	the	the	DET
easat-5378	247	4	said	say	VERB
easat-5378	247	5	study	study	NOUN
easat-5378	247	6	focuses	focus	VERB
easat-5378	247	7	only	only	ADV
easat-5378	247	8	on	on	ADP
easat-5378	247	9	the	the	DET
easat-5378	247	10	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	247	11	numbers	number	NOUN
easat-5378	247	12	.	.	PUNCT
easat-5378	248	1	in	in	ADP
easat-5378	248	2	the	the	DET
easat-5378	248	3	present	present	ADJ
easat-5378	248	4	study	study	NOUN
easat-5378	248	5	,	,	PUNCT
easat-5378	248	6	we	we	PRON
easat-5378	248	7	focus	focus	VERB
easat-5378	248	8	on	on	ADP
easat-5378	248	9	the	the	DET
easat-5378	248	10	(	(	PUNCT
easat-5378	248	11	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	248	12	)	)	PUNCT
easat-5378	248	13	,	,	PUNCT
easat-5378	248	14	𝑘(𝑚))factoriangular	𝑘(𝑚))factoriangular	ADJ
easat-5378	248	15	numbers	number	NOUN
easat-5378	248	16	and	and	CCONJ
easat-5378	248	17	on	on	ADP
easat-5378	248	18	the	the	DET
easat-5378	248	19	(	(	PUNCT
easat-5378	248	20	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	248	21	)	)	PUNCT
easat-5378	248	22	,	,	PUNCT
easat-5378	248	23	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	248	24	numbers	number	NOUN
easat-5378	248	25	.	.	PUNCT
easat-5378	249	1	3.2	3.2	NUM
easat-5378	249	2	.	.	PUNCT
easat-5378	250	1	main	main	ADJ
easat-5378	250	2	results	result	NOUN
easat-5378	250	3	3.2.1	3.2.1	NUM
easat-5378	250	4	.	.	PUNCT
easat-5378	251	1	on	on	ADP
easat-5378	251	2	the	the	DET
easat-5378	251	3	(	(	PUNCT
easat-5378	251	4	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	251	5	)	)	PUNCT
easat-5378	251	6	,	,	PUNCT
easat-5378	251	7	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	251	8	numbers	number	NOUN
easat-5378	251	9	we	we	PRON
easat-5378	251	10	integrate	integrate	VERB
easat-5378	251	11	the	the	DET
easat-5378	251	12	notions	notion	NOUN
easat-5378	251	13	of	of	ADP
easat-5378	251	14	(	(	PUNCT
easat-5378	251	15	𝑛	𝑛	ADJ
easat-5378	251	16	,	,	PUNCT
easat-5378	251	17	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	251	18	numbers	number	NOUN
easat-5378	251	19	and	and	CCONJ
easat-5378	251	20	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	251	21	numbers	number	NOUN
easat-5378	251	22	to	to	PART
easat-5378	251	23	form	form	VERB
easat-5378	251	24	the	the	DET
easat-5378	251	25	(	(	PUNCT
easat-5378	251	26	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	251	27	)	)	PUNCT
easat-5378	251	28	,	,	PUNCT
easat-5378	251	29	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	251	30	numbers	number	NOUN
easat-5378	251	31	(	(	PUNCT
easat-5378	251	32	see	see	VERB
easat-5378	251	33	definition	definition	NOUN
easat-5378	251	34	3.10	3.10	NUM
easat-5378	251	35	)	)	PUNCT
easat-5378	251	36	.	.	PUNCT
easat-5378	252	1	we	we	PRON
easat-5378	252	2	now	now	ADV
easat-5378	252	3	prove	prove	VERB
easat-5378	252	4	the	the	DET
easat-5378	252	5	succeeding	succeed	VERB
easat-5378	252	6	theorems	theorem	NOUN
easat-5378	252	7	and	and	CCONJ
easat-5378	252	8	give	give	VERB
easat-5378	252	9	examples	example	NOUN
easat-5378	252	10	of	of	ADP
easat-5378	252	11	sequences	sequence	NOUN
easat-5378	252	12	of	of	ADP
easat-5378	252	13	(	(	PUNCT
easat-5378	252	14	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	252	15	)	)	PUNCT
easat-5378	252	16	,	,	PUNCT
easat-5378	252	17	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	252	18	numbers	number	NOUN
easat-5378	252	19	.	.	PUNCT
easat-5378	253	1	for	for	ADP
easat-5378	253	2	simplicity	simplicity	NOUN
easat-5378	253	3	of	of	ADP
easat-5378	253	4	notation	notation	NOUN
easat-5378	253	5	and	and	CCONJ
easat-5378	253	6	to	to	PART
easat-5378	253	7	avoid	avoid	VERB
easat-5378	253	8	confusion	confusion	NOUN
easat-5378	253	9	between	between	ADP
easat-5378	253	10	a	a	DET
easat-5378	253	11	function	function	NOUN
easat-5378	253	12	and	and	CCONJ
easat-5378	253	13	a	a	DET
easat-5378	253	14	multiplication	multiplication	NOUN
easat-5378	253	15	,	,	PUNCT
easat-5378	253	16	we	we	PRON
easat-5378	253	17	use	use	VERB
easat-5378	253	18	𝑆𝑚	𝑆𝑚	PROPN
easat-5378	253	19	in	in	ADP
easat-5378	253	20	lieu	lieu	NOUN
easat-5378	253	21	of	of	ADP
easat-5378	253	22	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	253	23	)	)	PUNCT
easat-5378	253	24	in	in	ADP
easat-5378	253	25	the	the	DET
easat-5378	253	26	proofs	proof	NOUN
easat-5378	253	27	.	.	PUNCT
easat-5378	254	1	theorem	theorem	VERB
easat-5378	254	2	3.12	3.12	NUM
easat-5378	254	3	:	:	PUNCT
easat-5378	254	4	for	for	ADP
easat-5378	254	5	natural	natural	ADJ
easat-5378	254	6	numbers	number	NOUN
easat-5378	254	7	𝑛	𝑛	PROPN
easat-5378	254	8	,	,	PUNCT
easat-5378	254	9	𝑘	𝑘	PROPN
easat-5378	254	10	,	,	PUNCT
easat-5378	254	11	𝑚	𝑚	X
easat-5378	254	12	≥	≥	NUM
easat-5378	254	13	1	1	NUM
easat-5378	254	14	,	,	PUNCT
easat-5378	254	15	the	the	DET
easat-5378	254	16	(	(	PUNCT
easat-5378	254	17	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	254	18	)	)	PUNCT
easat-5378	254	19	,	,	PUNCT
easat-5378	254	20	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	254	21	numbers	number	NOUN
easat-5378	254	22	can	can	AUX
easat-5378	254	23	be	be	AUX
easat-5378	254	24	determined	determine	VERB
easat-5378	254	25	by	by	ADP
easat-5378	254	26	the	the	DET
easat-5378	254	27	formula	formula	NOUN
easat-5378	254	28	898	898	NUM
easat-5378	254	29	edelweiss	edelweiss	PROPN
easat-5378	254	30	applied	apply	VERB
easat-5378	254	31	science	science	NOUN
easat-5378	254	32	and	and	CCONJ
easat-5378	254	33	technology	technology	NOUN
easat-5378	254	34	issn	issn	PROPN
easat-5378	254	35	:	:	PUNCT
easat-5378	254	36	2576	2576	NUM
easat-5378	254	37	-	-	SYM
easat-5378	254	38	8484	8484	NUM
easat-5378	254	39	vol	vol	NOUN
easat-5378	254	40	.	.	PROPN
easat-5378	255	1	9	9	NUM
easat-5378	255	2	,	,	PUNCT
easat-5378	255	3	no	no	INTJ
easat-5378	255	4	.	.	NOUN
easat-5378	256	1	3	3	NUM
easat-5378	256	2	:	:	PUNCT
easat-5378	256	3	891	891	NUM
easat-5378	256	4	-	-	SYM
easat-5378	256	5	904	904	NUM
easat-5378	256	6	,	,	PUNCT
easat-5378	256	7	2025	2025	NUM
easat-5378	256	8	doi	doi	NOUN
easat-5378	256	9	:	:	PUNCT
easat-5378	256	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	256	11	©	©	PROPN
easat-5378	256	12	2025	2025	NUM
easat-5378	256	13	by	by	ADP
easat-5378	256	14	the	the	DET
easat-5378	256	15	author	author	NOUN
easat-5378	256	16	;	;	PUNCT
easat-5378	256	17	licensee	licensee	NOUN
easat-5378	256	18	learning	learn	VERB
easat-5378	256	19	gate	gate	NOUN
easat-5378	256	20	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	256	21	)	)	PUNCT
easat-5378	256	22	=	=	NOUN
easat-5378	256	23	(	(	PUNCT
easat-5378	256	24	𝑛!)𝑚	𝑛!)𝑚	VERB
easat-5378	256	25	+	+	CCONJ
easat-5378	256	26	1	1	NUM
easat-5378	256	27	𝑚	𝑚	X
easat-5378	256	28	+	+	NOUN
easat-5378	256	29	1	1	NUM
easat-5378	256	30	[	[	X
easat-5378	256	31	(	(	PUNCT
easat-5378	256	32	𝑘	𝑘	X
easat-5378	256	33	+	+	X
easat-5378	256	34	1)[(𝑘	1)[(𝑘	NUM
easat-5378	256	35	+	+	CCONJ
easat-5378	256	36	1)𝑚	1)𝑚	NUM
easat-5378	256	37	−	−	NOUN
easat-5378	256	38	1	1	NUM
easat-5378	256	39	]	]	PUNCT
easat-5378	256	40	−	−	PUNCT
easat-5378	256	41	∑	∑	INTJ
easat-5378	256	42	(	(	PUNCT
easat-5378	256	43	𝑚	𝑚	PROPN
easat-5378	256	44	+	+	PROPN
easat-5378	256	45	1	1	NUM
easat-5378	256	46	𝑖	𝑖	X
easat-5378	256	47	)	)	PUNCT
easat-5378	256	48	𝑚−1	𝑚−1	PROPN
easat-5378	256	49	𝑖=1	𝑖=1	PUNCT
easat-5378	256	50	𝑆𝑖(𝑘	𝑆𝑖(𝑘	PROPN
easat-5378	256	51	)	)	PUNCT
easat-5378	256	52	]	]	PUNCT
easat-5378	256	53	where	where	SCONJ
easat-5378	256	54	𝑛	𝑛	X
easat-5378	256	55	!	!	PUNCT
easat-5378	256	56	=	=	SYM
easat-5378	257	1	1	1	NUM
easat-5378	257	2	⋅	⋅	NUM
easat-5378	257	3	2	2	NUM
easat-5378	257	4	⋅	⋅	X
easat-5378	257	5	3	3	NUM
easat-5378	257	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	257	7	𝑛	𝑛	PROPN
easat-5378	257	8	and	and	CCONJ
easat-5378	257	9	𝑆𝑖(𝑘	𝑆𝑖(𝑘	NUM
easat-5378	257	10	)	)	PUNCT
easat-5378	257	11	=	=	SYM
easat-5378	257	12	1𝑖	1𝑖	NOUN
easat-5378	257	13	+	+	CCONJ
easat-5378	257	14	2𝑖	2𝑖	NOUN
easat-5378	257	15	+	+	CCONJ
easat-5378	257	16	3𝑖+	3𝑖+	NUM
easat-5378	257	17	.	.	PUNCT
easat-5378	257	18	.	.	PUNCT
easat-5378	257	19	.	.	PUNCT
easat-5378	258	1	+	+	PUNCT
easat-5378	258	2	𝑘𝑖.	𝑘𝑖.	ADJ
easat-5378	258	3	proof	proof	NOUN
easat-5378	258	4	:	:	PUNCT
easat-5378	258	5	from	from	ADP
easat-5378	258	6	definition	definition	NOUN
easat-5378	258	7	3.10	3.10	NUM
easat-5378	258	8	,	,	PUNCT
easat-5378	258	9	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	258	10	)	)	PUNCT
easat-5378	258	11	=	=	SYM
easat-5378	258	12	(	(	PUNCT
easat-5378	258	13	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	258	14	+	+	CCONJ
easat-5378	258	15	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	258	16	)	)	PUNCT
easat-5378	258	17	.	.	PUNCT
easat-5378	259	1	what	what	PRON
easat-5378	259	2	we	we	PRON
easat-5378	259	3	need	need	VERB
easat-5378	259	4	to	to	PART
easat-5378	259	5	show	show	VERB
easat-5378	259	6	is	be	AUX
easat-5378	259	7	that	that	PRON
easat-5378	259	8	𝑆𝑚(𝑘	𝑆𝑚(𝑘	ADJ
easat-5378	259	9	)	)	PUNCT
easat-5378	259	10	=	=	SYM
easat-5378	260	1	1	1	NUM
easat-5378	260	2	𝑚	𝑚	X
easat-5378	260	3	+	+	NOUN
easat-5378	260	4	1	1	NUM
easat-5378	260	5	[	[	X
easat-5378	260	6	(	(	PUNCT
easat-5378	260	7	𝑘	𝑘	X
easat-5378	260	8	+	+	X
easat-5378	260	9	1)[(𝑘	1)[(𝑘	NUM
easat-5378	260	10	+	+	CCONJ
easat-5378	260	11	1)𝑚	1)𝑚	NUM
easat-5378	260	12	−	−	NOUN
easat-5378	260	13	1	1	NUM
easat-5378	260	14	]	]	PUNCT
easat-5378	260	15	−	−	PUNCT
easat-5378	260	16	∑	∑	INTJ
easat-5378	260	17	(	(	PUNCT
easat-5378	260	18	𝑚	𝑚	PROPN
easat-5378	260	19	+	+	PROPN
easat-5378	260	20	1	1	NUM
easat-5378	260	21	𝑖	𝑖	X
easat-5378	260	22	)	)	PUNCT
easat-5378	260	23	𝑚−1	𝑚−1	PROPN
easat-5378	260	24	𝑖=1	𝑖=1	PUNCT
easat-5378	260	25	𝑆𝑖(𝑘	𝑆𝑖(𝑘	PROPN
easat-5378	260	26	)	)	PUNCT
easat-5378	260	27	]	]	PUNCT
easat-5378	260	28	from	from	ADP
easat-5378	260	29	a	a	DET
easat-5378	260	30	previous	previous	ADJ
easat-5378	260	31	paper	paper	NOUN
easat-5378	260	32	[	[	X
easat-5378	260	33	20	20	NUM
easat-5378	260	34	]	]	X
easat-5378	260	35	we	we	PRON
easat-5378	260	36	deduce	deduce	VERB
easat-5378	260	37	that	that	PRON
easat-5378	260	38	(	(	PUNCT
easat-5378	260	39	𝑘	𝑘	X
easat-5378	260	40	+	+	NOUN
easat-5378	260	41	1)𝑚+1	1)𝑚+1	NUM
easat-5378	260	42	−	−	PROPN
easat-5378	260	43	1	1	NUM
easat-5378	260	44	=	=	SYM
easat-5378	260	45	(	(	PUNCT
easat-5378	260	46	𝑚	𝑚	PROPN
easat-5378	260	47	+	+	PROPN
easat-5378	260	48	1	1	NUM
easat-5378	260	49	𝑚	𝑚	NOUN
easat-5378	260	50	)	)	PUNCT
easat-5378	261	1	𝑆𝑚	𝑆𝑚	NOUN
easat-5378	262	1	+	+	CCONJ
easat-5378	262	2	(	(	PUNCT
easat-5378	262	3	𝑚	𝑚	PROPN
easat-5378	262	4	+	+	PROPN
easat-5378	262	5	1	1	NUM
easat-5378	262	6	𝑚	𝑚	ADP
easat-5378	262	7	−	−	PROPN
easat-5378	262	8	1	1	NUM
easat-5378	262	9	)	)	PUNCT
easat-5378	262	10	𝑆𝑚−1	𝑆𝑚−1	PROPN
easat-5378	263	1	+	+	CCONJ
easat-5378	263	2	(	(	PUNCT
easat-5378	263	3	𝑚	𝑚	PROPN
easat-5378	263	4	+	+	PROPN
easat-5378	263	5	1	1	NUM
easat-5378	263	6	𝑚	𝑚	ADP
easat-5378	263	7	−	−	PROPN
easat-5378	263	8	2	2	NUM
easat-5378	263	9	)	)	PUNCT
easat-5378	263	10	𝑆𝑚−2	𝑆𝑚−2	PROPN
easat-5378	263	11	+	+	PROPN
easat-5378	263	12	.	.	PUNCT
easat-5378	263	13	.	.	PUNCT
easat-5378	263	14	.	.	PUNCT
easat-5378	264	1	+	+	CCONJ
easat-5378	264	2	(	(	PUNCT
easat-5378	264	3	𝑚	𝑚	X
easat-5378	264	4	+	+	NOUN
easat-5378	264	5	1	1	NUM
easat-5378	264	6	1	1	NUM
easat-5378	264	7	)	)	PUNCT
easat-5378	264	8	𝑆1	𝑆1	NOUN
easat-5378	264	9	+	+	CCONJ
easat-5378	264	10	𝑘	𝑘	PROPN
easat-5378	264	11	or	or	CCONJ
easat-5378	264	12	(	(	PUNCT
easat-5378	264	13	𝑘	𝑘	PROPN
easat-5378	264	14	+	+	NOUN
easat-5378	264	15	1)𝑚+1	1)𝑚+1	NUM
easat-5378	264	16	−	−	PROPN
easat-5378	265	1	(	(	PUNCT
easat-5378	265	2	𝑘	𝑘	PROPN
easat-5378	265	3	+	+	NOUN
easat-5378	265	4	1	1	NUM
easat-5378	265	5	)	)	PUNCT
easat-5378	265	6	=	=	NOUN
easat-5378	265	7	(	(	PUNCT
easat-5378	265	8	𝑚	𝑚	X
easat-5378	265	9	+	+	X
easat-5378	265	10	1)𝑆𝑚	1)𝑆𝑚	NUM
easat-5378	265	11	+	+	CCONJ
easat-5378	265	12	(	(	PUNCT
easat-5378	265	13	𝑚	𝑚	PROPN
easat-5378	265	14	+	+	NOUN
easat-5378	265	15	1	1	NUM
easat-5378	265	16	1	1	NUM
easat-5378	265	17	)	)	PUNCT
easat-5378	265	18	𝑆1	𝑆1	NOUN
easat-5378	265	19	+	+	CCONJ
easat-5378	265	20	(	(	PUNCT
easat-5378	265	21	𝑚	𝑚	PROPN
easat-5378	265	22	+	+	NOUN
easat-5378	265	23	1	1	NUM
easat-5378	265	24	2	2	NUM
easat-5378	265	25	)	)	PUNCT
easat-5378	265	26	𝑆2	𝑆2	PROPN
easat-5378	265	27	+	+	PROPN
easat-5378	265	28	.	.	PUNCT
easat-5378	265	29	.	.	PUNCT
easat-5378	265	30	.	.	PUNCT
easat-5378	266	1	+	+	CCONJ
easat-5378	266	2	(	(	PUNCT
easat-5378	266	3	𝑚	𝑚	X
easat-5378	266	4	+	+	NOUN
easat-5378	266	5	1	1	NUM
easat-5378	266	6	𝑚	𝑚	ADP
easat-5378	266	7	−	−	PROPN
easat-5378	266	8	1	1	NUM
easat-5378	266	9	)	)	PUNCT
easat-5378	266	10	𝑆𝑚−1	𝑆𝑚−1	PROPN
easat-5378	266	11	and	and	CCONJ
easat-5378	266	12	then	then	ADV
easat-5378	266	13	,	,	PUNCT
easat-5378	266	14	𝑆𝑚	𝑆𝑚	PROPN
easat-5378	266	15	=	=	NOUN
easat-5378	267	1	1	1	NUM
easat-5378	267	2	𝑚	𝑚	X
easat-5378	267	3	+	+	NOUN
easat-5378	267	4	1	1	NUM
easat-5378	268	1	[	[	X
easat-5378	268	2	(	(	PUNCT
easat-5378	268	3	𝑘	𝑘	X
easat-5378	268	4	+	+	X
easat-5378	268	5	1)[(𝑘	1)[(𝑘	NUM
easat-5378	268	6	+	+	CCONJ
easat-5378	268	7	1)𝑚	1)𝑚	NUM
easat-5378	268	8	−	−	NOUN
easat-5378	268	9	1	1	NUM
easat-5378	268	10	]	]	PUNCT
easat-5378	268	11	−	−	PUNCT
easat-5378	268	12	∑	∑	INTJ
easat-5378	268	13	(	(	PUNCT
easat-5378	268	14	𝑚	𝑚	PROPN
easat-5378	268	15	+	+	PROPN
easat-5378	268	16	1	1	NUM
easat-5378	268	17	𝑖	𝑖	X
easat-5378	268	18	)	)	PUNCT
easat-5378	268	19	𝑚−1	𝑚−1	PROPN
easat-5378	268	20	𝑖=1	𝑖=1	PUNCT
easat-5378	269	1	𝑆𝑖	𝑆𝑖	PROPN
easat-5378	269	2	]	]	X
easat-5378	269	3	it	it	PRON
easat-5378	269	4	follows	follow	VERB
easat-5378	269	5	shortly	shortly	ADV
easat-5378	269	6	that	that	SCONJ
easat-5378	269	7	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	269	8	)	)	PUNCT
easat-5378	269	9	=	=	NOUN
easat-5378	270	1	(	(	PUNCT
easat-5378	270	2	𝑛!)𝑚	𝑛!)𝑚	VERB
easat-5378	270	3	+	+	CCONJ
easat-5378	270	4	1	1	NUM
easat-5378	270	5	𝑚	𝑚	X
easat-5378	270	6	+	+	NOUN
easat-5378	270	7	1	1	NUM
easat-5378	271	1	[	[	X
easat-5378	271	2	(	(	PUNCT
easat-5378	271	3	𝑘	𝑘	X
easat-5378	271	4	+	+	X
easat-5378	271	5	1)[(𝑘	1)[(𝑘	NUM
easat-5378	271	6	+	+	CCONJ
easat-5378	271	7	1)𝑚	1)𝑚	NUM
easat-5378	271	8	−	−	NOUN
easat-5378	271	9	1	1	NUM
easat-5378	271	10	]	]	PUNCT
easat-5378	271	11	−	−	PUNCT
easat-5378	271	12	∑	∑	INTJ
easat-5378	271	13	(	(	PUNCT
easat-5378	271	14	𝑚	𝑚	PROPN
easat-5378	271	15	+	+	PROPN
easat-5378	271	16	1	1	NUM
easat-5378	271	17	𝑖	𝑖	X
easat-5378	271	18	)	)	PUNCT
easat-5378	271	19	𝑚−1	𝑚−1	PROPN
easat-5378	271	20	𝑖=1	𝑖=1	PUNCT
easat-5378	271	21	𝑆𝑖(𝑘	𝑆𝑖(𝑘	PROPN
easat-5378	271	22	)	)	PUNCT
easat-5378	271	23	]	]	PUNCT
easat-5378	271	24	theorem	theorem	VERB
easat-5378	271	25	3.13	3.13	NUM
easat-5378	271	26	:	:	PUNCT
easat-5378	271	27	the	the	DET
easat-5378	271	28	(	(	PUNCT
easat-5378	271	29	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	271	30	)	)	PUNCT
easat-5378	271	31	,	,	PUNCT
easat-5378	271	32	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	271	33	number	number	NOUN
easat-5378	271	34	for	for	ADP
easat-5378	271	35	even	even	ADV
easat-5378	271	36	𝑚	𝑚	NOUN
easat-5378	271	37	=	=	NOUN
easat-5378	271	38	2𝑗	2𝑗	NOUN
easat-5378	271	39	is	be	AUX
easat-5378	271	40	given	give	VERB
easat-5378	271	41	by	by	ADP
easat-5378	271	42	the	the	DET
easat-5378	271	43	formula	formula	NOUN
easat-5378	271	44	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	271	45	)	)	PUNCT
easat-5378	271	46	=	=	SYM
easat-5378	271	47	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	PROPN
easat-5378	271	48	)	)	PUNCT
easat-5378	271	49	=	=	PUNCT
easat-5378	272	1	(	(	PUNCT
easat-5378	272	2	𝑛!)2𝑗	𝑛!)2𝑗	NUM
easat-5378	272	3	+	+	CCONJ
easat-5378	272	4	2𝑘	2𝑘	NUM
easat-5378	272	5	+	+	CCONJ
easat-5378	272	6	1	1	NUM
easat-5378	272	7	2𝑗	2𝑗	NUM
easat-5378	272	8	+	+	NOUN
easat-5378	272	9	1	1	NUM
easat-5378	273	1	[	[	X
easat-5378	273	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	273	3	+	+	NUM
easat-5378	273	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	273	5	where	where	SCONJ
easat-5378	273	6	𝑛	𝑛	X
easat-5378	273	7	!	!	PUNCT
easat-5378	273	8	=	=	SYM
easat-5378	274	1	1	1	NUM
easat-5378	274	2	⋅	⋅	NUM
easat-5378	274	3	2	2	NUM
easat-5378	274	4	⋅	⋅	X
easat-5378	274	5	3	3	NUM
easat-5378	274	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	274	7	𝑛	𝑛	PROPN
easat-5378	274	8	,	,	PUNCT
easat-5378	274	9	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	274	10	=	=	SYM
easat-5378	274	11	1	1	NUM
easat-5378	274	12	+	+	NUM
easat-5378	274	13	2	2	NUM
easat-5378	274	14	+	+	SYM
easat-5378	274	15	3	3	NUM
easat-5378	274	16	+	+	NOUN
easat-5378	274	17	.	.	PUNCT
easat-5378	274	18	.	.	PUNCT
easat-5378	274	19	.	.	PUNCT
easat-5378	275	1	+	+	NOUN
easat-5378	275	2	𝑘	𝑘	X
easat-5378	275	3	=	=	PUNCT
easat-5378	275	4	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	275	5	+	+	CCONJ
easat-5378	275	6	1)/2	1)/2	NUM
easat-5378	275	7	,	,	PUNCT
easat-5378	275	8	and	and	CCONJ
easat-5378	275	9	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	275	10	)	)	PUNCT
easat-5378	275	11	is	be	AUX
easat-5378	275	12	a	a	DET
easat-5378	275	13	polynomial	polynomial	NOUN
easat-5378	275	14	in	in	ADP
easat-5378	275	15	𝑘	𝑘	PRON
easat-5378	275	16	of	of	ADP
easat-5378	275	17	degree	degree	NOUN
easat-5378	275	18	2𝑗	2𝑗	NOUN
easat-5378	275	19	−	−	PROPN
easat-5378	275	20	3	3	NUM
easat-5378	275	21	,	,	PUNCT
easat-5378	275	22	for	for	ADP
easat-5378	275	23	natural	natural	ADJ
easat-5378	275	24	numbers	number	NOUN
easat-5378	275	25	𝑛	𝑛	PROPN
easat-5378	275	26	,	,	PUNCT
easat-5378	275	27	𝑘	𝑘	PROPN
easat-5378	275	28	,	,	PUNCT
easat-5378	275	29	𝑗	𝑗	X
easat-5378	275	30	≥	≥	NOUN
easat-5378	275	31	1	1	NUM
easat-5378	275	32	.	.	PUNCT
easat-5378	276	1	proof	proof	NOUN
easat-5378	276	2	:	:	PUNCT
easat-5378	276	3	from	from	ADP
easat-5378	276	4	definition	definition	NOUN
easat-5378	276	5	3.10	3.10	NUM
easat-5378	276	6	,	,	PUNCT
easat-5378	276	7	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	276	8	)	)	PUNCT
easat-5378	276	9	=	=	SYM
easat-5378	276	10	(	(	PUNCT
easat-5378	276	11	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	276	12	+	+	CCONJ
easat-5378	276	13	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	276	14	)	)	PUNCT
easat-5378	276	15	.	.	PUNCT
easat-5378	277	1	we	we	PRON
easat-5378	277	2	need	need	VERB
easat-5378	277	3	to	to	PART
easat-5378	277	4	show	show	VERB
easat-5378	277	5	that	that	SCONJ
easat-5378	277	6	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	277	7	)	)	PUNCT
easat-5378	277	8	=	=	SYM
easat-5378	277	9	𝑆2𝑗(𝑘	𝑆2𝑗(𝑘	NOUN
easat-5378	277	10	)	)	PUNCT
easat-5378	277	11	=	=	NOUN
easat-5378	278	1	2𝑘	2𝑘	NUM
easat-5378	279	1	+	+	CCONJ
easat-5378	279	2	1	1	NUM
easat-5378	279	3	2𝑗	2𝑗	NUM
easat-5378	279	4	+	+	NOUN
easat-5378	279	5	1	1	NUM
easat-5378	279	6	[	[	X
easat-5378	279	7	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	279	8	+	+	NUM
easat-5378	279	9	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	279	10	it	it	PRON
easat-5378	279	11	is	be	AUX
easat-5378	279	12	similar	similar	ADJ
easat-5378	279	13	to	to	ADP
easat-5378	279	14	a	a	DET
easat-5378	279	15	previous	previous	ADJ
easat-5378	279	16	proof	proof	NOUN
easat-5378	279	17	[	[	X
easat-5378	279	18	20	20	NUM
easat-5378	279	19	]	]	PUNCT
easat-5378	279	20	that	that	DET
easat-5378	279	21	𝑆2𝑘(𝑛	𝑆2𝑘(𝑛	NOUN
easat-5378	279	22	)	)	PUNCT
easat-5378	279	23	=	=	PUNCT
easat-5378	279	24	2𝑛	2𝑛	NOUN
easat-5378	280	1	+	+	CCONJ
easat-5378	280	2	1	1	NUM
easat-5378	280	3	2𝑘	2𝑘	NUM
easat-5378	280	4	+	+	CCONJ
easat-5378	280	5	1	1	NUM
easat-5378	280	6	[	[	X
easat-5378	280	7	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	280	8	+	+	CCONJ
easat-5378	280	9	𝑃(𝑛2𝑘−3)]𝑇𝑛	𝑃(𝑛2𝑘−3)]𝑇𝑛	ADJ
easat-5378	280	10	we	we	PRON
easat-5378	280	11	simply	simply	ADV
easat-5378	280	12	have	have	VERB
easat-5378	280	13	𝑆2𝑗(𝑘	𝑆2𝑗(𝑘	NOUN
easat-5378	280	14	)	)	PUNCT
easat-5378	280	15	=	=	NOUN
easat-5378	281	1	2𝑘	2𝑘	NUM
easat-5378	282	1	+	+	CCONJ
easat-5378	282	2	1	1	NUM
easat-5378	282	3	2𝑗	2𝑗	NUM
easat-5378	282	4	+	+	NOUN
easat-5378	282	5	1	1	NUM
easat-5378	283	1	[	[	X
easat-5378	283	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	283	3	+	+	NUM
easat-5378	283	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	PROPN
easat-5378	283	5	and	and	CCONJ
easat-5378	283	6	then	then	ADV
easat-5378	283	7	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	283	8	)	)	PUNCT
easat-5378	283	9	=	=	SYM
easat-5378	283	10	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	PROPN
easat-5378	283	11	)	)	PUNCT
easat-5378	283	12	=	=	PUNCT
easat-5378	283	13	(	(	PUNCT
easat-5378	283	14	𝑛!)2𝑗	𝑛!)2𝑗	NUM
easat-5378	283	15	+	+	CCONJ
easat-5378	283	16	2𝑘	2𝑘	NUM
easat-5378	283	17	+	+	CCONJ
easat-5378	283	18	1	1	NUM
easat-5378	283	19	2𝑗	2𝑗	NUM
easat-5378	283	20	+	+	NOUN
easat-5378	283	21	1	1	NUM
easat-5378	284	1	[	[	X
easat-5378	284	2	𝑘2𝑗−2	𝑘2𝑗−2	INTJ
easat-5378	284	3	+	+	NUM
easat-5378	284	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	284	5	theorem	theorem	VERB
easat-5378	284	6	3.14	3.14	NUM
easat-5378	284	7	:	:	PUNCT
easat-5378	284	8	the	the	DET
easat-5378	284	9	(	(	PUNCT
easat-5378	284	10	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	284	11	)	)	PUNCT
easat-5378	284	12	,	,	PUNCT
easat-5378	284	13	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	284	14	number	number	NOUN
easat-5378	284	15	for	for	ADP
easat-5378	284	16	odd	odd	ADJ
easat-5378	284	17	𝑚	𝑚	NOUN
easat-5378	284	18	=	=	SYM
easat-5378	284	19	2𝑗	2𝑗	NOUN
easat-5378	284	20	+	+	CCONJ
easat-5378	284	21	1	1	NUM
easat-5378	284	22	is	be	AUX
easat-5378	284	23	given	give	VERB
easat-5378	284	24	by	by	ADP
easat-5378	284	25	the	the	DET
easat-5378	284	26	formula	formula	NOUN
easat-5378	284	27	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	284	28	)	)	PUNCT
easat-5378	284	29	=	=	SYM
easat-5378	284	30	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	NUM
easat-5378	284	31	)	)	PUNCT
easat-5378	284	32	=	=	NOUN
easat-5378	284	33	(	(	PUNCT
easat-5378	284	34	𝑛!)2𝑗+1	𝑛!)2𝑗+1	PROPN
easat-5378	284	35	+	+	CCONJ
easat-5378	284	36	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	284	37	+	+	CCONJ
easat-5378	284	38	1	1	X
easat-5378	284	39	)	)	PUNCT
easat-5378	284	40	𝑗	𝑗	NOUN
easat-5378	285	1	+	+	ADJ
easat-5378	285	2	1	1	NUM
easat-5378	286	1	[	[	X
easat-5378	286	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	286	3	+	+	NUM
easat-5378	286	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	286	5	where	where	SCONJ
easat-5378	286	6	𝑛	𝑛	X
easat-5378	286	7	!	!	PUNCT
easat-5378	286	8	=	=	SYM
easat-5378	287	1	1	1	NUM
easat-5378	287	2	⋅	⋅	NUM
easat-5378	287	3	2	2	NUM
easat-5378	287	4	⋅	⋅	X
easat-5378	287	5	3	3	NUM
easat-5378	287	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	287	7	𝑛	𝑛	PROPN
easat-5378	287	8	,	,	PUNCT
easat-5378	287	9	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	287	10	=	=	SYM
easat-5378	287	11	1	1	NUM
easat-5378	287	12	+	+	NUM
easat-5378	287	13	2	2	NUM
easat-5378	287	14	+	+	SYM
easat-5378	287	15	3	3	NUM
easat-5378	287	16	+	+	NOUN
easat-5378	287	17	.	.	PUNCT
easat-5378	287	18	.	.	PUNCT
easat-5378	287	19	.	.	PUNCT
easat-5378	288	1	+	+	NOUN
easat-5378	288	2	𝑘	𝑘	X
easat-5378	288	3	=	=	PUNCT
easat-5378	288	4	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	288	5	+	+	CCONJ
easat-5378	288	6	1)/2	1)/2	NUM
easat-5378	288	7	,	,	PUNCT
easat-5378	288	8	and	and	CCONJ
easat-5378	288	9	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	288	10	)	)	PUNCT
easat-5378	288	11	is	be	AUX
easat-5378	288	12	a	a	DET
easat-5378	288	13	polynomial	polynomial	NOUN
easat-5378	288	14	in	in	ADP
easat-5378	288	15	𝑘	𝑘	PRON
easat-5378	288	16	of	of	ADP
easat-5378	288	17	degree	degree	NOUN
easat-5378	288	18	2𝑗	2𝑗	NOUN
easat-5378	288	19	−	−	PROPN
easat-5378	288	20	3	3	NUM
easat-5378	288	21	,	,	PUNCT
easat-5378	288	22	for	for	ADP
easat-5378	288	23	natural	natural	ADJ
easat-5378	288	24	numbers	number	NOUN
easat-5378	288	25	𝑛	𝑛	PROPN
easat-5378	288	26	,	,	PUNCT
easat-5378	288	27	𝑘	𝑘	PROPN
easat-5378	288	28	,	,	PUNCT
easat-5378	288	29	𝑗	𝑗	X
easat-5378	288	30	≥	≥	NOUN
easat-5378	288	31	1	1	NUM
easat-5378	288	32	.	.	PUNCT
easat-5378	289	1	proof	proof	NOUN
easat-5378	289	2	:	:	PUNCT
easat-5378	289	3	from	from	ADP
easat-5378	289	4	definition	definition	NOUN
easat-5378	289	5	3.10	3.10	NUM
easat-5378	289	6	,	,	PUNCT
easat-5378	289	7	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	289	8	)	)	PUNCT
easat-5378	289	9	=	=	SYM
easat-5378	289	10	(	(	PUNCT
easat-5378	289	11	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	289	12	+	+	CCONJ
easat-5378	289	13	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	289	14	)	)	PUNCT
easat-5378	289	15	.	.	PUNCT
easat-5378	290	1	we	we	PRON
easat-5378	290	2	need	need	VERB
easat-5378	290	3	to	to	PART
easat-5378	290	4	show	show	VERB
easat-5378	290	5	that	that	SCONJ
easat-5378	290	6	899	899	NUM
easat-5378	290	7	edelweiss	edelweiss	PROPN
easat-5378	290	8	applied	apply	VERB
easat-5378	290	9	science	science	NOUN
easat-5378	290	10	and	and	CCONJ
easat-5378	290	11	technology	technology	NOUN
easat-5378	290	12	issn	issn	PROPN
easat-5378	290	13	:	:	PUNCT
easat-5378	290	14	2576	2576	NUM
easat-5378	290	15	-	-	SYM
easat-5378	290	16	8484	8484	NUM
easat-5378	290	17	vol	vol	NOUN
easat-5378	290	18	.	.	PROPN
easat-5378	291	1	9	9	NUM
easat-5378	291	2	,	,	PUNCT
easat-5378	291	3	no	no	INTJ
easat-5378	291	4	.	.	NOUN
easat-5378	292	1	3	3	NUM
easat-5378	292	2	:	:	PUNCT
easat-5378	292	3	891	891	NUM
easat-5378	292	4	-	-	SYM
easat-5378	292	5	904	904	NUM
easat-5378	292	6	,	,	PUNCT
easat-5378	292	7	2025	2025	NUM
easat-5378	292	8	doi	doi	NOUN
easat-5378	292	9	:	:	PUNCT
easat-5378	292	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	292	11	©	©	PROPN
easat-5378	292	12	2025	2025	NUM
easat-5378	292	13	by	by	ADP
easat-5378	292	14	the	the	DET
easat-5378	292	15	author	author	NOUN
easat-5378	292	16	;	;	PUNCT
easat-5378	292	17	licensee	licensee	PROPN
easat-5378	292	18	learning	learn	VERB
easat-5378	292	19	gate	gate	NOUN
easat-5378	292	20	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	292	21	)	)	PUNCT
easat-5378	292	22	=	=	SYM
easat-5378	292	23	𝑆2𝑗+1(𝑘	𝑆2𝑗+1(𝑘	PROPN
easat-5378	292	24	)	)	PUNCT
easat-5378	292	25	=	=	PUNCT
easat-5378	293	1	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	294	1	+	+	CCONJ
easat-5378	294	2	1	1	X
easat-5378	294	3	)	)	PUNCT
easat-5378	294	4	𝑗	𝑗	NOUN
easat-5378	295	1	+	+	ADJ
easat-5378	295	2	1	1	NUM
easat-5378	295	3	[	[	X
easat-5378	295	4	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	295	5	+	+	NUM
easat-5378	295	6	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	295	7	it	it	PRON
easat-5378	295	8	is	be	AUX
easat-5378	295	9	similar	similar	ADJ
easat-5378	295	10	to	to	ADP
easat-5378	295	11	a	a	DET
easat-5378	295	12	previous	previous	ADJ
easat-5378	295	13	proof	proof	NOUN
easat-5378	295	14	[	[	X
easat-5378	295	15	20	20	NUM
easat-5378	295	16	]	]	PUNCT
easat-5378	295	17	that	that	SCONJ
easat-5378	295	18	𝑆2𝑘+1(𝑛	𝑆2𝑘+1(𝑛	ADV
easat-5378	295	19	)	)	PUNCT
easat-5378	295	20	=	=	PUNCT
easat-5378	296	1	𝑛(𝑛	𝑛(𝑛	PROPN
easat-5378	296	2	+	+	CCONJ
easat-5378	296	3	1	1	X
easat-5378	296	4	)	)	PUNCT
easat-5378	296	5	𝑘	𝑘	NOUN
easat-5378	297	1	+	+	ADJ
easat-5378	297	2	1	1	NUM
easat-5378	297	3	[	[	X
easat-5378	297	4	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	297	5	+	+	CCONJ
easat-5378	297	6	𝑃(𝑛2𝑘−3)]𝑇𝑛	𝑃(𝑛2𝑘−3)]𝑇𝑛	NOUN
easat-5378	297	7	we	we	PRON
easat-5378	297	8	also	also	ADV
easat-5378	297	9	have	have	VERB
easat-5378	297	10	𝑆2𝑗+1(𝑘	𝑆2𝑗+1(𝑘	NUM
easat-5378	297	11	)	)	PUNCT
easat-5378	297	12	=	=	PUNCT
easat-5378	298	1	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	299	1	+	+	CCONJ
easat-5378	299	2	1	1	X
easat-5378	299	3	)	)	PUNCT
easat-5378	299	4	𝑗	𝑗	NOUN
easat-5378	300	1	+	+	ADJ
easat-5378	300	2	1	1	NUM
easat-5378	300	3	[	[	X
easat-5378	300	4	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	300	5	+	+	NUM
easat-5378	300	6	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	300	7	hence	hence	ADV
easat-5378	300	8	,	,	PUNCT
easat-5378	300	9	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	300	10	)	)	PUNCT
easat-5378	300	11	=	=	SYM
easat-5378	300	12	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	NUM
easat-5378	300	13	)	)	PUNCT
easat-5378	300	14	=	=	NOUN
easat-5378	300	15	(	(	PUNCT
easat-5378	301	1	𝑛!)2𝑗+1	𝑛!)2𝑗+1	PROPN
easat-5378	301	2	+	+	CCONJ
easat-5378	301	3	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	301	4	+	+	CCONJ
easat-5378	301	5	1	1	X
easat-5378	301	6	)	)	PUNCT
easat-5378	301	7	𝑗	𝑗	NOUN
easat-5378	302	1	+	+	ADJ
easat-5378	302	2	1	1	NUM
easat-5378	302	3	[	[	X
easat-5378	302	4	𝑘2𝑗−2	𝑘2𝑗−2	INTJ
easat-5378	302	5	+	+	NUM
easat-5378	302	6	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	302	7	theorem	theorem	VERB
easat-5378	302	8	3.15	3.15	NUM
easat-5378	302	9	:	:	PUNCT
easat-5378	302	10	the	the	DET
easat-5378	302	11	(	(	PUNCT
easat-5378	302	12	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	302	13	)	)	PUNCT
easat-5378	302	14	,	,	PUNCT
easat-5378	302	15	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	302	16	number	number	NOUN
easat-5378	302	17	for	for	ADP
easat-5378	302	18	even	even	ADV
easat-5378	302	19	𝑚	𝑚	NOUN
easat-5378	302	20	=	=	NOUN
easat-5378	302	21	2𝑗	2𝑗	NOUN
easat-5378	302	22	is	be	AUX
easat-5378	302	23	given	give	VERB
easat-5378	302	24	by	by	ADP
easat-5378	302	25	the	the	DET
easat-5378	302	26	formula	formula	NOUN
easat-5378	302	27	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	302	28	)	)	PUNCT
easat-5378	302	29	=	=	SYM
easat-5378	302	30	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	PROPN
easat-5378	302	31	)	)	PUNCT
easat-5378	302	32	=	=	PUNCT
easat-5378	303	1	(	(	PUNCT
easat-5378	303	2	𝑛!)2𝑗	𝑛!)2𝑗	NUM
easat-5378	303	3	+	+	CCONJ
easat-5378	303	4	3	3	NUM
easat-5378	303	5	2𝑗	2𝑗	NOUN
easat-5378	303	6	+	+	NOUN
easat-5378	303	7	1	1	NUM
easat-5378	304	1	[	[	X
easat-5378	304	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	304	3	+	+	NUM
easat-5378	304	4	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	NOUN
easat-5378	304	5	)	)	PUNCT
easat-5378	304	6	where	where	SCONJ
easat-5378	304	7	𝑛	𝑛	X
easat-5378	304	8	!	!	PUNCT
easat-5378	304	9	=	=	SYM
easat-5378	305	1	1	1	NUM
easat-5378	305	2	⋅	⋅	NUM
easat-5378	305	3	2	2	NUM
easat-5378	305	4	⋅	⋅	X
easat-5378	305	5	3	3	NUM
easat-5378	305	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	305	7	𝑛	𝑛	PROPN
easat-5378	305	8	,	,	PUNCT
easat-5378	305	9	𝑆2(𝑘	𝑆2(𝑘	PROPN
easat-5378	305	10	)	)	PUNCT
easat-5378	305	11	=	=	PUNCT
easat-5378	306	1	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	307	1	+	+	NUM
easat-5378	307	2	1)(2𝑘	1)(2𝑘	NUM
easat-5378	307	3	+	+	CCONJ
easat-5378	307	4	1)/6	1)/6	NUM
easat-5378	307	5	,	,	PUNCT
easat-5378	307	6	and	and	CCONJ
easat-5378	307	7	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	307	8	)	)	PUNCT
easat-5378	307	9	is	be	AUX
easat-5378	307	10	a	a	DET
easat-5378	307	11	polynomial	polynomial	NOUN
easat-5378	307	12	in	in	ADP
easat-5378	307	13	𝑘	𝑘	PRON
easat-5378	307	14	of	of	ADP
easat-5378	307	15	degree	degree	NOUN
easat-5378	307	16	2𝑗	2𝑗	NOUN
easat-5378	307	17	−	−	PROPN
easat-5378	307	18	3	3	NUM
easat-5378	307	19	,	,	PUNCT
easat-5378	307	20	for	for	ADP
easat-5378	307	21	natural	natural	ADJ
easat-5378	307	22	numbers	number	NOUN
easat-5378	307	23	𝑛	𝑛	PROPN
easat-5378	307	24	,	,	PUNCT
easat-5378	307	25	𝑘	𝑘	DET
easat-5378	307	26	≥	≥	NOUN
easat-5378	307	27	1and	1and	NUM
easat-5378	307	28	𝑗	𝑗	NOUN
easat-5378	307	29	>	>	X
easat-5378	307	30	1	1	NUM
easat-5378	307	31	.	.	PUNCT
easat-5378	308	1	proof	proof	NOUN
easat-5378	308	2	:	:	PUNCT
easat-5378	308	3	from	from	ADP
easat-5378	308	4	definition	definition	NOUN
easat-5378	308	5	3.10	3.10	NUM
easat-5378	308	6	,	,	PUNCT
easat-5378	308	7	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	308	8	)	)	PUNCT
easat-5378	308	9	=	=	SYM
easat-5378	308	10	(	(	PUNCT
easat-5378	308	11	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	308	12	+	+	CCONJ
easat-5378	308	13	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	308	14	)	)	PUNCT
easat-5378	308	15	.	.	PUNCT
easat-5378	309	1	we	we	PRON
easat-5378	309	2	need	need	VERB
easat-5378	309	3	to	to	PART
easat-5378	309	4	show	show	VERB
easat-5378	309	5	that	that	SCONJ
easat-5378	309	6	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	309	7	)	)	PUNCT
easat-5378	309	8	=	=	SYM
easat-5378	309	9	𝑆2𝑗(𝑘	𝑆2𝑗(𝑘	NOUN
easat-5378	309	10	)	)	PUNCT
easat-5378	309	11	=	=	SYM
easat-5378	309	12	3	3	NUM
easat-5378	309	13	2𝑗	2𝑗	NOUN
easat-5378	309	14	+	+	NOUN
easat-5378	309	15	1	1	NUM
easat-5378	310	1	[	[	X
easat-5378	310	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	310	3	+	+	NUM
easat-5378	310	4	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	NOUN
easat-5378	310	5	)	)	PUNCT
easat-5378	310	6	it	it	PRON
easat-5378	310	7	is	be	AUX
easat-5378	310	8	similar	similar	ADJ
easat-5378	310	9	to	to	ADP
easat-5378	310	10	a	a	DET
easat-5378	310	11	previous	previous	ADJ
easat-5378	310	12	proof	proof	NOUN
easat-5378	310	13	[	[	X
easat-5378	310	14	20	20	NUM
easat-5378	310	15	]	]	PUNCT
easat-5378	310	16	that	that	DET
easat-5378	310	17	𝑆2𝑘(𝑛	𝑆2𝑘(𝑛	NOUN
easat-5378	310	18	)	)	PUNCT
easat-5378	310	19	=	=	PUNCT
easat-5378	310	20	3	3	NUM
easat-5378	310	21	2𝑘	2𝑘	NUM
easat-5378	311	1	+	+	CCONJ
easat-5378	311	2	1	1	NUM
easat-5378	311	3	[	[	X
easat-5378	311	4	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	311	5	+	+	NUM
easat-5378	311	6	𝑃(𝑛2𝑘−3)]𝑆2(𝑛	𝑃(𝑛2𝑘−3)]𝑆2(𝑛	NOUN
easat-5378	311	7	)	)	PUNCT
easat-5378	311	8	we	we	PRON
easat-5378	311	9	have	have	VERB
easat-5378	311	10	𝑆2𝑗(𝑘	𝑆2𝑗(𝑘	NOUN
easat-5378	311	11	)	)	PUNCT
easat-5378	311	12	=	=	SYM
easat-5378	311	13	3	3	NUM
easat-5378	311	14	2𝑗	2𝑗	NOUN
easat-5378	311	15	+	+	NOUN
easat-5378	311	16	1	1	NUM
easat-5378	312	1	[	[	X
easat-5378	312	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	312	3	+	+	NUM
easat-5378	312	4	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	NOUN
easat-5378	312	5	)	)	PUNCT
easat-5378	312	6	and	and	CCONJ
easat-5378	312	7	then	then	ADV
easat-5378	312	8	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	312	9	)	)	PUNCT
easat-5378	312	10	=	=	SYM
easat-5378	312	11	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	PROPN
easat-5378	312	12	)	)	PUNCT
easat-5378	312	13	=	=	PUNCT
easat-5378	313	1	(	(	PUNCT
easat-5378	313	2	𝑛!)2𝑗	𝑛!)2𝑗	NUM
easat-5378	313	3	+	+	CCONJ
easat-5378	313	4	3	3	NUM
easat-5378	313	5	2𝑗	2𝑗	NOUN
easat-5378	313	6	+	+	NOUN
easat-5378	313	7	1	1	NUM
easat-5378	314	1	[	[	X
easat-5378	314	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	314	3	+	+	CCONJ
easat-5378	314	4	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	NOUN
easat-5378	314	5	)	)	PUNCT
easat-5378	314	6	theorem	theorem	VERB
easat-5378	314	7	3.16	3.16	NUM
easat-5378	314	8	:	:	PUNCT
easat-5378	314	9	the	the	DET
easat-5378	314	10	(	(	PUNCT
easat-5378	314	11	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	314	12	)	)	PUNCT
easat-5378	314	13	,	,	PUNCT
easat-5378	314	14	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	314	15	number	number	NOUN
easat-5378	314	16	for	for	ADP
easat-5378	314	17	odd	odd	ADJ
easat-5378	314	18	𝑚	𝑚	NOUN
easat-5378	314	19	=	=	SYM
easat-5378	314	20	2𝑗	2𝑗	NOUN
easat-5378	314	21	+	+	CCONJ
easat-5378	314	22	1	1	NUM
easat-5378	314	23	is	be	AUX
easat-5378	314	24	given	give	VERB
easat-5378	314	25	by	by	ADP
easat-5378	314	26	the	the	DET
easat-5378	314	27	formula	formula	NOUN
easat-5378	314	28	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	314	29	)	)	PUNCT
easat-5378	314	30	=	=	SYM
easat-5378	314	31	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	NUM
easat-5378	314	32	)	)	PUNCT
easat-5378	314	33	=	=	NOUN
easat-5378	315	1	(	(	PUNCT
easat-5378	315	2	𝑛!)2𝑗+1	𝑛!)2𝑗+1	NUM
easat-5378	315	3	+	+	CCONJ
easat-5378	315	4	2	2	NUM
easat-5378	315	5	𝑗	𝑗	NOUN
easat-5378	315	6	+	+	NOUN
easat-5378	315	7	1	1	NUM
easat-5378	316	1	[	[	X
easat-5378	316	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	316	3	+	+	CCONJ
easat-5378	316	4	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	NOUN
easat-5378	316	5	)	)	PUNCT
easat-5378	316	6	where	where	SCONJ
easat-5378	316	7	𝑛	𝑛	X
easat-5378	316	8	!	!	PUNCT
easat-5378	316	9	=	=	SYM
easat-5378	317	1	1	1	NUM
easat-5378	317	2	⋅	⋅	NUM
easat-5378	317	3	2	2	NUM
easat-5378	317	4	⋅	⋅	X
easat-5378	317	5	3	3	NUM
easat-5378	317	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	317	7	𝑛	𝑛	PROPN
easat-5378	317	8	,	,	PUNCT
easat-5378	317	9	𝑆3(𝑘	𝑆3(𝑘	PROPN
easat-5378	317	10	)	)	PUNCT
easat-5378	317	11	=	=	X
easat-5378	318	1	𝑘2(𝑘	𝑘2(𝑘	PROPN
easat-5378	319	1	+	+	NOUN
easat-5378	319	2	1)2/4	1)2/4	NUM
easat-5378	319	3	,	,	PUNCT
easat-5378	319	4	and	and	CCONJ
easat-5378	319	5	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	319	6	)	)	PUNCT
easat-5378	319	7	is	be	AUX
easat-5378	319	8	a	a	DET
easat-5378	319	9	polynomial	polynomial	NOUN
easat-5378	319	10	in	in	ADP
easat-5378	319	11	𝑘	𝑘	PRON
easat-5378	319	12	of	of	ADP
easat-5378	319	13	degree	degree	NOUN
easat-5378	319	14	2𝑗	2𝑗	NOUN
easat-5378	319	15	−	−	PROPN
easat-5378	319	16	3	3	NUM
easat-5378	319	17	,	,	PUNCT
easat-5378	319	18	for	for	ADP
easat-5378	319	19	natural	natural	ADJ
easat-5378	319	20	numbers	number	NOUN
easat-5378	319	21	𝑛	𝑛	PROPN
easat-5378	319	22	,	,	PUNCT
easat-5378	319	23	𝑘	𝑘	DET
easat-5378	319	24	≥	≥	NOUN
easat-5378	319	25	1and	1and	NUM
easat-5378	319	26	𝑗	𝑗	NOUN
easat-5378	319	27	>	>	X
easat-5378	319	28	1	1	NUM
easat-5378	319	29	.	.	PUNCT
easat-5378	320	1	proof	proof	NOUN
easat-5378	320	2	:	:	PUNCT
easat-5378	320	3	from	from	ADP
easat-5378	320	4	definition	definition	NOUN
easat-5378	320	5	3.10	3.10	NUM
easat-5378	320	6	,	,	PUNCT
easat-5378	320	7	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	320	8	)	)	PUNCT
easat-5378	320	9	=	=	SYM
easat-5378	320	10	(	(	PUNCT
easat-5378	320	11	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	320	12	+	+	CCONJ
easat-5378	320	13	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	320	14	)	)	PUNCT
easat-5378	320	15	.	.	PUNCT
easat-5378	321	1	we	we	PRON
easat-5378	321	2	need	need	VERB
easat-5378	321	3	to	to	PART
easat-5378	321	4	show	show	VERB
easat-5378	321	5	that	that	SCONJ
easat-5378	321	6	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	321	7	)	)	PUNCT
easat-5378	321	8	=	=	SYM
easat-5378	321	9	𝑆2𝑗+1(𝑘	𝑆2𝑗+1(𝑘	PROPN
easat-5378	321	10	)	)	PUNCT
easat-5378	321	11	=	=	SYM
easat-5378	322	1	2	2	NUM
easat-5378	322	2	𝑗	𝑗	NOUN
easat-5378	322	3	+	+	NUM
easat-5378	322	4	1	1	NUM
easat-5378	323	1	[	[	X
easat-5378	323	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	323	3	+	+	CCONJ
easat-5378	323	4	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	NOUN
easat-5378	323	5	)	)	PUNCT
easat-5378	323	6	it	it	PRON
easat-5378	323	7	is	be	AUX
easat-5378	323	8	similar	similar	ADJ
easat-5378	323	9	to	to	ADP
easat-5378	323	10	a	a	DET
easat-5378	323	11	previous	previous	ADJ
easat-5378	323	12	proof	proof	NOUN
easat-5378	323	13	[	[	X
easat-5378	323	14	20	20	NUM
easat-5378	323	15	]	]	PUNCT
easat-5378	323	16	that	that	SCONJ
easat-5378	323	17	𝑆2𝑘+1(𝑚	𝑆2𝑘+1(𝑚	NOUN
easat-5378	323	18	)	)	PUNCT
easat-5378	323	19	=	=	SYM
easat-5378	324	1	2	2	NUM
easat-5378	324	2	𝑘	𝑘	PRON
easat-5378	324	3	+	+	NOUN
easat-5378	324	4	1	1	NUM
easat-5378	324	5	[	[	X
easat-5378	324	6	𝑛2𝑘−2	𝑛2𝑘−2	X
easat-5378	324	7	+	+	X
easat-5378	324	8	𝑃(𝑛2𝑘−3)]𝑆3(𝑚	𝑃(𝑛2𝑘−3)]𝑆3(𝑚	X
easat-5378	324	9	)	)	PUNCT
easat-5378	324	10	we	we	PRON
easat-5378	324	11	have	have	VERB
easat-5378	324	12	𝑆2𝑗+1(𝑘	𝑆2𝑗+1(𝑘	NUM
easat-5378	324	13	)	)	PUNCT
easat-5378	324	14	=	=	SYM
easat-5378	325	1	2	2	NUM
easat-5378	325	2	𝑗	𝑗	NOUN
easat-5378	325	3	+	+	NUM
easat-5378	325	4	1	1	NUM
easat-5378	325	5	[	[	X
easat-5378	325	6	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	325	7	+	+	CCONJ
easat-5378	325	8	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	NOUN
easat-5378	325	9	)	)	PUNCT
easat-5378	325	10	thus	thus	ADV
easat-5378	325	11	,	,	PUNCT
easat-5378	325	12	900	900	NUM
easat-5378	325	13	edelweiss	edelweiss	PROPN
easat-5378	325	14	applied	apply	VERB
easat-5378	325	15	science	science	NOUN
easat-5378	325	16	and	and	CCONJ
easat-5378	325	17	technology	technology	NOUN
easat-5378	325	18	issn	issn	PROPN
easat-5378	325	19	:	:	PUNCT
easat-5378	325	20	2576	2576	NUM
easat-5378	325	21	-	-	SYM
easat-5378	325	22	8484	8484	NUM
easat-5378	325	23	vol	vol	NOUN
easat-5378	325	24	.	.	PROPN
easat-5378	326	1	9	9	NUM
easat-5378	326	2	,	,	PUNCT
easat-5378	326	3	no	no	INTJ
easat-5378	326	4	.	.	NOUN
easat-5378	327	1	3	3	NUM
easat-5378	327	2	:	:	PUNCT
easat-5378	327	3	891	891	NUM
easat-5378	327	4	-	-	SYM
easat-5378	327	5	904	904	NUM
easat-5378	327	6	,	,	PUNCT
easat-5378	327	7	2025	2025	NUM
easat-5378	327	8	doi	doi	NOUN
easat-5378	327	9	:	:	PUNCT
easat-5378	327	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	327	11	©	©	PROPN
easat-5378	327	12	2025	2025	NUM
easat-5378	327	13	by	by	ADP
easat-5378	327	14	the	the	DET
easat-5378	327	15	author	author	NOUN
easat-5378	327	16	;	;	PUNCT
easat-5378	327	17	licensee	licensee	NOUN
easat-5378	327	18	learning	learn	VERB
easat-5378	327	19	gate	gate	NOUN
easat-5378	327	20	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	327	21	)	)	PUNCT
easat-5378	327	22	=	=	SYM
easat-5378	327	23	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	NUM
easat-5378	327	24	)	)	PUNCT
easat-5378	327	25	=	=	NOUN
easat-5378	327	26	(	(	PUNCT
easat-5378	327	27	𝑛!)2𝑗+1	𝑛!)2𝑗+1	NUM
easat-5378	327	28	+	+	CCONJ
easat-5378	327	29	2	2	NUM
easat-5378	327	30	𝑗	𝑗	NOUN
easat-5378	327	31	+	+	NOUN
easat-5378	327	32	1	1	NUM
easat-5378	328	1	[	[	X
easat-5378	328	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	328	3	+	+	CCONJ
easat-5378	328	4	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	NOUN
easat-5378	328	5	)	)	PUNCT
easat-5378	328	6	two	two	NUM
easat-5378	328	7	examples	example	NOUN
easat-5378	328	8	of	of	ADP
easat-5378	328	9	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	328	10	)	)	PUNCT
easat-5378	328	11	,	,	PUNCT
easat-5378	328	12	with	with	ADP
easat-5378	328	13	𝑚	𝑚	PROPN
easat-5378	328	14	>	>	SYM
easat-5378	328	15	1	1	NUM
easat-5378	328	16	,	,	PUNCT
easat-5378	328	17	are	be	AUX
easat-5378	328	18	presented	present	VERB
easat-5378	328	19	here	here	ADV
easat-5378	328	20	.	.	PUNCT
easat-5378	329	1	the	the	DET
easat-5378	329	2	first	first	ADJ
easat-5378	329	3	example	example	NOUN
easat-5378	329	4	is	be	AUX
easat-5378	329	5	when	when	SCONJ
easat-5378	329	6	𝑚	𝑚	PROPN
easat-5378	329	7	=	=	SYM
easat-5378	329	8	2	2	NUM
easat-5378	329	9	and	and	CCONJ
easat-5378	329	10	the	the	DET
easat-5378	329	11	(	(	PUNCT
easat-5378	329	12	𝑛(2	𝑛(2	NOUN
easat-5378	329	13	)	)	PUNCT
easat-5378	329	14	,	,	PUNCT
easat-5378	329	15	𝑘(2))-factoriangular	𝑘(2))-factoriangular	ADJ
easat-5378	329	16	numbers	number	NOUN
easat-5378	329	17	are	be	AUX
easat-5378	329	18	given	give	VERB
easat-5378	329	19	in	in	ADP
easat-5378	329	20	table	table	NOUN
easat-5378	329	21	2	2	NUM
easat-5378	329	22	.	.	PUNCT
easat-5378	329	23	table	table	NOUN
easat-5378	329	24	2	2	NUM
easat-5378	329	25	.	.	PUNCT
easat-5378	329	26	table	table	NOUN
easat-5378	329	27	of	of	ADP
easat-5378	329	28	(	(	PUNCT
easat-5378	329	29	n(2	n(2	PROPN
easat-5378	329	30	)	)	PUNCT
easat-5378	329	31	,	,	PUNCT
easat-5378	329	32	k(2))-factoriangular	k(2))-factoriangular	ADJ
easat-5378	329	33	numbers	number	NOUN
easat-5378	329	34	.	.	PUNCT
easat-5378	330	1	𝒏	𝒏	PROPN
easat-5378	330	2	\	\	PROPN
easat-5378	330	3	𝒌	𝒌	DET
easat-5378	330	4	1	1	NUM
easat-5378	330	5	2	2	NUM
easat-5378	330	6	3	3	NUM
easat-5378	330	7	4	4	NUM
easat-5378	330	8	5	5	NUM
easat-5378	330	9	𝒌	𝒌	SYM
easat-5378	330	10	1	1	NUM
easat-5378	330	11	2	2	NUM
easat-5378	330	12	6	6	NUM
easat-5378	330	13	15	15	NUM
easat-5378	330	14	31	31	NUM
easat-5378	330	15	56	56	NUM
easat-5378	330	16	1	1	NUM
easat-5378	330	17	+	+	CCONJ
easat-5378	330	18	𝑆2(𝑘	𝑆2(𝑘	NOUN
easat-5378	330	19	)	)	PUNCT
easat-5378	330	20	2	2	NUM
easat-5378	330	21	5	5	NUM
easat-5378	330	22	9	9	NUM
easat-5378	330	23	18	18	NUM
easat-5378	330	24	34	34	NUM
easat-5378	330	25	59	59	NUM
easat-5378	330	26	4	4	NUM
easat-5378	330	27	+	+	CCONJ
easat-5378	330	28	𝑆2(𝑘	𝑆2(𝑘	PRON
easat-5378	330	29	)	)	PUNCT
easat-5378	330	30	3	3	NUM
easat-5378	330	31	37	37	NUM
easat-5378	330	32	41	41	NUM
easat-5378	330	33	50	50	NUM
easat-5378	330	34	66	66	NUM
easat-5378	330	35	91	91	NUM
easat-5378	330	36	36	36	NUM
easat-5378	330	37	+	+	CCONJ
easat-5378	330	38	𝑆2(𝑘	𝑆2(𝑘	PRON
easat-5378	330	39	)	)	PUNCT
easat-5378	330	40	4	4	NUM
easat-5378	330	41	577	577	NUM
easat-5378	330	42	581	581	NUM
easat-5378	330	43	590	590	NUM
easat-5378	330	44	606	606	NUM
easat-5378	330	45	631	631	NUM
easat-5378	330	46	576	576	NUM
easat-5378	330	47	+	+	CCONJ
easat-5378	330	48	𝑆2(𝑘	𝑆2(𝑘	PRON
easat-5378	330	49	)	)	PUNCT
easat-5378	330	50	5	5	NUM
easat-5378	330	51	14401	14401	NUM
easat-5378	330	52	14405	14405	NUM
easat-5378	330	53	14414	14414	NUM
easat-5378	330	54	14430	14430	NUM
easat-5378	330	55	14455	14455	NUM
easat-5378	330	56	14400	14400	NUM
easat-5378	330	57	+	+	CCONJ
easat-5378	330	58	𝑆2(𝑘	𝑆2(𝑘	PROPN
easat-5378	330	59	)	)	PUNCT
easat-5378	331	1	𝑛	𝑛	PROPN
easat-5378	331	2	(	(	PUNCT
easat-5378	331	3	𝑛!)2	𝑛!)2	PROPN
easat-5378	331	4	+	+	NUM
easat-5378	331	5	1	1	NUM
easat-5378	331	6	(	(	PUNCT
easat-5378	331	7	𝑛!)2	𝑛!)2	PROPN
easat-5378	331	8	+	+	NUM
easat-5378	331	9	5	5	NUM
easat-5378	331	10	(	(	PUNCT
easat-5378	331	11	𝑛!)2	𝑛!)2	PROPN
easat-5378	331	12	+	+	NUM
easat-5378	331	13	14	14	NUM
easat-5378	331	14	(	(	PUNCT
easat-5378	331	15	𝑛!)2	𝑛!)2	PROPN
easat-5378	331	16	+	+	NUM
easat-5378	331	17	30	30	NUM
easat-5378	331	18	(	(	PUNCT
easat-5378	331	19	𝑛!)2	𝑛!)2	PROPN
easat-5378	331	20	+	+	NUM
easat-5378	331	21	55	55	NUM
easat-5378	331	22	(	(	PUNCT
easat-5378	331	23	𝑛!)2	𝑛!)2	PROPN
easat-5378	331	24	+	+	NUM
easat-5378	331	25	𝑆2(𝑘	𝑆2(𝑘	PROPN
easat-5378	331	26	)	)	PUNCT
easat-5378	331	27	then	then	ADV
easat-5378	331	28	,	,	PUNCT
easat-5378	331	29	the	the	DET
easat-5378	331	30	sequence	sequence	NOUN
easat-5378	331	31	of	of	ADP
easat-5378	331	32	(	(	PUNCT
easat-5378	331	33	𝑛(2	𝑛(2	NOUN
easat-5378	331	34	)	)	PUNCT
easat-5378	331	35	,	,	PUNCT
easat-5378	331	36	𝑘(2))-factoriangular	𝑘(2))-factoriangular	ADJ
easat-5378	331	37	numbers	number	NOUN
easat-5378	331	38	is	be	AUX
easat-5378	331	39	given	give	VERB
easat-5378	331	40	by	by	ADP
easat-5378	331	41	{	{	PUNCT
easat-5378	331	42	𝐹𝑡𝑛(2),𝑘(2	𝐹𝑡𝑛(2),𝑘(2	ADV
easat-5378	331	43	)	)	PUNCT
easat-5378	331	44	}	}	PUNCT
easat-5378	331	45	=	=	SYM
easat-5378	331	46	{	{	PUNCT
easat-5378	331	47	2	2	NUM
easat-5378	331	48	,	,	PUNCT
easat-5378	331	49	5	5	NUM
easat-5378	331	50	,	,	PUNCT
easat-5378	331	51	6	6	NUM
easat-5378	331	52	,	,	PUNCT
easat-5378	331	53	9	9	NUM
easat-5378	331	54	,	,	PUNCT
easat-5378	331	55	37	37	NUM
easat-5378	331	56	,	,	PUNCT
easat-5378	331	57	15	15	NUM
easat-5378	331	58	,	,	PUNCT
easat-5378	331	59	41	41	NUM
easat-5378	331	60	,	,	PUNCT
easat-5378	331	61	18	18	NUM
easat-5378	331	62	,	,	PUNCT
easat-5378	331	63	50	50	NUM
easat-5378	331	64	,	,	PUNCT
easat-5378	331	65	.	.	PUNCT
easat-5378	331	66	.	.	PUNCT
easat-5378	331	67	.	.	PUNCT
easat-5378	331	68	}	}	PUNCT
easat-5378	332	1	for	for	ADP
easat-5378	332	2	(	(	PUNCT
easat-5378	332	3	n	n	CCONJ
easat-5378	332	4	,	,	PUNCT
easat-5378	332	5	k	k	NOUN
easat-5378	332	6	)	)	PUNCT
easat-5378	332	7	=	=	SYM
easat-5378	332	8	(	(	PUNCT
easat-5378	332	9	1,1	1,1	NUM
easat-5378	332	10	)	)	PUNCT
easat-5378	332	11	,	,	PUNCT
easat-5378	332	12	(	(	PUNCT
easat-5378	332	13	2,1	2,1	NUM
easat-5378	332	14	)	)	PUNCT
easat-5378	332	15	,	,	PUNCT
easat-5378	332	16	(	(	PUNCT
easat-5378	332	17	1,2	1,2	NUM
easat-5378	332	18	)	)	PUNCT
easat-5378	332	19	,	,	PUNCT
easat-5378	332	20	(	(	PUNCT
easat-5378	332	21	2,2	2,2	NUM
easat-5378	332	22	)	)	PUNCT
easat-5378	332	23	,	,	PUNCT
easat-5378	332	24	(	(	PUNCT
easat-5378	332	25	3,1	3,1	NUM
easat-5378	332	26	)	)	PUNCT
easat-5378	332	27	,	,	PUNCT
easat-5378	332	28	(	(	PUNCT
easat-5378	332	29	1,3	1,3	NUM
easat-5378	332	30	)	)	PUNCT
easat-5378	332	31	,	,	PUNCT
easat-5378	332	32	(	(	PUNCT
easat-5378	332	33	3,2	3,2	NUM
easat-5378	332	34	)	)	PUNCT
easat-5378	332	35	,	,	PUNCT
easat-5378	332	36	(	(	PUNCT
easat-5378	332	37	2,3	2,3	NUM
easat-5378	332	38	)	)	PUNCT
easat-5378	332	39	,	,	PUNCT
easat-5378	332	40	(	(	PUNCT
easat-5378	332	41	3,3	3,3	NOUN
easat-5378	332	42	)	)	PUNCT
easat-5378	332	43	,	,	PUNCT
easat-5378	332	44	…	…	PUNCT
easat-5378	332	45	.	.	PUNCT
easat-5378	333	1	the	the	DET
easat-5378	333	2	second	second	ADJ
easat-5378	333	3	example	example	NOUN
easat-5378	333	4	is	be	AUX
easat-5378	333	5	when	when	SCONJ
easat-5378	333	6	𝑚	𝑚	PROPN
easat-5378	333	7	=	=	SYM
easat-5378	333	8	3	3	NUM
easat-5378	333	9	and	and	CCONJ
easat-5378	333	10	the	the	DET
easat-5378	333	11	(	(	PUNCT
easat-5378	333	12	𝑛(3	𝑛(3	NOUN
easat-5378	333	13	)	)	PUNCT
easat-5378	333	14	,	,	PUNCT
easat-5378	333	15	𝑘(3))-factoriangular	𝑘(3))-factoriangular	ADJ
easat-5378	333	16	numbers	number	NOUN
easat-5378	333	17	are	be	AUX
easat-5378	333	18	given	give	VERB
easat-5378	333	19	in	in	ADP
easat-5378	333	20	table	table	NOUN
easat-5378	333	21	3	3	NUM
easat-5378	333	22	.	.	PUNCT
easat-5378	333	23	table	table	NOUN
easat-5378	333	24	3	3	NUM
easat-5378	333	25	.	.	PUNCT
easat-5378	333	26	table	table	NOUN
easat-5378	333	27	of	of	ADP
easat-5378	333	28	(	(	PUNCT
easat-5378	333	29	n(3	n(3	PROPN
easat-5378	333	30	)	)	PUNCT
easat-5378	333	31	,	,	PUNCT
easat-5378	333	32	k(3))-factoriangular	k(3))-factoriangular	ADJ
easat-5378	333	33	numbers	number	NOUN
easat-5378	333	34	.	.	PUNCT
easat-5378	334	1	𝒏	𝒏	PROPN
easat-5378	334	2	\	\	PROPN
easat-5378	334	3	𝒌	𝒌	DET
easat-5378	334	4	1	1	NUM
easat-5378	334	5	2	2	NUM
easat-5378	334	6	3	3	NUM
easat-5378	334	7	4	4	NUM
easat-5378	334	8	𝒌	𝒌	SYM
easat-5378	334	9	1	1	NUM
easat-5378	334	10	2	2	NUM
easat-5378	334	11	10	10	NUM
easat-5378	334	12	37	37	NUM
easat-5378	334	13	101	101	NUM
easat-5378	334	14	1	1	NUM
easat-5378	334	15	+	+	NUM
easat-5378	334	16	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	334	17	)	)	PUNCT
easat-5378	334	18	2	2	NUM
easat-5378	334	19	9	9	NUM
easat-5378	334	20	17	17	NUM
easat-5378	334	21	44	44	NUM
easat-5378	334	22	108	108	NUM
easat-5378	334	23	8	8	NUM
easat-5378	334	24	+	+	NUM
easat-5378	334	25	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	334	26	)	)	PUNCT
easat-5378	334	27	3	3	NUM
easat-5378	334	28	217	217	NUM
easat-5378	334	29	225	225	NUM
easat-5378	334	30	252	252	NUM
easat-5378	334	31	316	316	NUM
easat-5378	334	32	216	216	NUM
easat-5378	334	33	+	+	NUM
easat-5378	334	34	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	334	35	)	)	PUNCT
easat-5378	334	36	4	4	NUM
easat-5378	334	37	13825	13825	NUM
easat-5378	334	38	13833	13833	NUM
easat-5378	334	39	13860	13860	NUM
easat-5378	334	40	13924	13924	NUM
easat-5378	334	41	13824	13824	NUM
easat-5378	334	42	+	+	CCONJ
easat-5378	334	43	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	334	44	)	)	PUNCT
easat-5378	334	45	𝒏	𝒏	PROPN
easat-5378	334	46	(	(	PUNCT
easat-5378	334	47	𝑛!)3	𝑛!)3	PROPN
easat-5378	334	48	+	+	NOUN
easat-5378	334	49	1	1	NUM
easat-5378	334	50	(	(	PUNCT
easat-5378	334	51	𝑛!)3	𝑛!)3	PROPN
easat-5378	334	52	+	+	NOUN
easat-5378	334	53	9	9	NUM
easat-5378	334	54	(	(	PUNCT
easat-5378	334	55	𝑛!)3	𝑛!)3	PROPN
easat-5378	334	56	+	+	NUM
easat-5378	334	57	36	36	NUM
easat-5378	334	58	(	(	PUNCT
easat-5378	334	59	𝑛!)3	𝑛!)3	PROPN
easat-5378	334	60	+	+	NOUN
easat-5378	334	61	100	100	NUM
easat-5378	334	62	(	(	PUNCT
easat-5378	334	63	𝑛!)3	𝑛!)3	PROPN
easat-5378	334	64	+	+	NUM
easat-5378	334	65	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	334	66	)	)	PUNCT
easat-5378	334	67	then	then	ADV
easat-5378	334	68	,	,	PUNCT
easat-5378	334	69	the	the	DET
easat-5378	334	70	sequence	sequence	NOUN
easat-5378	334	71	of	of	ADP
easat-5378	334	72	(	(	PUNCT
easat-5378	334	73	𝑛(3	𝑛(3	NOUN
easat-5378	334	74	)	)	PUNCT
easat-5378	334	75	,	,	PUNCT
easat-5378	334	76	𝑘(3))-factoriangular	𝑘(3))-factoriangular	ADJ
easat-5378	334	77	numbers	number	NOUN
easat-5378	334	78	is	be	AUX
easat-5378	334	79	given	give	VERB
easat-5378	334	80	by	by	ADP
easat-5378	334	81	{	{	PUNCT
easat-5378	334	82	𝐹𝑡𝑛(3),𝑘(3	𝐹𝑡𝑛(3),𝑘(3	NOUN
easat-5378	334	83	)	)	PUNCT
easat-5378	334	84	}	}	PUNCT
easat-5378	334	85	=	=	SYM
easat-5378	334	86	{	{	PUNCT
easat-5378	334	87	2	2	NUM
easat-5378	334	88	,	,	PUNCT
easat-5378	334	89	9	9	NUM
easat-5378	334	90	,	,	PUNCT
easat-5378	334	91	10	10	NUM
easat-5378	334	92	,	,	PUNCT
easat-5378	334	93	17	17	NUM
easat-5378	334	94	,	,	PUNCT
easat-5378	334	95	217	217	NUM
easat-5378	334	96	,	,	PUNCT
easat-5378	334	97	37	37	NUM
easat-5378	334	98	,	,	PUNCT
easat-5378	334	99	225	225	NUM
easat-5378	334	100	,	,	PUNCT
easat-5378	334	101	44	44	NUM
easat-5378	334	102	,	,	PUNCT
easat-5378	334	103	252	252	NUM
easat-5378	334	104	,	,	PUNCT
easat-5378	334	105	.	.	PUNCT
easat-5378	334	106	.	.	PUNCT
easat-5378	334	107	.	.	PUNCT
easat-5378	335	1	}	}	PUNCT
easat-5378	336	1	for	for	ADP
easat-5378	336	2	(	(	PUNCT
easat-5378	336	3	n	n	CCONJ
easat-5378	336	4	,	,	PUNCT
easat-5378	336	5	k	k	NOUN
easat-5378	336	6	)	)	PUNCT
easat-5378	336	7	=	=	SYM
easat-5378	336	8	(	(	PUNCT
easat-5378	336	9	1,1	1,1	NUM
easat-5378	336	10	)	)	PUNCT
easat-5378	336	11	,	,	PUNCT
easat-5378	336	12	(	(	PUNCT
easat-5378	336	13	2,1	2,1	NUM
easat-5378	336	14	)	)	PUNCT
easat-5378	336	15	,	,	PUNCT
easat-5378	336	16	(	(	PUNCT
easat-5378	336	17	1,2	1,2	NUM
easat-5378	336	18	)	)	PUNCT
easat-5378	336	19	,	,	PUNCT
easat-5378	336	20	(	(	PUNCT
easat-5378	336	21	2,2	2,2	NUM
easat-5378	336	22	)	)	PUNCT
easat-5378	336	23	,	,	PUNCT
easat-5378	336	24	(	(	PUNCT
easat-5378	336	25	3,1	3,1	NUM
easat-5378	336	26	)	)	PUNCT
easat-5378	336	27	,	,	PUNCT
easat-5378	336	28	(	(	PUNCT
easat-5378	336	29	1,3	1,3	NUM
easat-5378	336	30	)	)	PUNCT
easat-5378	336	31	,	,	PUNCT
easat-5378	336	32	(	(	PUNCT
easat-5378	336	33	3,2	3,2	NUM
easat-5378	336	34	)	)	PUNCT
easat-5378	336	35	,	,	PUNCT
easat-5378	336	36	(	(	PUNCT
easat-5378	336	37	2,3	2,3	NUM
easat-5378	336	38	)	)	PUNCT
easat-5378	336	39	,	,	PUNCT
easat-5378	336	40	(	(	PUNCT
easat-5378	336	41	3,3	3,3	NOUN
easat-5378	336	42	)	)	PUNCT
easat-5378	336	43	,	,	PUNCT
easat-5378	336	44	…	…	PUNCT
easat-5378	336	45	.	.	PUNCT
easat-5378	337	1	3.2.2	3.2.2	X
easat-5378	337	2	.	.	PUNCT
easat-5378	338	1	on	on	ADP
easat-5378	338	2	the	the	DET
easat-5378	338	3	(	(	PUNCT
easat-5378	338	4	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	338	5	)	)	PUNCT
easat-5378	338	6	,	,	PUNCT
easat-5378	338	7	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	338	8	numbers	number	NOUN
easat-5378	338	9	we	we	PRON
easat-5378	338	10	further	far	ADV
easat-5378	338	11	generalized	generalize	VERB
easat-5378	338	12	(	(	PUNCT
easat-5378	338	13	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	338	14	)	)	PUNCT
easat-5378	338	15	,	,	PUNCT
easat-5378	338	16	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	338	17	numbers	number	NOUN
easat-5378	338	18	to	to	PART
easat-5378	338	19	have	have	VERB
easat-5378	338	20	the	the	DET
easat-5378	338	21	(	(	PUNCT
easat-5378	338	22	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	338	23	)	)	PUNCT
easat-5378	338	24	,	,	PUNCT
easat-5378	338	25	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	338	26	numbers	number	NOUN
easat-5378	338	27	(	(	PUNCT
easat-5378	338	28	see	see	VERB
easat-5378	338	29	definition	definition	NOUN
easat-5378	338	30	3.11	3.11	NUM
easat-5378	338	31	)	)	PUNCT
easat-5378	338	32	.	.	PUNCT
easat-5378	339	1	we	we	PRON
easat-5378	339	2	now	now	ADV
easat-5378	339	3	present	present	VERB
easat-5378	339	4	the	the	DET
easat-5378	339	5	following	follow	VERB
easat-5378	339	6	theorems	theorem	NOUN
easat-5378	339	7	whose	whose	DET
easat-5378	339	8	proofs	proof	NOUN
easat-5378	339	9	are	be	AUX
easat-5378	339	10	similar	similar	ADJ
easat-5378	339	11	to	to	ADP
easat-5378	339	12	the	the	DET
easat-5378	339	13	previous	previous	ADJ
easat-5378	339	14	theorems	theorem	NOUN
easat-5378	339	15	on	on	ADP
easat-5378	339	16	(	(	PUNCT
easat-5378	339	17	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	339	18	)	)	PUNCT
easat-5378	339	19	,	,	PUNCT
easat-5378	339	20	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	339	21	numbers	number	NOUN
easat-5378	339	22	.	.	PUNCT
easat-5378	340	1	we	we	PRON
easat-5378	340	2	also	also	ADV
easat-5378	340	3	give	give	VERB
easat-5378	340	4	examples	example	NOUN
easat-5378	340	5	of	of	ADP
easat-5378	340	6	sequences	sequence	NOUN
easat-5378	340	7	of	of	ADP
easat-5378	340	8	(	(	PUNCT
easat-5378	340	9	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	340	10	)	)	PUNCT
easat-5378	340	11	,	,	PUNCT
easat-5378	340	12	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	340	13	numbers	number	NOUN
easat-5378	340	14	.	.	PUNCT
easat-5378	341	1	theorem	theorem	VERB
easat-5378	341	2	3.17	3.17	NUM
easat-5378	341	3	:	:	PUNCT
easat-5378	341	4	for	for	ADP
easat-5378	341	5	natural	natural	ADJ
easat-5378	341	6	numbers	number	NOUN
easat-5378	341	7	𝑛	𝑛	PROPN
easat-5378	341	8	,	,	PUNCT
easat-5378	341	9	𝑘	𝑘	PROPN
easat-5378	341	10	,	,	PUNCT
easat-5378	341	11	𝑎	𝑎	NOUN
easat-5378	341	12	,	,	PUNCT
easat-5378	341	13	𝑏	𝑏	DET
easat-5378	341	14	≥	≥	NUM
easat-5378	341	15	1	1	NUM
easat-5378	341	16	,	,	PUNCT
easat-5378	341	17	the	the	DET
easat-5378	341	18	(	(	PUNCT
easat-5378	341	19	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	341	20	)	)	PUNCT
easat-5378	341	21	,	,	PUNCT
easat-5378	341	22	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	341	23	numbers	number	NOUN
easat-5378	341	24	can	can	AUX
easat-5378	341	25	be	be	AUX
easat-5378	341	26	determined	determine	VERB
easat-5378	341	27	by	by	ADP
easat-5378	341	28	the	the	DET
easat-5378	341	29	formula	formula	NOUN
easat-5378	341	30	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	341	31	)	)	PUNCT
easat-5378	342	1	=	=	PUNCT
easat-5378	342	2	(	(	PUNCT
easat-5378	342	3	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	342	4	+	+	X
easat-5378	342	5	1	1	NUM
easat-5378	342	6	𝑏	𝑏	NOUN
easat-5378	342	7	+	+	NOUN
easat-5378	342	8	1	1	NUM
easat-5378	342	9	[	[	X
easat-5378	342	10	(	(	PUNCT
easat-5378	342	11	𝑘	𝑘	X
easat-5378	342	12	+	+	X
easat-5378	342	13	1)[(𝑘	1)[(𝑘	NUM
easat-5378	342	14	+	+	CCONJ
easat-5378	342	15	1)𝑏	1)𝑏	NUM
easat-5378	342	16	−	−	ADP
easat-5378	342	17	1	1	NUM
easat-5378	342	18	]	]	PUNCT
easat-5378	342	19	−	−	PUNCT
easat-5378	342	20	∑	∑	PROPN
easat-5378	342	21	(	(	PUNCT
easat-5378	342	22	𝑏	𝑏	PROPN
easat-5378	342	23	+	+	NOUN
easat-5378	342	24	1	1	NUM
easat-5378	342	25	𝑖	𝑖	NOUN
easat-5378	342	26	)	)	PUNCT
easat-5378	342	27	𝑏−1	𝑏−1	PROPN
easat-5378	342	28	𝑖=1	𝑖=1	PUNCT
easat-5378	343	1	𝑆𝑖(𝑘	𝑆𝑖(𝑘	PROPN
easat-5378	343	2	)	)	PUNCT
easat-5378	343	3	]	]	PUNCT
easat-5378	343	4	where	where	SCONJ
easat-5378	343	5	𝑛	𝑛	X
easat-5378	343	6	!	!	PUNCT
easat-5378	343	7	=	=	SYM
easat-5378	344	1	1	1	NUM
easat-5378	344	2	⋅	⋅	NUM
easat-5378	344	3	2	2	NUM
easat-5378	344	4	⋅	⋅	X
easat-5378	344	5	3	3	NUM
easat-5378	344	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	344	7	𝑛	𝑛	PROPN
easat-5378	344	8	and	and	CCONJ
easat-5378	344	9	𝑆𝑖(𝑘	𝑆𝑖(𝑘	NUM
easat-5378	344	10	)	)	PUNCT
easat-5378	344	11	=	=	SYM
easat-5378	344	12	1𝑖	1𝑖	NOUN
easat-5378	344	13	+	+	CCONJ
easat-5378	344	14	2𝑖	2𝑖	NOUN
easat-5378	344	15	+	+	CCONJ
easat-5378	344	16	3𝑖+	3𝑖+	NUM
easat-5378	344	17	.	.	PUNCT
easat-5378	344	18	.	.	PUNCT
easat-5378	344	19	.	.	PUNCT
easat-5378	345	1	+	+	CCONJ
easat-5378	345	2	𝑘𝑖.	𝑘𝑖.	VERB
easat-5378	345	3	the	the	DET
easat-5378	345	4	proof	proof	NOUN
easat-5378	345	5	of	of	ADP
easat-5378	345	6	theorem	theorem	ADJ
easat-5378	345	7	3.17	3.17	NUM
easat-5378	345	8	is	be	AUX
easat-5378	345	9	similar	similar	ADJ
easat-5378	345	10	to	to	ADP
easat-5378	345	11	the	the	DET
easat-5378	345	12	proof	proof	NOUN
easat-5378	345	13	of	of	ADP
easat-5378	345	14	theorem	theorem	ADJ
easat-5378	345	15	3.12	3.12	NUM
easat-5378	345	16	.	.	PUNCT
easat-5378	345	17	901	901	NUM
easat-5378	345	18	edelweiss	edelweiss	PROPN
easat-5378	345	19	applied	apply	VERB
easat-5378	345	20	science	science	NOUN
easat-5378	345	21	and	and	CCONJ
easat-5378	345	22	technology	technology	NOUN
easat-5378	345	23	issn	issn	PROPN
easat-5378	345	24	:	:	PUNCT
easat-5378	345	25	2576	2576	NUM
easat-5378	345	26	-	-	SYM
easat-5378	345	27	8484	8484	NUM
easat-5378	345	28	vol	vol	NOUN
easat-5378	345	29	.	.	PROPN
easat-5378	346	1	9	9	NUM
easat-5378	346	2	,	,	PUNCT
easat-5378	346	3	no	no	INTJ
easat-5378	346	4	.	.	NOUN
easat-5378	347	1	3	3	NUM
easat-5378	347	2	:	:	PUNCT
easat-5378	347	3	891	891	NUM
easat-5378	347	4	-	-	SYM
easat-5378	347	5	904	904	NUM
easat-5378	347	6	,	,	PUNCT
easat-5378	347	7	2025	2025	NUM
easat-5378	347	8	doi	doi	NOUN
easat-5378	347	9	:	:	PUNCT
easat-5378	347	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	347	11	©	©	PROPN
easat-5378	347	12	2025	2025	NUM
easat-5378	347	13	by	by	ADP
easat-5378	347	14	the	the	DET
easat-5378	347	15	author	author	NOUN
easat-5378	347	16	;	;	PUNCT
easat-5378	347	17	licensee	licensee	PROPN
easat-5378	347	18	learning	learning	PROPN
easat-5378	347	19	gate	gate	PROPN
easat-5378	347	20	theorem	theorem	VERB
easat-5378	347	21	3.18	3.18	NUM
easat-5378	347	22	:	:	PUNCT
easat-5378	347	23	the	the	DET
easat-5378	347	24	(	(	PUNCT
easat-5378	347	25	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	347	26	)	)	PUNCT
easat-5378	347	27	,	,	PUNCT
easat-5378	347	28	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	347	29	number	number	NOUN
easat-5378	347	30	for	for	ADP
easat-5378	347	31	even	even	ADV
easat-5378	347	32	𝑏	𝑏	NOUN
easat-5378	347	33	=	=	NOUN
easat-5378	347	34	2𝑗	2𝑗	NOUN
easat-5378	347	35	is	be	AUX
easat-5378	347	36	given	give	VERB
easat-5378	347	37	by	by	ADP
easat-5378	347	38	the	the	DET
easat-5378	347	39	formula	formula	NOUN
easat-5378	347	40	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	347	41	)	)	PUNCT
easat-5378	348	1	=	=	SYM
easat-5378	348	2	𝐹𝑡𝑛(𝑎),𝑘(2𝑗	𝐹𝑡𝑛(𝑎),𝑘(2𝑗	NOUN
easat-5378	348	3	)	)	PUNCT
easat-5378	348	4	=	=	NOUN
easat-5378	348	5	(	(	PUNCT
easat-5378	348	6	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	348	7	+	+	CCONJ
easat-5378	349	1	2𝑘	2𝑘	NUM
easat-5378	349	2	+	+	SYM
easat-5378	349	3	1	1	NUM
easat-5378	349	4	2𝑗	2𝑗	NUM
easat-5378	349	5	+	+	NOUN
easat-5378	349	6	1	1	NUM
easat-5378	350	1	[	[	X
easat-5378	350	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	350	3	+	+	NUM
easat-5378	350	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	350	5	where	where	SCONJ
easat-5378	350	6	𝑛	𝑛	X
easat-5378	350	7	!	!	PUNCT
easat-5378	350	8	=	=	SYM
easat-5378	351	1	1	1	NUM
easat-5378	351	2	⋅	⋅	NUM
easat-5378	351	3	2	2	NUM
easat-5378	351	4	⋅	⋅	X
easat-5378	351	5	3	3	NUM
easat-5378	351	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	351	7	𝑛	𝑛	PROPN
easat-5378	351	8	,	,	PUNCT
easat-5378	351	9	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	351	10	=	=	SYM
easat-5378	351	11	1	1	NUM
easat-5378	351	12	+	+	NUM
easat-5378	351	13	2	2	NUM
easat-5378	351	14	+	+	SYM
easat-5378	351	15	3	3	NUM
easat-5378	351	16	+	+	NOUN
easat-5378	351	17	.	.	PUNCT
easat-5378	351	18	.	.	PUNCT
easat-5378	351	19	.	.	PUNCT
easat-5378	352	1	+	+	NOUN
easat-5378	352	2	𝑘	𝑘	X
easat-5378	352	3	=	=	PUNCT
easat-5378	352	4	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	352	5	+	+	CCONJ
easat-5378	352	6	1)/2	1)/2	NUM
easat-5378	352	7	,	,	PUNCT
easat-5378	352	8	and	and	CCONJ
easat-5378	352	9	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	352	10	)	)	PUNCT
easat-5378	352	11	is	be	AUX
easat-5378	352	12	a	a	DET
easat-5378	352	13	polynomial	polynomial	NOUN
easat-5378	352	14	in	in	ADP
easat-5378	352	15	𝑘	𝑘	PRON
easat-5378	352	16	of	of	ADP
easat-5378	352	17	degree	degree	NOUN
easat-5378	352	18	2𝑗	2𝑗	NOUN
easat-5378	352	19	−	−	PROPN
easat-5378	352	20	3	3	NUM
easat-5378	352	21	,	,	PUNCT
easat-5378	352	22	for	for	ADP
easat-5378	352	23	natural	natural	ADJ
easat-5378	352	24	numbers	number	NOUN
easat-5378	352	25	𝑛	𝑛	PROPN
easat-5378	352	26	,	,	PUNCT
easat-5378	352	27	𝑘	𝑘	PROPN
easat-5378	352	28	,	,	PUNCT
easat-5378	352	29	𝑗	𝑗	PRON
easat-5378	352	30	≥	≥	NOUN
easat-5378	352	31	1	1	NUM
easat-5378	352	32	.	.	PUNCT
easat-5378	353	1	the	the	DET
easat-5378	353	2	proof	proof	NOUN
easat-5378	353	3	of	of	ADP
easat-5378	353	4	theorem	theorem	ADJ
easat-5378	353	5	3.18	3.18	NUM
easat-5378	353	6	is	be	AUX
easat-5378	353	7	similar	similar	ADJ
easat-5378	353	8	to	to	ADP
easat-5378	353	9	the	the	DET
easat-5378	353	10	proof	proof	NOUN
easat-5378	353	11	of	of	ADP
easat-5378	353	12	theorem	theorem	NOUN
easat-5378	353	13	3.13	3.13	NUM
easat-5378	353	14	.	.	PUNCT
easat-5378	354	1	theorem	theorem	VERB
easat-5378	354	2	3.19	3.19	NUM
easat-5378	354	3	:	:	PUNCT
easat-5378	354	4	the	the	DET
easat-5378	354	5	(	(	PUNCT
easat-5378	354	6	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	354	7	)	)	PUNCT
easat-5378	354	8	,	,	PUNCT
easat-5378	354	9	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	354	10	number	number	NOUN
easat-5378	354	11	for	for	ADP
easat-5378	354	12	odd	odd	ADJ
easat-5378	354	13	𝑏	𝑏	NOUN
easat-5378	354	14	=	=	SYM
easat-5378	354	15	2𝑗	2𝑗	NOUN
easat-5378	354	16	+	+	CCONJ
easat-5378	354	17	1	1	NUM
easat-5378	354	18	is	be	AUX
easat-5378	354	19	given	give	VERB
easat-5378	354	20	by	by	ADP
easat-5378	354	21	the	the	DET
easat-5378	354	22	formula	formula	NOUN
easat-5378	354	23	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	354	24	)	)	PUNCT
easat-5378	354	25	=	=	SYM
easat-5378	355	1	𝐹𝑡𝑛(𝑎),𝑘(2𝑗+1	𝐹𝑡𝑛(𝑎),𝑘(2𝑗+1	X
easat-5378	355	2	)	)	PUNCT
easat-5378	355	3	=	=	NOUN
easat-5378	355	4	(	(	PUNCT
easat-5378	355	5	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	355	6	+	+	CCONJ
easat-5378	355	7	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	355	8	+	+	NOUN
easat-5378	355	9	1	1	X
easat-5378	355	10	)	)	PUNCT
easat-5378	355	11	𝑗	𝑗	NOUN
easat-5378	356	1	+	+	ADJ
easat-5378	356	2	1	1	NUM
easat-5378	357	1	[	[	X
easat-5378	357	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	357	3	+	+	NUM
easat-5378	357	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	357	5	where	where	SCONJ
easat-5378	357	6	𝑛	𝑛	X
easat-5378	357	7	!	!	PUNCT
easat-5378	357	8	=	=	SYM
easat-5378	358	1	1	1	NUM
easat-5378	358	2	⋅	⋅	NUM
easat-5378	358	3	2	2	NUM
easat-5378	358	4	⋅	⋅	X
easat-5378	358	5	3	3	NUM
easat-5378	358	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	358	7	𝑛	𝑛	PROPN
easat-5378	358	8	,	,	PUNCT
easat-5378	358	9	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	358	10	=	=	SYM
easat-5378	358	11	1	1	NUM
easat-5378	358	12	+	+	NUM
easat-5378	358	13	2	2	NUM
easat-5378	358	14	+	+	SYM
easat-5378	358	15	3	3	NUM
easat-5378	358	16	+	+	NOUN
easat-5378	358	17	.	.	PUNCT
easat-5378	358	18	.	.	PUNCT
easat-5378	358	19	.	.	PUNCT
easat-5378	359	1	+	+	NOUN
easat-5378	359	2	𝑘	𝑘	X
easat-5378	359	3	=	=	PUNCT
easat-5378	359	4	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	359	5	+	+	CCONJ
easat-5378	359	6	1)/2	1)/2	NUM
easat-5378	359	7	,	,	PUNCT
easat-5378	359	8	and	and	CCONJ
easat-5378	359	9	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	359	10	)	)	PUNCT
easat-5378	359	11	is	be	AUX
easat-5378	359	12	a	a	DET
easat-5378	359	13	polynomial	polynomial	NOUN
easat-5378	359	14	in	in	ADP
easat-5378	359	15	𝑘	𝑘	PRON
easat-5378	359	16	of	of	ADP
easat-5378	359	17	degree	degree	NOUN
easat-5378	359	18	2𝑗	2𝑗	NOUN
easat-5378	359	19	−	−	PROPN
easat-5378	359	20	3	3	NUM
easat-5378	359	21	,	,	PUNCT
easat-5378	359	22	for	for	ADP
easat-5378	359	23	natural	natural	ADJ
easat-5378	359	24	numbers	number	NOUN
easat-5378	359	25	𝑛	𝑛	PROPN
easat-5378	359	26	,	,	PUNCT
easat-5378	359	27	𝑘	𝑘	PROPN
easat-5378	359	28	,	,	PUNCT
easat-5378	359	29	𝑗	𝑗	PRON
easat-5378	359	30	≥	≥	NOUN
easat-5378	359	31	1	1	NUM
easat-5378	359	32	.	.	PUNCT
easat-5378	360	1	the	the	DET
easat-5378	360	2	proof	proof	NOUN
easat-5378	360	3	of	of	ADP
easat-5378	360	4	theorem	theorem	ADJ
easat-5378	360	5	3.19	3.19	NUM
easat-5378	360	6	is	be	AUX
easat-5378	360	7	similar	similar	ADJ
easat-5378	360	8	to	to	ADP
easat-5378	360	9	the	the	DET
easat-5378	360	10	proof	proof	NOUN
easat-5378	360	11	of	of	ADP
easat-5378	360	12	theorem	theorem	ADJ
easat-5378	360	13	3.14	3.14	NUM
easat-5378	360	14	.	.	PUNCT
easat-5378	361	1	theorem	theorem	VERB
easat-5378	361	2	3.20	3.20	NUM
easat-5378	361	3	:	:	PUNCT
easat-5378	361	4	the	the	DET
easat-5378	361	5	(	(	PUNCT
easat-5378	361	6	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	361	7	)	)	PUNCT
easat-5378	361	8	,	,	PUNCT
easat-5378	361	9	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	361	10	number	number	NOUN
easat-5378	361	11	for	for	ADP
easat-5378	361	12	even	even	ADV
easat-5378	361	13	𝑏	𝑏	NOUN
easat-5378	361	14	=	=	NOUN
easat-5378	361	15	2𝑗	2𝑗	NOUN
easat-5378	361	16	is	be	AUX
easat-5378	361	17	given	give	VERB
easat-5378	361	18	by	by	ADP
easat-5378	361	19	the	the	DET
easat-5378	361	20	formula	formula	NOUN
easat-5378	361	21	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	361	22	)	)	PUNCT
easat-5378	362	1	=	=	SYM
easat-5378	362	2	𝐹𝑡𝑛(𝑎),𝑘(2𝑗	𝐹𝑡𝑛(𝑎),𝑘(2𝑗	NOUN
easat-5378	362	3	)	)	PUNCT
easat-5378	362	4	=	=	NOUN
easat-5378	362	5	(	(	PUNCT
easat-5378	362	6	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	362	7	+	+	CCONJ
easat-5378	362	8	3	3	NUM
easat-5378	362	9	2𝑗	2𝑗	NOUN
easat-5378	362	10	+	+	NOUN
easat-5378	362	11	1	1	NUM
easat-5378	363	1	[	[	X
easat-5378	363	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	363	3	+	+	NUM
easat-5378	363	4	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	NOUN
easat-5378	363	5	)	)	PUNCT
easat-5378	363	6	where	where	SCONJ
easat-5378	363	7	𝑛	𝑛	X
easat-5378	363	8	!	!	PUNCT
easat-5378	363	9	=	=	SYM
easat-5378	364	1	1	1	NUM
easat-5378	364	2	⋅	⋅	NUM
easat-5378	364	3	2	2	NUM
easat-5378	364	4	⋅	⋅	X
easat-5378	364	5	3	3	NUM
easat-5378	364	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	364	7	𝑛	𝑛	PROPN
easat-5378	364	8	,	,	PUNCT
easat-5378	364	9	𝑆2(𝑘	𝑆2(𝑘	PROPN
easat-5378	364	10	)	)	PUNCT
easat-5378	364	11	=	=	PUNCT
easat-5378	365	1	𝑘(𝑘	𝑘(𝑘	PROPN
easat-5378	366	1	+	+	NUM
easat-5378	366	2	1)(2𝑘	1)(2𝑘	NUM
easat-5378	366	3	+	+	CCONJ
easat-5378	366	4	1)/6	1)/6	NUM
easat-5378	366	5	,	,	PUNCT
easat-5378	366	6	and	and	CCONJ
easat-5378	366	7	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	366	8	)	)	PUNCT
easat-5378	366	9	is	be	AUX
easat-5378	366	10	a	a	DET
easat-5378	366	11	polynomial	polynomial	NOUN
easat-5378	366	12	in	in	ADP
easat-5378	366	13	𝑘	𝑘	PRON
easat-5378	366	14	of	of	ADP
easat-5378	366	15	degree	degree	NOUN
easat-5378	366	16	2𝑗	2𝑗	NOUN
easat-5378	366	17	−	−	PROPN
easat-5378	366	18	3	3	NUM
easat-5378	366	19	,	,	PUNCT
easat-5378	366	20	for	for	ADP
easat-5378	366	21	natural	natural	ADJ
easat-5378	366	22	numbers	number	NOUN
easat-5378	366	23	𝑛	𝑛	PROPN
easat-5378	366	24	,	,	PUNCT
easat-5378	366	25	𝑘	𝑘	DET
easat-5378	366	26	≥	≥	NOUN
easat-5378	366	27	1and	1and	NUM
easat-5378	366	28	𝑗	𝑗	NOUN
easat-5378	366	29	>	>	X
easat-5378	366	30	1	1	NUM
easat-5378	366	31	.	.	PUNCT
easat-5378	367	1	the	the	DET
easat-5378	367	2	proof	proof	NOUN
easat-5378	367	3	of	of	ADP
easat-5378	367	4	theorem	theorem	ADJ
easat-5378	367	5	3.20	3.20	NUM
easat-5378	367	6	is	be	AUX
easat-5378	367	7	similar	similar	ADJ
easat-5378	367	8	to	to	ADP
easat-5378	367	9	the	the	DET
easat-5378	367	10	proof	proof	NOUN
easat-5378	367	11	of	of	ADP
easat-5378	367	12	theorem	theorem	ADJ
easat-5378	367	13	3.15	3.15	NUM
easat-5378	367	14	.	.	PUNCT
easat-5378	368	1	theorem	theorem	VERB
easat-5378	368	2	3.21	3.21	NUM
easat-5378	368	3	:	:	PUNCT
easat-5378	368	4	the	the	DET
easat-5378	368	5	(	(	PUNCT
easat-5378	368	6	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	368	7	)	)	PUNCT
easat-5378	368	8	,	,	PUNCT
easat-5378	368	9	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	368	10	number	number	NOUN
easat-5378	368	11	for	for	ADP
easat-5378	368	12	odd	odd	ADJ
easat-5378	368	13	𝑏	𝑏	NOUN
easat-5378	368	14	=	=	SYM
easat-5378	368	15	2𝑗	2𝑗	NOUN
easat-5378	368	16	+	+	CCONJ
easat-5378	368	17	1	1	NUM
easat-5378	368	18	is	be	AUX
easat-5378	368	19	given	give	VERB
easat-5378	368	20	by	by	ADP
easat-5378	368	21	the	the	DET
easat-5378	368	22	formula	formula	NOUN
easat-5378	368	23	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	368	24	)	)	PUNCT
easat-5378	368	25	=	=	SYM
easat-5378	369	1	𝐹𝑡𝑛(𝑎),𝑘(2𝑗+1	𝐹𝑡𝑛(𝑎),𝑘(2𝑗+1	X
easat-5378	369	2	)	)	PUNCT
easat-5378	369	3	=	=	NOUN
easat-5378	369	4	(	(	PUNCT
easat-5378	369	5	𝑛!)𝑎	𝑛!)𝑎	NOUN
easat-5378	369	6	+	+	X
easat-5378	369	7	2	2	NUM
easat-5378	369	8	𝑗	𝑗	NOUN
easat-5378	370	1	+	+	NOUN
easat-5378	370	2	1	1	NUM
easat-5378	371	1	[	[	X
easat-5378	371	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	371	3	+	+	CCONJ
easat-5378	371	4	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	NOUN
easat-5378	371	5	)	)	PUNCT
easat-5378	371	6	where	where	SCONJ
easat-5378	371	7	𝑛	𝑛	X
easat-5378	371	8	!	!	PUNCT
easat-5378	371	9	=	=	SYM
easat-5378	372	1	1	1	NUM
easat-5378	372	2	⋅	⋅	NUM
easat-5378	372	3	2	2	NUM
easat-5378	372	4	⋅	⋅	X
easat-5378	372	5	3	3	NUM
easat-5378	372	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	372	7	𝑛	𝑛	PROPN
easat-5378	372	8	,	,	PUNCT
easat-5378	372	9	𝑆3(𝑘	𝑆3(𝑘	PROPN
easat-5378	372	10	)	)	PUNCT
easat-5378	372	11	=	=	X
easat-5378	373	1	𝑘2(𝑘	𝑘2(𝑘	PROPN
easat-5378	374	1	+	+	NOUN
easat-5378	374	2	1)2/4	1)2/4	NUM
easat-5378	374	3	,	,	PUNCT
easat-5378	374	4	and	and	CCONJ
easat-5378	374	5	𝑃(𝑘2𝑗−3	𝑃(𝑘2𝑗−3	NOUN
easat-5378	374	6	)	)	PUNCT
easat-5378	374	7	is	be	AUX
easat-5378	374	8	a	a	DET
easat-5378	374	9	polynomial	polynomial	NOUN
easat-5378	374	10	in	in	ADP
easat-5378	374	11	𝑘	𝑘	PRON
easat-5378	374	12	of	of	ADP
easat-5378	374	13	degree	degree	NOUN
easat-5378	374	14	2𝑗	2𝑗	NOUN
easat-5378	374	15	−	−	PROPN
easat-5378	374	16	3	3	NUM
easat-5378	374	17	,	,	PUNCT
easat-5378	374	18	for	for	ADP
easat-5378	374	19	natural	natural	ADJ
easat-5378	374	20	numbers	number	NOUN
easat-5378	374	21	𝑛	𝑛	PROPN
easat-5378	374	22	,	,	PUNCT
easat-5378	374	23	𝑘	𝑘	DET
easat-5378	374	24	≥	≥	NOUN
easat-5378	374	25	1and	1and	NUM
easat-5378	374	26	𝑗	𝑗	NOUN
easat-5378	374	27	>	>	X
easat-5378	374	28	1	1	NUM
easat-5378	374	29	.	.	PUNCT
easat-5378	375	1	the	the	DET
easat-5378	375	2	proof	proof	NOUN
easat-5378	375	3	of	of	ADP
easat-5378	375	4	theorem	theorem	ADJ
easat-5378	375	5	3.21	3.21	NUM
easat-5378	375	6	is	be	AUX
easat-5378	375	7	similar	similar	ADJ
easat-5378	375	8	to	to	ADP
easat-5378	375	9	the	the	DET
easat-5378	375	10	proof	proof	NOUN
easat-5378	375	11	of	of	ADP
easat-5378	375	12	theorem	theorem	NOUN
easat-5378	375	13	3.16	3.16	NUM
easat-5378	375	14	.	.	PUNCT
easat-5378	376	1	three	three	NUM
easat-5378	376	2	examples	example	NOUN
easat-5378	376	3	of	of	ADP
easat-5378	376	4	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	376	5	)	)	PUNCT
easat-5378	376	6	,	,	PUNCT
easat-5378	376	7	with	with	ADP
easat-5378	376	8	𝑎	𝑎	X
easat-5378	376	9	,	,	PUNCT
easat-5378	376	10	𝑏	𝑏	PRON
easat-5378	376	11	≥	≥	NOUN
easat-5378	376	12	1	1	NUM
easat-5378	376	13	but	but	CCONJ
easat-5378	376	14	not	not	PART
easat-5378	376	15	both	both	PRON
easat-5378	376	16	𝑎	𝑎	NOUN
easat-5378	376	17	,	,	PUNCT
easat-5378	376	18	𝑏	𝑏	PROPN
easat-5378	376	19	=	=	SYM
easat-5378	376	20	1	1	NUM
easat-5378	376	21	,	,	PUNCT
easat-5378	376	22	are	be	AUX
easat-5378	376	23	presented	present	VERB
easat-5378	376	24	here	here	ADV
easat-5378	376	25	.	.	PUNCT
easat-5378	377	1	when	when	SCONJ
easat-5378	377	2	𝑎	𝑎	X
easat-5378	377	3	=	=	SYM
easat-5378	377	4	1	1	NUM
easat-5378	377	5	and	and	CCONJ
easat-5378	377	6	𝑏	𝑏	NOUN
easat-5378	377	7	=	=	SYM
easat-5378	377	8	2	2	NUM
easat-5378	377	9	,	,	PUNCT
easat-5378	377	10	the	the	DET
easat-5378	377	11	(	(	PUNCT
easat-5378	377	12	𝑛(1	𝑛(1	PROPN
easat-5378	377	13	)	)	PUNCT
easat-5378	377	14	,	,	PUNCT
easat-5378	377	15	𝑘(2))-factoriangular	𝑘(2))-factoriangular	ADJ
easat-5378	377	16	numbers	number	NOUN
easat-5378	377	17	are	be	AUX
easat-5378	377	18	given	give	VERB
easat-5378	377	19	in	in	ADP
easat-5378	377	20	table	table	NOUN
easat-5378	377	21	4	4	NUM
easat-5378	377	22	.	.	PUNCT
easat-5378	377	23	table	table	NOUN
easat-5378	377	24	4	4	NUM
easat-5378	377	25	.	.	PUNCT
easat-5378	377	26	table	table	NOUN
easat-5378	377	27	of	of	ADP
easat-5378	377	28	(	(	PUNCT
easat-5378	377	29	n(1	n(1	NOUN
easat-5378	377	30	)	)	PUNCT
easat-5378	377	31	,	,	PUNCT
easat-5378	377	32	k(2))-factoriangular	k(2))-factoriangular	ADJ
easat-5378	377	33	numbers	number	NOUN
easat-5378	377	34	.	.	PUNCT
easat-5378	378	1	𝒏	𝒏	PROPN
easat-5378	378	2	\	\	PROPN
easat-5378	378	3	𝒌	𝒌	DET
easat-5378	378	4	1	1	NUM
easat-5378	378	5	2	2	NUM
easat-5378	378	6	3	3	NUM
easat-5378	378	7	4	4	NUM
easat-5378	378	8	5	5	NUM
easat-5378	378	9	𝒌	𝒌	SYM
easat-5378	378	10	1	1	NUM
easat-5378	378	11	2	2	NUM
easat-5378	378	12	6	6	NUM
easat-5378	378	13	15	15	NUM
easat-5378	378	14	31	31	NUM
easat-5378	378	15	56	56	NUM
easat-5378	378	16	1	1	NUM
easat-5378	378	17	+	+	CCONJ
easat-5378	378	18	𝑆2(𝑘	𝑆2(𝑘	NOUN
easat-5378	378	19	)	)	PUNCT
easat-5378	378	20	2	2	NUM
easat-5378	378	21	3	3	NUM
easat-5378	378	22	7	7	NUM
easat-5378	378	23	16	16	NUM
easat-5378	378	24	32	32	NUM
easat-5378	378	25	57	57	NUM
easat-5378	378	26	2	2	NUM
easat-5378	378	27	+	+	CCONJ
easat-5378	378	28	𝑆2(𝑘	𝑆2(𝑘	PRON
easat-5378	378	29	)	)	PUNCT
easat-5378	378	30	3	3	NUM
easat-5378	378	31	7	7	NUM
easat-5378	378	32	11	11	NUM
easat-5378	378	33	20	20	NUM
easat-5378	378	34	36	36	NUM
easat-5378	378	35	61	61	NUM
easat-5378	378	36	6	6	NUM
easat-5378	378	37	+	+	CCONJ
easat-5378	378	38	𝑆2(𝑘	𝑆2(𝑘	PRON
easat-5378	378	39	)	)	PUNCT
easat-5378	378	40	4	4	NUM
easat-5378	378	41	25	25	NUM
easat-5378	378	42	29	29	NUM
easat-5378	378	43	38	38	NUM
easat-5378	378	44	54	54	NUM
easat-5378	378	45	79	79	NUM
easat-5378	378	46	24	24	NUM
easat-5378	378	47	+	+	CCONJ
easat-5378	378	48	𝑆2(𝑘	𝑆2(𝑘	PRON
easat-5378	378	49	)	)	PUNCT
easat-5378	378	50	5	5	NUM
easat-5378	378	51	121	121	NUM
easat-5378	378	52	125	125	NUM
easat-5378	378	53	134	134	NUM
easat-5378	378	54	150	150	NUM
easat-5378	378	55	175	175	NUM
easat-5378	378	56	120	120	NUM
easat-5378	378	57	+	+	CCONJ
easat-5378	378	58	𝑆2(𝑘	𝑆2(𝑘	PROPN
easat-5378	378	59	)	)	PUNCT
easat-5378	378	60	𝒏	𝒏	PROPN
easat-5378	378	61	𝑛	𝑛	X
easat-5378	378	62	!	!	PUNCT
easat-5378	379	1	+	+	CCONJ
easat-5378	379	2	1	1	NUM
easat-5378	379	3	𝑛	𝑛	NOUN
easat-5378	379	4	!	!	PUNCT
easat-5378	380	1	+	+	CCONJ
easat-5378	380	2	5	5	NUM
easat-5378	380	3	𝑛	𝑛	NOUN
easat-5378	380	4	!	!	PUNCT
easat-5378	381	1	+	+	CCONJ
easat-5378	381	2	14	14	NUM
easat-5378	381	3	𝑛	𝑛	NOUN
easat-5378	381	4	!	!	PUNCT
easat-5378	382	1	+	+	CCONJ
easat-5378	382	2	30	30	NUM
easat-5378	382	3	𝑛	𝑛	NOUN
easat-5378	382	4	!	!	PUNCT
easat-5378	383	1	+	+	CCONJ
easat-5378	383	2	55	55	NUM
easat-5378	383	3	𝑛	𝑛	NOUN
easat-5378	383	4	!	!	PUNCT
easat-5378	384	1	+	+	CCONJ
easat-5378	384	2	𝑆2(𝑘	𝑆2(𝑘	NOUN
easat-5378	384	3	)	)	PUNCT
easat-5378	384	4	then	then	ADV
easat-5378	384	5	,	,	PUNCT
easat-5378	384	6	the	the	DET
easat-5378	384	7	sequence	sequence	NOUN
easat-5378	384	8	of	of	ADP
easat-5378	384	9	(	(	PUNCT
easat-5378	384	10	𝑛(1	𝑛(1	PROPN
easat-5378	384	11	)	)	PUNCT
easat-5378	384	12	,	,	PUNCT
easat-5378	384	13	𝑘(2))-factoriangular	𝑘(2))-factoriangular	ADJ
easat-5378	384	14	numbers	number	NOUN
easat-5378	384	15	is	be	AUX
easat-5378	384	16	given	give	VERB
easat-5378	384	17	by	by	ADP
easat-5378	384	18	{	{	PUNCT
easat-5378	384	19	𝐹𝑡𝑛(1),𝑘(2	𝐹𝑡𝑛(1),𝑘(2	ADJ
easat-5378	384	20	)	)	PUNCT
easat-5378	384	21	}	}	PUNCT
easat-5378	384	22	=	=	SYM
easat-5378	384	23	{	{	PUNCT
easat-5378	384	24	2	2	NUM
easat-5378	384	25	,	,	PUNCT
easat-5378	384	26	3	3	NUM
easat-5378	384	27	,	,	PUNCT
easat-5378	384	28	6	6	NUM
easat-5378	384	29	,	,	PUNCT
easat-5378	384	30	7	7	NUM
easat-5378	384	31	,	,	PUNCT
easat-5378	384	32	7	7	NUM
easat-5378	384	33	,	,	PUNCT
easat-5378	384	34	15	15	NUM
easat-5378	384	35	,	,	PUNCT
easat-5378	384	36	11	11	NUM
easat-5378	384	37	,	,	PUNCT
easat-5378	384	38	16	16	NUM
easat-5378	384	39	,	,	PUNCT
easat-5378	384	40	20	20	NUM
easat-5378	384	41	,	,	PUNCT
easat-5378	384	42	.	.	PUNCT
easat-5378	384	43	.	.	PUNCT
easat-5378	384	44	.	.	PUNCT
easat-5378	385	1	}	}	PUNCT
easat-5378	386	1	for	for	ADP
easat-5378	386	2	(	(	PUNCT
easat-5378	386	3	n	n	CCONJ
easat-5378	386	4	,	,	PUNCT
easat-5378	386	5	k	k	NOUN
easat-5378	386	6	)	)	PUNCT
easat-5378	386	7	=	=	SYM
easat-5378	386	8	(	(	PUNCT
easat-5378	386	9	1,1	1,1	NUM
easat-5378	386	10	)	)	PUNCT
easat-5378	386	11	,	,	PUNCT
easat-5378	386	12	(	(	PUNCT
easat-5378	386	13	2,1	2,1	NUM
easat-5378	386	14	)	)	PUNCT
easat-5378	386	15	,	,	PUNCT
easat-5378	386	16	(	(	PUNCT
easat-5378	386	17	1,2	1,2	NUM
easat-5378	386	18	)	)	PUNCT
easat-5378	386	19	,	,	PUNCT
easat-5378	386	20	(	(	PUNCT
easat-5378	386	21	2,2	2,2	NUM
easat-5378	386	22	)	)	PUNCT
easat-5378	386	23	,	,	PUNCT
easat-5378	386	24	(	(	PUNCT
easat-5378	386	25	3,1	3,1	NUM
easat-5378	386	26	)	)	PUNCT
easat-5378	386	27	,	,	PUNCT
easat-5378	386	28	(	(	PUNCT
easat-5378	386	29	1,3	1,3	NUM
easat-5378	386	30	)	)	PUNCT
easat-5378	386	31	,	,	PUNCT
easat-5378	386	32	(	(	PUNCT
easat-5378	386	33	3,2	3,2	NUM
easat-5378	386	34	)	)	PUNCT
easat-5378	386	35	,	,	PUNCT
easat-5378	386	36	(	(	PUNCT
easat-5378	386	37	2,3	2,3	NUM
easat-5378	386	38	)	)	PUNCT
easat-5378	386	39	,	,	PUNCT
easat-5378	386	40	(	(	PUNCT
easat-5378	386	41	3,3	3,3	NOUN
easat-5378	386	42	)	)	PUNCT
easat-5378	386	43	,	,	PUNCT
easat-5378	386	44	…	…	PUNCT
easat-5378	386	45	.	.	PUNCT
easat-5378	387	1	when	when	SCONJ
easat-5378	387	2	𝑎	𝑎	X
easat-5378	387	3	=	=	SYM
easat-5378	387	4	2	2	NUM
easat-5378	387	5	and	and	CCONJ
easat-5378	387	6	𝑏	𝑏	NOUN
easat-5378	387	7	=	=	SYM
easat-5378	387	8	1	1	NUM
easat-5378	387	9	,	,	PUNCT
easat-5378	387	10	the	the	DET
easat-5378	387	11	(	(	PUNCT
easat-5378	387	12	𝑛(2	𝑛(2	NOUN
easat-5378	387	13	)	)	PUNCT
easat-5378	387	14	,	,	PUNCT
easat-5378	387	15	𝑘(1))-factoriangular	𝑘(1))-factoriangular	NOUN
easat-5378	387	16	numbers	number	NOUN
easat-5378	387	17	are	be	AUX
easat-5378	387	18	given	give	VERB
easat-5378	387	19	in	in	ADP
easat-5378	387	20	table	table	NOUN
easat-5378	387	21	5	5	NUM
easat-5378	387	22	.	.	PUNCT
easat-5378	387	23	902	902	NUM
easat-5378	387	24	edelweiss	edelweiss	PROPN
easat-5378	387	25	applied	apply	VERB
easat-5378	387	26	science	science	NOUN
easat-5378	387	27	and	and	CCONJ
easat-5378	387	28	technology	technology	NOUN
easat-5378	387	29	issn	issn	PROPN
easat-5378	387	30	:	:	PUNCT
easat-5378	387	31	2576	2576	NUM
easat-5378	387	32	-	-	SYM
easat-5378	387	33	8484	8484	NUM
easat-5378	387	34	vol	vol	NOUN
easat-5378	387	35	.	.	PROPN
easat-5378	388	1	9	9	NUM
easat-5378	388	2	,	,	PUNCT
easat-5378	388	3	no	no	INTJ
easat-5378	388	4	.	.	NOUN
easat-5378	389	1	3	3	NUM
easat-5378	389	2	:	:	PUNCT
easat-5378	389	3	891	891	NUM
easat-5378	389	4	-	-	SYM
easat-5378	389	5	904	904	NUM
easat-5378	389	6	,	,	PUNCT
easat-5378	389	7	2025	2025	NUM
easat-5378	389	8	doi	doi	NOUN
easat-5378	389	9	:	:	PUNCT
easat-5378	389	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	389	11	©	©	PROPN
easat-5378	389	12	2025	2025	NUM
easat-5378	389	13	by	by	ADP
easat-5378	389	14	the	the	DET
easat-5378	389	15	author	author	NOUN
easat-5378	389	16	;	;	PUNCT
easat-5378	389	17	licensee	licensee	PROPN
easat-5378	389	18	learning	learn	VERB
easat-5378	389	19	gate	gate	NOUN
easat-5378	389	20	table	table	NOUN
easat-5378	389	21	5	5	NUM
easat-5378	389	22	.	.	PUNCT
easat-5378	389	23	table	table	NOUN
easat-5378	389	24	of	of	ADP
easat-5378	389	25	(	(	PUNCT
easat-5378	389	26	n(2	n(2	PROPN
easat-5378	389	27	)	)	PUNCT
easat-5378	389	28	,	,	PUNCT
easat-5378	389	29	k(1))-factoriangular	k(1))-factoriangular	ADJ
easat-5378	389	30	numbers	number	NOUN
easat-5378	389	31	.	.	PUNCT
easat-5378	390	1	𝒏\𝒌	𝒏\𝒌	CCONJ
easat-5378	390	2	1	1	NUM
easat-5378	390	3	2	2	NUM
easat-5378	390	4	3	3	NUM
easat-5378	390	5	4	4	NUM
easat-5378	390	6	5	5	NUM
easat-5378	390	7	𝒌	𝒌	SYM
easat-5378	390	8	1	1	NUM
easat-5378	390	9	2	2	NUM
easat-5378	390	10	4	4	NUM
easat-5378	390	11	7	7	NUM
easat-5378	390	12	11	11	NUM
easat-5378	390	13	16	16	NUM
easat-5378	390	14	1	1	NUM
easat-5378	390	15	+	+	NUM
easat-5378	390	16	𝑆1(𝑘	𝑆1(𝑘	NOUN
easat-5378	390	17	)	)	PUNCT
easat-5378	390	18	2	2	NUM
easat-5378	390	19	5	5	NUM
easat-5378	390	20	7	7	NUM
easat-5378	390	21	10	10	NUM
easat-5378	390	22	14	14	NUM
easat-5378	390	23	19	19	NUM
easat-5378	390	24	4	4	NUM
easat-5378	390	25	+	+	NUM
easat-5378	390	26	𝑆1(𝑘	𝑆1(𝑘	NOUN
easat-5378	390	27	)	)	PUNCT
easat-5378	390	28	3	3	NUM
easat-5378	390	29	37	37	NUM
easat-5378	390	30	39	39	NUM
easat-5378	390	31	42	42	NUM
easat-5378	390	32	46	46	NUM
easat-5378	390	33	51	51	NUM
easat-5378	390	34	36	36	NUM
easat-5378	390	35	+	+	NUM
easat-5378	390	36	𝑆1(𝑘	𝑆1(𝑘	NOUN
easat-5378	390	37	)	)	PUNCT
easat-5378	390	38	4	4	NUM
easat-5378	390	39	577	577	NUM
easat-5378	390	40	579	579	NUM
easat-5378	390	41	582	582	NUM
easat-5378	390	42	586	586	NUM
easat-5378	390	43	591	591	NUM
easat-5378	390	44	576	576	NUM
easat-5378	390	45	+	+	NUM
easat-5378	390	46	𝑆1(𝑘	𝑆1(𝑘	NOUN
easat-5378	390	47	)	)	PUNCT
easat-5378	390	48	5	5	NUM
easat-5378	390	49	14401	14401	NUM
easat-5378	390	50	14403	14403	NUM
easat-5378	390	51	14406	14406	NUM
easat-5378	390	52	14410	14410	NUM
easat-5378	390	53	14415	14415	NUM
easat-5378	390	54	14400	14400	NUM
easat-5378	390	55	+	+	CCONJ
easat-5378	390	56	𝑆1(𝑘	𝑆1(𝑘	X
easat-5378	390	57	)	)	PUNCT
easat-5378	390	58	𝒏	𝒏	PROPN
easat-5378	390	59	(	(	PUNCT
easat-5378	390	60	𝑛!)2	𝑛!)2	PROPN
easat-5378	390	61	+	+	NUM
easat-5378	390	62	1	1	NUM
easat-5378	390	63	(	(	PUNCT
easat-5378	390	64	𝑛!)2	𝑛!)2	PROPN
easat-5378	390	65	+	+	NUM
easat-5378	390	66	3	3	NUM
easat-5378	390	67	(	(	PUNCT
easat-5378	390	68	𝑛!)2	𝑛!)2	PROPN
easat-5378	390	69	+	+	NUM
easat-5378	390	70	6	6	NUM
easat-5378	390	71	(	(	PUNCT
easat-5378	390	72	𝑛!)2	𝑛!)2	PROPN
easat-5378	390	73	+	+	NUM
easat-5378	390	74	10	10	NUM
easat-5378	390	75	(	(	PUNCT
easat-5378	390	76	𝑛!)2	𝑛!)2	PROPN
easat-5378	390	77	+	+	NUM
easat-5378	390	78	15	15	NUM
easat-5378	390	79	(	(	PUNCT
easat-5378	390	80	𝑛!)2	𝑛!)2	PROPN
easat-5378	390	81	+	+	NUM
easat-5378	390	82	𝑆1(𝑘	𝑆1(𝑘	PROPN
easat-5378	390	83	)	)	PUNCT
easat-5378	390	84	then	then	ADV
easat-5378	390	85	,	,	PUNCT
easat-5378	390	86	the	the	DET
easat-5378	390	87	sequence	sequence	NOUN
easat-5378	390	88	of	of	ADP
easat-5378	390	89	(	(	PUNCT
easat-5378	390	90	𝑛(2	𝑛(2	NOUN
easat-5378	390	91	)	)	PUNCT
easat-5378	390	92	,	,	PUNCT
easat-5378	390	93	𝑘(1))-factoriangular	𝑘(1))-factoriangular	NOUN
easat-5378	390	94	numbers	number	NOUN
easat-5378	390	95	is	be	AUX
easat-5378	390	96	given	give	VERB
easat-5378	390	97	by	by	ADP
easat-5378	390	98	{	{	PUNCT
easat-5378	390	99	𝐹𝑡𝑛(2),𝑘(1	𝐹𝑡𝑛(2),𝑘(1	NOUN
easat-5378	390	100	)	)	PUNCT
easat-5378	390	101	}	}	PUNCT
easat-5378	390	102	=	=	SYM
easat-5378	390	103	{	{	PUNCT
easat-5378	390	104	2	2	NUM
easat-5378	390	105	,	,	PUNCT
easat-5378	390	106	5	5	NUM
easat-5378	390	107	,	,	PUNCT
easat-5378	390	108	4	4	NUM
easat-5378	390	109	,	,	PUNCT
easat-5378	390	110	7	7	NUM
easat-5378	390	111	,	,	PUNCT
easat-5378	390	112	37	37	NUM
easat-5378	390	113	,	,	PUNCT
easat-5378	390	114	7	7	NUM
easat-5378	390	115	,	,	PUNCT
easat-5378	390	116	39	39	NUM
easat-5378	390	117	,	,	PUNCT
easat-5378	390	118	10	10	NUM
easat-5378	390	119	,	,	PUNCT
easat-5378	390	120	42	42	NUM
easat-5378	390	121	,	,	PUNCT
easat-5378	390	122	.	.	PUNCT
easat-5378	390	123	.	.	PUNCT
easat-5378	390	124	.	.	PUNCT
easat-5378	391	1	}	}	PUNCT
easat-5378	392	1	for	for	ADP
easat-5378	392	2	(	(	PUNCT
easat-5378	392	3	n	n	CCONJ
easat-5378	392	4	,	,	PUNCT
easat-5378	392	5	k	k	NOUN
easat-5378	392	6	)	)	PUNCT
easat-5378	392	7	=	=	SYM
easat-5378	392	8	(	(	PUNCT
easat-5378	392	9	1,1	1,1	NUM
easat-5378	392	10	)	)	PUNCT
easat-5378	392	11	,	,	PUNCT
easat-5378	392	12	(	(	PUNCT
easat-5378	392	13	2,1	2,1	NUM
easat-5378	392	14	)	)	PUNCT
easat-5378	392	15	,	,	PUNCT
easat-5378	392	16	(	(	PUNCT
easat-5378	392	17	1,2	1,2	NUM
easat-5378	392	18	)	)	PUNCT
easat-5378	392	19	,	,	PUNCT
easat-5378	392	20	(	(	PUNCT
easat-5378	392	21	2,2	2,2	NUM
easat-5378	392	22	)	)	PUNCT
easat-5378	392	23	,	,	PUNCT
easat-5378	392	24	(	(	PUNCT
easat-5378	392	25	3,1	3,1	NUM
easat-5378	392	26	)	)	PUNCT
easat-5378	392	27	,	,	PUNCT
easat-5378	392	28	(	(	PUNCT
easat-5378	392	29	1,3	1,3	NUM
easat-5378	392	30	)	)	PUNCT
easat-5378	392	31	,	,	PUNCT
easat-5378	392	32	(	(	PUNCT
easat-5378	392	33	3,2	3,2	NUM
easat-5378	392	34	)	)	PUNCT
easat-5378	392	35	,	,	PUNCT
easat-5378	392	36	(	(	PUNCT
easat-5378	392	37	2,3	2,3	NUM
easat-5378	392	38	)	)	PUNCT
easat-5378	392	39	,	,	PUNCT
easat-5378	392	40	(	(	PUNCT
easat-5378	392	41	3,3	3,3	NOUN
easat-5378	392	42	)	)	PUNCT
easat-5378	392	43	,	,	PUNCT
easat-5378	392	44	…	…	PUNCT
easat-5378	392	45	.	.	PUNCT
easat-5378	393	1	when	when	SCONJ
easat-5378	393	2	𝑎	𝑎	X
easat-5378	393	3	=	=	SYM
easat-5378	393	4	2	2	NUM
easat-5378	393	5	and	and	CCONJ
easat-5378	393	6	𝑏	𝑏	NOUN
easat-5378	393	7	=	=	SYM
easat-5378	393	8	3	3	NUM
easat-5378	393	9	,	,	PUNCT
easat-5378	393	10	the	the	DET
easat-5378	393	11	(	(	PUNCT
easat-5378	393	12	𝑛(2	𝑛(2	NOUN
easat-5378	393	13	)	)	PUNCT
easat-5378	393	14	,	,	PUNCT
easat-5378	393	15	𝑘(3))-factoriangular	𝑘(3))-factoriangular	ADJ
easat-5378	393	16	numbers	number	NOUN
easat-5378	393	17	are	be	AUX
easat-5378	393	18	given	give	VERB
easat-5378	393	19	in	in	ADP
easat-5378	393	20	table	table	NOUN
easat-5378	393	21	6	6	NUM
easat-5378	393	22	.	.	PUNCT
easat-5378	393	23	table	table	NOUN
easat-5378	393	24	6	6	NUM
easat-5378	393	25	.	.	PUNCT
easat-5378	393	26	table	table	NOUN
easat-5378	393	27	of	of	ADP
easat-5378	393	28	(	(	PUNCT
easat-5378	393	29	n(2	n(2	PROPN
easat-5378	393	30	)	)	PUNCT
easat-5378	393	31	,	,	PUNCT
easat-5378	393	32	k(3))-factoriangular	k(3))-factoriangular	ADJ
easat-5378	393	33	numbers	number	NOUN
easat-5378	393	34	𝒏\𝒌	𝒏\𝒌	PROPN
easat-5378	394	1	1	1	NUM
easat-5378	394	2	2	2	NUM
easat-5378	394	3	3	3	NUM
easat-5378	394	4	4	4	NUM
easat-5378	394	5	5	5	NUM
easat-5378	394	6	𝒌	𝒌	SYM
easat-5378	394	7	1	1	NUM
easat-5378	394	8	2	2	NUM
easat-5378	394	9	10	10	NUM
easat-5378	394	10	37	37	NUM
easat-5378	394	11	101	101	NUM
easat-5378	394	12	226	226	NUM
easat-5378	394	13	1	1	NUM
easat-5378	394	14	+	+	NUM
easat-5378	394	15	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	394	16	)	)	PUNCT
easat-5378	394	17	2	2	NUM
easat-5378	394	18	5	5	NUM
easat-5378	394	19	13	13	NUM
easat-5378	394	20	40	40	NUM
easat-5378	394	21	104	104	NUM
easat-5378	394	22	229	229	NUM
easat-5378	394	23	4	4	NUM
easat-5378	394	24	+	+	NUM
easat-5378	394	25	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	394	26	)	)	PUNCT
easat-5378	394	27	3	3	NUM
easat-5378	394	28	37	37	NUM
easat-5378	394	29	45	45	NUM
easat-5378	394	30	72	72	NUM
easat-5378	394	31	136	136	NUM
easat-5378	394	32	261	261	NUM
easat-5378	394	33	36	36	NUM
easat-5378	394	34	+	+	NUM
easat-5378	394	35	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	394	36	)	)	PUNCT
easat-5378	394	37	4	4	NUM
easat-5378	394	38	577	577	NUM
easat-5378	394	39	585	585	NUM
easat-5378	394	40	612	612	NUM
easat-5378	394	41	676	676	NUM
easat-5378	394	42	801	801	NUM
easat-5378	394	43	576	576	NUM
easat-5378	394	44	+	+	CCONJ
easat-5378	394	45	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	394	46	)	)	PUNCT
easat-5378	394	47	5	5	NUM
easat-5378	394	48	14401	14401	NUM
easat-5378	394	49	14409	14409	NUM
easat-5378	394	50	14436	14436	NUM
easat-5378	394	51	14500	14500	NUM
easat-5378	394	52	14625	14625	NUM
easat-5378	394	53	14400	14400	NUM
easat-5378	394	54	+	+	CCONJ
easat-5378	394	55	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	394	56	)	)	PUNCT
easat-5378	394	57	𝒏	𝒏	PROPN
easat-5378	394	58	(	(	PUNCT
easat-5378	394	59	𝑛!)2	𝑛!)2	PROPN
easat-5378	394	60	+	+	NUM
easat-5378	394	61	1	1	NUM
easat-5378	394	62	(	(	PUNCT
easat-5378	394	63	𝑛!)2	𝑛!)2	PROPN
easat-5378	394	64	+	+	NUM
easat-5378	394	65	9	9	NUM
easat-5378	394	66	(	(	PUNCT
easat-5378	394	67	𝑛!)2	𝑛!)2	PROPN
easat-5378	394	68	+	+	NUM
easat-5378	394	69	36	36	NUM
easat-5378	394	70	(	(	PUNCT
easat-5378	394	71	𝑛!)2	𝑛!)2	PROPN
easat-5378	394	72	+	+	NUM
easat-5378	394	73	100	100	NUM
easat-5378	394	74	(	(	PUNCT
easat-5378	394	75	𝑛!)2	𝑛!)2	PROPN
easat-5378	394	76	+	+	NUM
easat-5378	394	77	225	225	NUM
easat-5378	394	78	(	(	PUNCT
easat-5378	394	79	𝑛!)2	𝑛!)2	PROPN
easat-5378	394	80	+	+	NUM
easat-5378	394	81	𝑆3(𝑘	𝑆3(𝑘	NOUN
easat-5378	394	82	)	)	PUNCT
easat-5378	394	83	then	then	ADV
easat-5378	394	84	,	,	PUNCT
easat-5378	394	85	the	the	DET
easat-5378	394	86	sequence	sequence	NOUN
easat-5378	394	87	of	of	ADP
easat-5378	394	88	(	(	PUNCT
easat-5378	394	89	𝑛(2	𝑛(2	NOUN
easat-5378	394	90	)	)	PUNCT
easat-5378	394	91	,	,	PUNCT
easat-5378	394	92	𝑘(3))-factoriangular	𝑘(3))-factoriangular	ADJ
easat-5378	394	93	numbers	number	NOUN
easat-5378	394	94	is	be	AUX
easat-5378	394	95	given	give	VERB
easat-5378	394	96	by	by	ADP
easat-5378	394	97	{	{	PUNCT
easat-5378	394	98	𝐹𝑡𝑛(2),𝑘(3	𝐹𝑡𝑛(2),𝑘(3	NOUN
easat-5378	394	99	)	)	PUNCT
easat-5378	394	100	}	}	PUNCT
easat-5378	394	101	=	=	SYM
easat-5378	394	102	{	{	PUNCT
easat-5378	394	103	2	2	NUM
easat-5378	394	104	,	,	PUNCT
easat-5378	394	105	5	5	NUM
easat-5378	394	106	,	,	PUNCT
easat-5378	394	107	10	10	NUM
easat-5378	394	108	,	,	PUNCT
easat-5378	394	109	13	13	NUM
easat-5378	394	110	,	,	PUNCT
easat-5378	394	111	37	37	NUM
easat-5378	394	112	,	,	PUNCT
easat-5378	394	113	37	37	NUM
easat-5378	394	114	,	,	PUNCT
easat-5378	394	115	45	45	NUM
easat-5378	394	116	,	,	PUNCT
easat-5378	394	117	40	40	NUM
easat-5378	394	118	,	,	PUNCT
easat-5378	394	119	72	72	NUM
easat-5378	394	120	,	,	PUNCT
easat-5378	394	121	.	.	PUNCT
easat-5378	394	122	.	.	PUNCT
easat-5378	394	123	.	.	PUNCT
easat-5378	395	1	}	}	PUNCT
easat-5378	396	1	for	for	ADP
easat-5378	396	2	(	(	PUNCT
easat-5378	396	3	n	n	CCONJ
easat-5378	396	4	,	,	PUNCT
easat-5378	396	5	k	k	NOUN
easat-5378	396	6	)	)	PUNCT
easat-5378	396	7	=	=	SYM
easat-5378	396	8	(	(	PUNCT
easat-5378	396	9	1,1	1,1	NUM
easat-5378	396	10	)	)	PUNCT
easat-5378	396	11	,	,	PUNCT
easat-5378	396	12	(	(	PUNCT
easat-5378	396	13	2,1	2,1	NUM
easat-5378	396	14	)	)	PUNCT
easat-5378	396	15	,	,	PUNCT
easat-5378	396	16	(	(	PUNCT
easat-5378	396	17	1,2	1,2	NUM
easat-5378	396	18	)	)	PUNCT
easat-5378	396	19	,	,	PUNCT
easat-5378	396	20	(	(	PUNCT
easat-5378	396	21	2,2	2,2	NUM
easat-5378	396	22	)	)	PUNCT
easat-5378	396	23	,	,	PUNCT
easat-5378	396	24	(	(	PUNCT
easat-5378	396	25	3,1	3,1	NUM
easat-5378	396	26	)	)	PUNCT
easat-5378	396	27	,	,	PUNCT
easat-5378	396	28	(	(	PUNCT
easat-5378	396	29	1,3	1,3	NUM
easat-5378	396	30	)	)	PUNCT
easat-5378	396	31	,	,	PUNCT
easat-5378	396	32	(	(	PUNCT
easat-5378	396	33	3,2	3,2	NUM
easat-5378	396	34	)	)	PUNCT
easat-5378	396	35	,	,	PUNCT
easat-5378	396	36	(	(	PUNCT
easat-5378	396	37	2,3	2,3	NUM
easat-5378	396	38	)	)	PUNCT
easat-5378	396	39	,	,	PUNCT
easat-5378	396	40	(	(	PUNCT
easat-5378	396	41	3,3	3,3	NOUN
easat-5378	396	42	)	)	PUNCT
easat-5378	396	43	,	,	PUNCT
easat-5378	396	44	…	…	PUNCT
easat-5378	396	45	.	.	PUNCT
easat-5378	397	1	4	4	X
easat-5378	397	2	.	.	X
easat-5378	397	3	conclusions	conclusion	NOUN
easat-5378	397	4	research	research	NOUN
easat-5378	397	5	on	on	ADP
easat-5378	397	6	factoriangular	factoriangular	ADJ
easat-5378	397	7	numbers	number	NOUN
easat-5378	397	8	is	be	AUX
easat-5378	397	9	still	still	ADV
easat-5378	397	10	relatively	relatively	ADV
easat-5378	397	11	new	new	ADJ
easat-5378	397	12	with	with	ADP
easat-5378	397	13	its	its	PRON
easat-5378	397	14	introduction	introduction	NOUN
easat-5378	397	15	to	to	ADP
easat-5378	397	16	the	the	DET
easat-5378	397	17	number	number	NOUN
easat-5378	397	18	theory	theory	NOUN
easat-5378	397	19	literature	literature	NOUN
easat-5378	397	20	only	only	ADV
easat-5378	397	21	in	in	ADP
easat-5378	397	22	2015	2015	NUM
easat-5378	397	23	.	.	PUNCT
easat-5378	398	1	a	a	DET
easat-5378	398	2	factoriangular	factoriangular	ADJ
easat-5378	398	3	number	number	NOUN
easat-5378	398	4	is	be	AUX
easat-5378	398	5	formed	form	VERB
easat-5378	398	6	by	by	ADP
easat-5378	398	7	adding	add	VERB
easat-5378	398	8	a	a	DET
easat-5378	398	9	factorial	factorial	NOUN
easat-5378	398	10	and	and	CCONJ
easat-5378	398	11	its	its	PRON
easat-5378	398	12	additive	additive	ADJ
easat-5378	398	13	analog	analog	NOUN
easat-5378	398	14	,	,	PUNCT
easat-5378	398	15	a	a	DET
easat-5378	398	16	triangular	triangular	ADJ
easat-5378	398	17	number	number	NOUN
easat-5378	398	18	.	.	PUNCT
easat-5378	399	1	when	when	SCONJ
easat-5378	399	2	a	a	DET
easat-5378	399	3	triangular	triangular	NOUN
easat-5378	399	4	number	number	NOUN
easat-5378	399	5	is	be	AUX
easat-5378	399	6	added	add	VERB
easat-5378	399	7	to	to	ADP
easat-5378	399	8	its	its	PRON
easat-5378	399	9	corresponding	correspond	VERB
easat-5378	399	10	factorial	factorial	NOUN
easat-5378	399	11	,	,	PUNCT
easat-5378	399	12	the	the	DET
easat-5378	399	13	result	result	NOUN
easat-5378	399	14	is	be	AUX
easat-5378	399	15	an	an	DET
easat-5378	399	16	n	n	CCONJ
easat-5378	399	17	-	-	PUNCT
easat-5378	399	18	factoriangular	factoriangular	ADJ
easat-5378	399	19	number	number	NOUN
easat-5378	399	20	.	.	PUNCT
easat-5378	400	1	the	the	DET
easat-5378	400	2	sequence	sequence	NOUN
easat-5378	400	3	{	{	PUNCT
easat-5378	400	4	𝐹𝑡𝑛	𝐹𝑡𝑛	PROPN
easat-5378	400	5	}	}	PUNCT
easat-5378	400	6	=	=	SYM
easat-5378	400	7	{	{	PUNCT
easat-5378	400	8	2	2	NUM
easat-5378	400	9	,	,	PUNCT
easat-5378	400	10	5	5	NUM
easat-5378	400	11	,	,	PUNCT
easat-5378	400	12	12	12	NUM
easat-5378	400	13	,	,	PUNCT
easat-5378	400	14	34	34	NUM
easat-5378	400	15	,	,	PUNCT
easat-5378	400	16	135	135	NUM
easat-5378	400	17	,	,	PUNCT
easat-5378	400	18	741	741	NUM
easat-5378	400	19	,	,	PUNCT
easat-5378	400	20	5068	5068	NUM
easat-5378	400	21	,	,	PUNCT
easat-5378	400	22	…	…	PUNCT
easat-5378	400	23	}	}	PUNCT
easat-5378	400	24	is	be	AUX
easat-5378	400	25	the	the	DET
easat-5378	400	26	sequence	sequence	NOUN
easat-5378	400	27	of	of	ADP
easat-5378	400	28	n	n	CCONJ
easat-5378	400	29	-	-	PUNCT
easat-5378	400	30	factoriangular	factoriangular	ADJ
easat-5378	400	31	numbers	number	NOUN
easat-5378	400	32	.	.	PUNCT
easat-5378	401	1	the	the	DET
easat-5378	401	2	terms	term	NOUN
easat-5378	401	3	in	in	ADP
easat-5378	401	4	the	the	DET
easat-5378	401	5	sequence	sequence	NOUN
easat-5378	401	6	can	can	AUX
easat-5378	401	7	be	be	AUX
easat-5378	401	8	generated	generate	VERB
easat-5378	401	9	by	by	ADP
easat-5378	401	10	using	use	VERB
easat-5378	401	11	the	the	DET
easat-5378	401	12	formula	formula	NOUN
easat-5378	401	13	𝐹𝑡𝑛	𝐹𝑡𝑛	PROPN
easat-5378	401	14	=	=	SYM
easat-5378	401	15	𝑛	𝑛	PROPN
easat-5378	401	16	!	!	PUNCT
easat-5378	402	1	+	+	CCONJ
easat-5378	403	1	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	403	2	,	,	PUNCT
easat-5378	403	3	where	where	SCONJ
easat-5378	403	4	𝑛	𝑛	X
easat-5378	403	5	!	!	PUNCT
easat-5378	403	6	=	=	SYM
easat-5378	403	7	1	1	NUM
easat-5378	403	8	⋅	⋅	NUM
easat-5378	403	9	2	2	NUM
easat-5378	403	10	⋅	⋅	X
easat-5378	403	11	3	3	NUM
easat-5378	403	12	⋅⋅⋅	⋅⋅⋅	X
easat-5378	403	13	𝑛	𝑛	PROPN
easat-5378	403	14	and	and	CCONJ
easat-5378	403	15	𝑇𝑛	𝑇𝑛	PROPN
easat-5378	403	16	=	=	NOUN
easat-5378	403	17	1	1	NUM
easat-5378	403	18	+	+	NUM
easat-5378	403	19	2	2	NUM
easat-5378	403	20	+	+	SYM
easat-5378	403	21	3	3	NUM
easat-5378	403	22	+	+	NOUN
easat-5378	403	23	.	.	PUNCT
easat-5378	403	24	.	.	PUNCT
easat-5378	403	25	.	.	PUNCT
easat-5378	404	1	+	+	PUNCT
easat-5378	404	2	𝑛.	𝑛.	NOUN
easat-5378	404	3	this	this	DET
easat-5378	404	4	sequence	sequence	NOUN
easat-5378	404	5	of	of	ADP
easat-5378	404	6	n	n	CCONJ
easat-5378	404	7	-	-	PUNCT
easat-5378	404	8	factoriangular	factoriangular	ADJ
easat-5378	404	9	numbers	number	NOUN
easat-5378	404	10	can	can	AUX
easat-5378	404	11	be	be	AUX
easat-5378	404	12	generalized	generalize	VERB
easat-5378	404	13	in	in	ADP
easat-5378	404	14	several	several	ADJ
easat-5378	404	15	ways	way	NOUN
easat-5378	404	16	.	.	PUNCT
easat-5378	405	1	one	one	NUM
easat-5378	405	2	generalization	generalization	NOUN
easat-5378	405	3	is	be	AUX
easat-5378	405	4	the	the	DET
easat-5378	405	5	sequence	sequence	NOUN
easat-5378	405	6	of	of	ADP
easat-5378	405	7	(	(	PUNCT
easat-5378	405	8	𝑛	𝑛	ADJ
easat-5378	405	9	,	,	PUNCT
easat-5378	405	10	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	405	11	numbers	number	NOUN
easat-5378	405	12	of	of	ADP
easat-5378	405	13	the	the	DET
easat-5378	405	14	form	form	NOUN
easat-5378	405	15	𝐹𝑡𝑛,𝑘	𝐹𝑡𝑛,𝑘	NUM
easat-5378	405	16	=	=	SYM
easat-5378	405	17	𝑛	𝑛	NOUN
easat-5378	405	18	!	!	PUNCT
easat-5378	406	1	+	+	PUNCT
easat-5378	407	1	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	407	2	,	,	PUNCT
easat-5378	407	3	where	where	SCONJ
easat-5378	407	4	𝑛	𝑛	X
easat-5378	407	5	!	!	PUNCT
easat-5378	407	6	=	=	SYM
easat-5378	408	1	1	1	NUM
easat-5378	408	2	⋅	⋅	NUM
easat-5378	408	3	2	2	NUM
easat-5378	408	4	⋅	⋅	X
easat-5378	408	5	3	3	NUM
easat-5378	408	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	408	7	𝑛	𝑛	PROPN
easat-5378	408	8	and	and	CCONJ
easat-5378	408	9	𝑇𝑘	𝑇𝑘	PROPN
easat-5378	408	10	=	=	SYM
easat-5378	408	11	1	1	NUM
easat-5378	408	12	+	+	NUM
easat-5378	408	13	2	2	NUM
easat-5378	408	14	+	+	SYM
easat-5378	408	15	3	3	NUM
easat-5378	408	16	+	+	NOUN
easat-5378	408	17	.	.	PUNCT
easat-5378	408	18	.	.	PUNCT
easat-5378	408	19	.	.	PUNCT
easat-5378	409	1	+	+	ADP
easat-5378	409	2	𝑘.	𝑘.	NOUN
easat-5378	409	3	another	another	DET
easat-5378	409	4	generalization	generalization	NOUN
easat-5378	409	5	is	be	AUX
easat-5378	409	6	the	the	DET
easat-5378	409	7	sequence	sequence	NOUN
easat-5378	409	8	of	of	ADP
easat-5378	409	9	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	409	10	numbers	number	NOUN
easat-5378	409	11	of	of	ADP
easat-5378	409	12	the	the	DET
easat-5378	409	13	form	form	NOUN
easat-5378	409	14	𝐹𝑡𝑛(𝑚	𝐹𝑡𝑛(𝑚	ADJ
easat-5378	409	15	)	)	PUNCT
easat-5378	409	16	=	=	SYM
easat-5378	409	17	(	(	PUNCT
easat-5378	409	18	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	409	19	+	+	CCONJ
easat-5378	409	20	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NOUN
easat-5378	409	21	)	)	PUNCT
easat-5378	409	22	,	,	PUNCT
easat-5378	409	23	where	where	SCONJ
easat-5378	409	24	(	(	PUNCT
easat-5378	409	25	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	409	26	=	=	SYM
easat-5378	409	27	1𝑚	1𝑚	NUM
easat-5378	409	28	⋅	⋅	PROPN
easat-5378	409	29	2𝑚	2𝑚	NOUN
easat-5378	409	30	⋅	⋅	PROPN
easat-5378	409	31	3𝑚	3𝑚	NUM
easat-5378	409	32	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	409	33	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	409	34	and	and	CCONJ
easat-5378	409	35	𝑆𝑚(𝑛	𝑆𝑚(𝑛	NUM
easat-5378	409	36	)	)	PUNCT
easat-5378	409	37	=	=	SYM
easat-5378	409	38	1𝑚	1𝑚	NOUN
easat-5378	409	39	+	+	CCONJ
easat-5378	409	40	2𝑚	2𝑚	NOUN
easat-5378	409	41	+	+	CCONJ
easat-5378	409	42	3𝑚+	3𝑚+	NUM
easat-5378	409	43	.	.	PUNCT
easat-5378	409	44	.	.	PUNCT
easat-5378	409	45	.	.	PUNCT
easat-5378	410	1	+	+	PUNCT
easat-5378	410	2	𝑛𝑚.	𝑛𝑚.	X
easat-5378	410	3	more	more	ADJ
easat-5378	410	4	generalizations	generalization	NOUN
easat-5378	410	5	can	can	AUX
easat-5378	410	6	be	be	AUX
easat-5378	410	7	made	make	VERB
easat-5378	410	8	by	by	ADP
easat-5378	410	9	integrating	integrate	VERB
easat-5378	410	10	the	the	DET
easat-5378	410	11	concepts	concept	NOUN
easat-5378	410	12	of	of	ADP
easat-5378	410	13	(	(	PUNCT
easat-5378	410	14	𝑛	𝑛	ADJ
easat-5378	410	15	,	,	PUNCT
easat-5378	410	16	𝑘)-factoriangular	𝑘)-factoriangular	ADJ
easat-5378	410	17	numbers	number	NOUN
easat-5378	410	18	and	and	CCONJ
easat-5378	410	19	𝑛(𝑚)-factoriangular	𝑛(𝑚)-factoriangular	PROPN
easat-5378	410	20	numbers	number	NOUN
easat-5378	410	21	to	to	PART
easat-5378	410	22	produce	produce	VERB
easat-5378	410	23	the	the	DET
easat-5378	410	24	sequence	sequence	NOUN
easat-5378	410	25	of	of	ADP
easat-5378	410	26	(	(	PUNCT
easat-5378	410	27	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	410	28	)	)	PUNCT
easat-5378	410	29	,	,	PUNCT
easat-5378	410	30	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	410	31	numbers	number	NOUN
easat-5378	410	32	of	of	ADP
easat-5378	410	33	the	the	DET
easat-5378	410	34	form	form	NOUN
easat-5378	410	35	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	410	36	)	)	PUNCT
easat-5378	410	37	=	=	SYM
easat-5378	410	38	(	(	PUNCT
easat-5378	410	39	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	410	40	+	+	CCONJ
easat-5378	410	41	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	410	42	)	)	PUNCT
easat-5378	410	43	,	,	PUNCT
easat-5378	410	44	where	where	SCONJ
easat-5378	410	45	(	(	PUNCT
easat-5378	410	46	𝑛!)𝑚	𝑛!)𝑚	NOUN
easat-5378	410	47	=	=	SYM
easat-5378	410	48	1𝑚	1𝑚	NUM
easat-5378	410	49	⋅	⋅	PROPN
easat-5378	410	50	2𝑚	2𝑚	NOUN
easat-5378	410	51	⋅	⋅	PROPN
easat-5378	410	52	3𝑚	3𝑚	NUM
easat-5378	410	53	⋅⋅⋅	⋅⋅⋅	PROPN
easat-5378	410	54	𝑛𝑚	𝑛𝑚	NOUN
easat-5378	410	55	and	and	CCONJ
easat-5378	410	56	𝑆𝑚(𝑘	𝑆𝑚(𝑘	NOUN
easat-5378	410	57	)	)	PUNCT
easat-5378	410	58	=	=	SYM
easat-5378	410	59	1𝑚	1𝑚	NOUN
easat-5378	411	1	+	+	CCONJ
easat-5378	411	2	2𝑚	2𝑚	NOUN
easat-5378	411	3	+	+	CCONJ
easat-5378	411	4	3𝑚+	3𝑚+	NUM
easat-5378	411	5	.	.	PUNCT
easat-5378	411	6	.	.	PUNCT
easat-5378	411	7	.	.	PUNCT
easat-5378	412	1	+	+	X
easat-5378	412	2	𝑘𝑚.	𝑘𝑚.	VERB
easat-5378	412	3	the	the	DET
easat-5378	412	4	sequence	sequence	NOUN
easat-5378	412	5	of	of	ADP
easat-5378	412	6	(	(	PUNCT
easat-5378	412	7	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	412	8	)	)	PUNCT
easat-5378	412	9	,	,	PUNCT
easat-5378	412	10	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	412	11	numbers	number	NOUN
easat-5378	412	12	can	can	AUX
easat-5378	412	13	be	be	AUX
easat-5378	412	14	further	far	ADV
easat-5378	412	15	generalized	generalize	VERB
easat-5378	412	16	into	into	ADP
easat-5378	412	17	the	the	DET
easat-5378	412	18	sequence	sequence	NOUN
easat-5378	412	19	of	of	ADP
easat-5378	412	20	(	(	PUNCT
easat-5378	412	21	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	412	22	)	)	PUNCT
easat-5378	412	23	,	,	PUNCT
easat-5378	412	24	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	412	25	numbers	number	NOUN
easat-5378	412	26	of	of	ADP
easat-5378	412	27	the	the	DET
easat-5378	412	28	form	form	NOUN
easat-5378	412	29	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	412	30	)	)	PUNCT
easat-5378	412	31	=	=	PUNCT
easat-5378	412	32	(	(	PUNCT
easat-5378	412	33	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	412	34	+	+	CCONJ
easat-5378	412	35	𝑆𝑏(𝑘	𝑆𝑏(𝑘	NOUN
easat-5378	412	36	)	)	PUNCT
easat-5378	412	37	,	,	PUNCT
easat-5378	412	38	where	where	SCONJ
easat-5378	412	39	(	(	PUNCT
easat-5378	412	40	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	412	41	=	=	SYM
easat-5378	413	1	1𝑎	1𝑎	NUM
easat-5378	413	2	⋅	⋅	NUM
easat-5378	413	3	2𝑎	2𝑎	NUM
easat-5378	413	4	⋅	⋅	PROPN
easat-5378	413	5	3𝑎	3𝑎	NUM
easat-5378	413	6	⋅⋅⋅	⋅⋅⋅	X
easat-5378	413	7	𝑛𝑎	𝑛𝑎	PROPN
easat-5378	413	8	and	and	CCONJ
easat-5378	413	9	𝑆𝑏(𝑘	𝑆𝑏(𝑘	NOUN
easat-5378	413	10	)	)	PUNCT
easat-5378	414	1	=	=	SYM
easat-5378	414	2	1𝑏	1𝑏	NOUN
easat-5378	414	3	+	+	CCONJ
easat-5378	414	4	2𝑏	2𝑏	NUM
easat-5378	414	5	+	+	X
easat-5378	414	6	3𝑏+	3𝑏+	NUM
easat-5378	414	7	.	.	PUNCT
easat-5378	414	8	.	.	PUNCT
easat-5378	414	9	.	.	PUNCT
easat-5378	415	1	+	+	PUNCT
easat-5378	415	2	𝑘𝑏.	𝑘𝑏.	ADJ
easat-5378	415	3	903	903	NUM
easat-5378	415	4	edelweiss	edelweiss	PROPN
easat-5378	415	5	applied	apply	VERB
easat-5378	415	6	science	science	NOUN
easat-5378	415	7	and	and	CCONJ
easat-5378	415	8	technology	technology	NOUN
easat-5378	415	9	issn	issn	PROPN
easat-5378	415	10	:	:	PUNCT
easat-5378	415	11	2576	2576	NUM
easat-5378	415	12	-	-	SYM
easat-5378	415	13	8484	8484	NUM
easat-5378	415	14	vol	vol	NOUN
easat-5378	415	15	.	.	PROPN
easat-5378	416	1	9	9	NUM
easat-5378	416	2	,	,	PUNCT
easat-5378	416	3	no	no	INTJ
easat-5378	416	4	.	.	NOUN
easat-5378	417	1	3	3	NUM
easat-5378	417	2	:	:	PUNCT
easat-5378	417	3	891	891	NUM
easat-5378	417	4	-	-	SYM
easat-5378	417	5	904	904	NUM
easat-5378	417	6	,	,	PUNCT
easat-5378	417	7	2025	2025	NUM
easat-5378	417	8	doi	doi	NOUN
easat-5378	417	9	:	:	PUNCT
easat-5378	417	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	417	11	©	©	PROPN
easat-5378	417	12	2025	2025	NUM
easat-5378	417	13	by	by	ADP
easat-5378	417	14	the	the	DET
easat-5378	417	15	author	author	NOUN
easat-5378	417	16	;	;	PUNCT
easat-5378	417	17	licensee	licensee	PROPN
easat-5378	417	18	learning	learning	NOUN
easat-5378	417	19	gate	gate	VERB
easat-5378	417	20	the	the	DET
easat-5378	417	21	(	(	PUNCT
easat-5378	417	22	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	417	23	)	)	PUNCT
easat-5378	417	24	,	,	PUNCT
easat-5378	417	25	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	417	26	numbers	number	NOUN
easat-5378	417	27	can	can	AUX
easat-5378	417	28	also	also	ADV
easat-5378	417	29	be	be	AUX
easat-5378	417	30	determined	determine	VERB
easat-5378	417	31	by	by	ADP
easat-5378	417	32	the	the	DET
easat-5378	417	33	formula	formula	NOUN
easat-5378	417	34	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	417	35	)	)	PUNCT
easat-5378	418	1	=	=	NOUN
easat-5378	418	2	(	(	PUNCT
easat-5378	418	3	𝑛!)𝑚	𝑛!)𝑚	VERB
easat-5378	418	4	+	+	CCONJ
easat-5378	418	5	1	1	NUM
easat-5378	418	6	𝑚+1	𝑚+1	PART
easat-5378	419	1	[	[	X
easat-5378	419	2	(	(	PUNCT
easat-5378	419	3	𝑘	𝑘	X
easat-5378	419	4	+	+	X
easat-5378	419	5	1)[(𝑘	1)[(𝑘	NUM
easat-5378	419	6	+	+	CCONJ
easat-5378	419	7	1)𝑚	1)𝑚	NUM
easat-5378	419	8	−	−	NOUN
easat-5378	419	9	1	1	NUM
easat-5378	419	10	]	]	PUNCT
easat-5378	419	11	−	−	PUNCT
easat-5378	419	12	∑	∑	INTJ
easat-5378	419	13	(	(	PUNCT
easat-5378	419	14	𝑚	𝑚	PROPN
easat-5378	419	15	+	+	PROPN
easat-5378	419	16	1	1	NUM
easat-5378	419	17	𝑖	𝑖	X
easat-5378	419	18	)	)	PUNCT
easat-5378	419	19	𝑚−1	𝑚−1	PROPN
easat-5378	419	20	𝑖=1	𝑖=1	PUNCT
easat-5378	419	21	𝑆𝑖(𝑘	𝑆𝑖(𝑘	PROPN
easat-5378	419	22	)	)	PUNCT
easat-5378	419	23	]	]	PUNCT
easat-5378	419	24	and	and	CCONJ
easat-5378	419	25	the	the	DET
easat-5378	419	26	(	(	PUNCT
easat-5378	419	27	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	419	28	)	)	PUNCT
easat-5378	419	29	,	,	PUNCT
easat-5378	419	30	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	419	31	numbers	number	NOUN
easat-5378	419	32	by	by	ADP
easat-5378	419	33	the	the	DET
easat-5378	419	34	formula	formula	NOUN
easat-5378	419	35	𝐹𝑡𝑛(𝑎),𝑘(𝑏	𝐹𝑡𝑛(𝑎),𝑘(𝑏	NOUN
easat-5378	419	36	)	)	PUNCT
easat-5378	419	37	=	=	PUNCT
easat-5378	419	38	(	(	PUNCT
easat-5378	419	39	𝑛!)𝑎	𝑛!)𝑎	X
easat-5378	419	40	+	+	X
easat-5378	419	41	1	1	NUM
easat-5378	419	42	𝑏+1	𝑏+1	X
easat-5378	419	43	[	[	X
easat-5378	419	44	(	(	PUNCT
easat-5378	419	45	𝑘	𝑘	X
easat-5378	419	46	+	+	X
easat-5378	419	47	1)[(𝑘	1)[(𝑘	NUM
easat-5378	419	48	+	+	CCONJ
easat-5378	419	49	1)𝑏	1)𝑏	NUM
easat-5378	419	50	−	−	ADP
easat-5378	419	51	1	1	NUM
easat-5378	419	52	]	]	PUNCT
easat-5378	419	53	−	−	PUNCT
easat-5378	419	54	∑	∑	PROPN
easat-5378	419	55	(	(	PUNCT
easat-5378	419	56	𝑏	𝑏	PROPN
easat-5378	419	57	+	+	NOUN
easat-5378	419	58	1	1	NUM
easat-5378	419	59	𝑖	𝑖	NOUN
easat-5378	419	60	)	)	PUNCT
easat-5378	419	61	𝑏−1	𝑏−1	NOUN
easat-5378	419	62	𝑖=1	𝑖=1	PUNCT
easat-5378	419	63	𝑆𝑖(𝑘	𝑆𝑖(𝑘	PROPN
easat-5378	419	64	)	)	PUNCT
easat-5378	419	65	]	]	PUNCT
easat-5378	419	66	.	.	PUNCT
easat-5378	420	1	the	the	DET
easat-5378	420	2	formulas	formula	NOUN
easat-5378	420	3	for	for	ADP
easat-5378	420	4	(	(	PUNCT
easat-5378	420	5	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	420	6	)	)	PUNCT
easat-5378	420	7	,	,	PUNCT
easat-5378	420	8	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	420	9	number	number	NOUN
easat-5378	420	10	for	for	ADP
easat-5378	420	11	even	even	ADV
easat-5378	420	12	𝑚	𝑚	NOUN
easat-5378	420	13	=	=	NOUN
easat-5378	420	14	2𝑗	2𝑗	NOUN
easat-5378	420	15	are	be	AUX
easat-5378	420	16	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	420	17	)	)	PUNCT
easat-5378	420	18	=	=	SYM
easat-5378	420	19	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	PROPN
easat-5378	420	20	)	)	PUNCT
easat-5378	420	21	=	=	PUNCT
easat-5378	421	1	(	(	PUNCT
easat-5378	421	2	𝑛!)2𝑗	𝑛!)2𝑗	NUM
easat-5378	421	3	+	+	NUM
easat-5378	421	4	2𝑘+1	2𝑘+1	NUM
easat-5378	421	5	2𝑗+1	2𝑗+1	NUM
easat-5378	422	1	[	[	X
easat-5378	422	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	422	3	+	+	NUM
easat-5378	422	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	422	5	and	and	CCONJ
easat-5378	422	6	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	422	7	)	)	PUNCT
easat-5378	422	8	=	=	SYM
easat-5378	422	9	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	𝐹𝑡𝑛(2𝑗),𝑘(2𝑗	PROPN
easat-5378	422	10	)	)	PUNCT
easat-5378	422	11	=	=	PUNCT
easat-5378	423	1	(	(	PUNCT
easat-5378	423	2	𝑛!)2𝑗	𝑛!)2𝑗	NUM
easat-5378	423	3	+	+	CCONJ
easat-5378	423	4	3	3	NUM
easat-5378	423	5	2𝑗+1	2𝑗+1	PROPN
easat-5378	424	1	[	[	X
easat-5378	424	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	424	3	+	+	NUM
easat-5378	424	4	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	𝑃(𝑘2𝑗−3)]𝑆2(𝑘	NOUN
easat-5378	424	5	)	)	PUNCT
easat-5378	424	6	.	.	PUNCT
easat-5378	425	1	the	the	DET
easat-5378	425	2	formulas	formula	NOUN
easat-5378	425	3	for	for	ADP
easat-5378	425	4	(	(	PUNCT
easat-5378	425	5	𝑛(𝑚	𝑛(𝑚	NOUN
easat-5378	425	6	)	)	PUNCT
easat-5378	425	7	,	,	PUNCT
easat-5378	425	8	𝑘(𝑚))-factoriangular	𝑘(𝑚))-factoriangular	ADJ
easat-5378	425	9	number	number	NOUN
easat-5378	425	10	for	for	ADP
easat-5378	425	11	odd	odd	ADJ
easat-5378	425	12	𝑚	𝑚	NOUN
easat-5378	425	13	=	=	SYM
easat-5378	425	14	2𝑗	2𝑗	NOUN
easat-5378	425	15	+	+	CCONJ
easat-5378	425	16	1	1	NUM
easat-5378	425	17	are	be	AUX
easat-5378	425	18	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	NOUN
easat-5378	425	19	)	)	PUNCT
easat-5378	425	20	=	=	SYM
easat-5378	425	21	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	NUM
easat-5378	425	22	)	)	PUNCT
easat-5378	425	23	=	=	NOUN
easat-5378	425	24	(	(	PUNCT
easat-5378	425	25	𝑛!)2𝑗+1	𝑛!)2𝑗+1	PROPN
easat-5378	425	26	+	+	NUM
easat-5378	425	27	𝑘(𝑘+1	𝑘(𝑘+1	NOUN
easat-5378	425	28	)	)	PUNCT
easat-5378	425	29	𝑗+1	𝑗+1	PUNCT
easat-5378	426	1	[	[	X
easat-5378	426	2	𝑘2𝑗−2	𝑘2𝑗−2	X
easat-5378	426	3	+	+	NUM
easat-5378	426	4	𝑃(𝑘2𝑗−3)]𝑇𝑘	𝑃(𝑘2𝑗−3)]𝑇𝑘	NOUN
easat-5378	426	5	and	and	CCONJ
easat-5378	426	6	𝐹𝑡𝑛(𝑚),𝑘(𝑚	𝐹𝑡𝑛(𝑚),𝑘(𝑚	PROPN
easat-5378	426	7	)	)	PUNCT
easat-5378	426	8	=	=	SYM
easat-5378	426	9	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	𝐹𝑡𝑛(2𝑗+1),𝑘(2𝑗+1	NUM
easat-5378	426	10	)	)	PUNCT
easat-5378	426	11	=	=	NOUN
easat-5378	426	12	(	(	PUNCT
easat-5378	427	1	𝑛!)2𝑗+1	𝑛!)2𝑗+1	NUM
easat-5378	427	2	+	+	CCONJ
easat-5378	427	3	2	2	NUM
easat-5378	427	4	𝑗+1	𝑗+1	PUNCT
easat-5378	428	1	[	[	X
easat-5378	428	2	𝑘2𝑗−2	𝑘2𝑗−2	INTJ
easat-5378	428	3	+	+	NOUN
easat-5378	428	4	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	𝑃(𝑘2𝑗−3)]𝑆3(𝑘	PROPN
easat-5378	428	5	)	)	PUNCT
easat-5378	428	6	.	.	PUNCT
easat-5378	429	1	similar	similar	ADJ
easat-5378	429	2	formulas	formula	NOUN
easat-5378	429	3	can	can	AUX
easat-5378	429	4	be	be	AUX
easat-5378	429	5	provided	provide	VERB
easat-5378	429	6	for	for	ADP
easat-5378	429	7	(	(	PUNCT
easat-5378	429	8	𝑛(𝑎	𝑛(𝑎	PROPN
easat-5378	429	9	)	)	PUNCT
easat-5378	429	10	,	,	PUNCT
easat-5378	429	11	𝑘(𝑏))-factoriangular	𝑘(𝑏))-factoriangular	PROPN
easat-5378	429	12	number	number	NOUN
easat-5378	429	13	for	for	ADP
easat-5378	429	14	even	even	ADV
easat-5378	429	15	𝑚	𝑚	NOUN
easat-5378	429	16	=	=	SYM
easat-5378	429	17	2𝑗	2𝑗	NOUN
easat-5378	429	18	and	and	CCONJ
easat-5378	429	19	for	for	ADP
easat-5378	429	20	odd	odd	ADJ
easat-5378	429	21	𝑚	𝑚	NOUN
easat-5378	429	22	=	=	SYM
easat-5378	429	23	2𝑗	2𝑗	NOUN
easat-5378	429	24	+	+	NOUN
easat-5378	429	25	1	1	X
easat-5378	429	26	.	.	X
easat-5378	429	27	triangular	triangular	NOUN
easat-5378	429	28	arrays	array	NOUN
easat-5378	429	29	of	of	ADP
easat-5378	429	30	factoriangular	factoriangular	ADJ
easat-5378	429	31	numbers	number	NOUN
easat-5378	429	32	may	may	AUX
easat-5378	429	33	be	be	AUX
easat-5378	429	34	formed	form	VERB
easat-5378	429	35	from	from	ADP
easat-5378	429	36	the	the	DET
easat-5378	429	37	tables	table	NOUN
easat-5378	429	38	of	of	ADP
easat-5378	429	39	generalized	generalized	ADJ
easat-5378	429	40	factoriangular	factoriangular	ADJ
easat-5378	429	41	numbers	number	NOUN
easat-5378	429	42	.	.	PUNCT
easat-5378	430	1	this	this	PRON
easat-5378	430	2	will	will	AUX
easat-5378	430	3	be	be	AUX
easat-5378	430	4	of	of	ADP
easat-5378	430	5	future	future	ADJ
easat-5378	430	6	interest	interest	NOUN
easat-5378	430	7	to	to	ADP
easat-5378	430	8	other	other	ADJ
easat-5378	430	9	mathematical	mathematical	ADJ
easat-5378	430	10	explorers	explorer	NOUN
easat-5378	430	11	,	,	PUNCT
easat-5378	430	12	especially	especially	ADV
easat-5378	430	13	those	those	PRON
easat-5378	430	14	in	in	ADP
easat-5378	430	15	the	the	DET
easat-5378	430	16	field	field	NOUN
easat-5378	430	17	of	of	ADP
easat-5378	430	18	recreational	recreational	ADJ
easat-5378	430	19	mathematics	mathematic	NOUN
easat-5378	430	20	.	.	PUNCT
easat-5378	431	1	transparency	transparency	NOUN
easat-5378	431	2	:	:	PUNCT
easat-5378	431	3	the	the	DET
easat-5378	431	4	author	author	NOUN
easat-5378	431	5	confirms	confirm	VERB
easat-5378	431	6	that	that	SCONJ
easat-5378	431	7	the	the	DET
easat-5378	431	8	manuscript	manuscript	NOUN
easat-5378	431	9	is	be	AUX
easat-5378	431	10	an	an	DET
easat-5378	431	11	honest	honest	ADJ
easat-5378	431	12	,	,	PUNCT
easat-5378	431	13	accurate	accurate	ADJ
easat-5378	431	14	,	,	PUNCT
easat-5378	431	15	and	and	CCONJ
easat-5378	431	16	transparent	transparent	ADJ
easat-5378	431	17	account	account	NOUN
easat-5378	431	18	of	of	ADP
easat-5378	431	19	the	the	DET
easat-5378	431	20	study	study	NOUN
easat-5378	431	21	;	;	PUNCT
easat-5378	431	22	that	that	SCONJ
easat-5378	431	23	no	no	DET
easat-5378	431	24	vital	vital	ADJ
easat-5378	431	25	features	feature	NOUN
easat-5378	431	26	of	of	ADP
easat-5378	431	27	the	the	DET
easat-5378	431	28	study	study	NOUN
easat-5378	431	29	have	have	AUX
easat-5378	431	30	been	be	AUX
easat-5378	431	31	omitted	omit	VERB
easat-5378	431	32	;	;	PUNCT
easat-5378	431	33	and	and	CCONJ
easat-5378	431	34	that	that	SCONJ
easat-5378	431	35	any	any	DET
easat-5378	431	36	discrepancies	discrepancy	NOUN
easat-5378	431	37	from	from	ADP
easat-5378	431	38	the	the	DET
easat-5378	431	39	study	study	NOUN
easat-5378	431	40	as	as	SCONJ
easat-5378	431	41	planned	plan	VERB
easat-5378	431	42	have	have	AUX
easat-5378	431	43	been	be	AUX
easat-5378	431	44	explained	explain	VERB
easat-5378	431	45	.	.	PUNCT
easat-5378	432	1	this	this	DET
easat-5378	432	2	study	study	NOUN
easat-5378	432	3	followed	follow	VERB
easat-5378	432	4	all	all	DET
easat-5378	432	5	ethical	ethical	ADJ
easat-5378	432	6	practices	practice	NOUN
easat-5378	432	7	during	during	ADP
easat-5378	432	8	writing	writing	NOUN
easat-5378	432	9	.	.	PUNCT
easat-5378	433	1	copyright	copyright	NOUN
easat-5378	433	2	:	:	PUNCT
easat-5378	433	3	©	©	PROPN
easat-5378	433	4	2025	2025	NUM
easat-5378	433	5	by	by	ADP
easat-5378	433	6	the	the	DET
easat-5378	433	7	authors	author	NOUN
easat-5378	433	8	.	.	PUNCT
easat-5378	434	1	this	this	DET
easat-5378	434	2	open	open	ADJ
easat-5378	434	3	-	-	PUNCT
easat-5378	434	4	access	access	NOUN
easat-5378	434	5	article	article	NOUN
easat-5378	434	6	is	be	AUX
easat-5378	434	7	distributed	distribute	VERB
easat-5378	434	8	under	under	ADP
easat-5378	434	9	the	the	DET
easat-5378	434	10	terms	term	NOUN
easat-5378	434	11	and	and	CCONJ
easat-5378	434	12	conditions	condition	NOUN
easat-5378	434	13	of	of	ADP
easat-5378	434	14	the	the	DET
easat-5378	434	15	creative	creative	ADJ
easat-5378	434	16	commons	common	NOUN
easat-5378	434	17	attribution	attribution	NOUN
easat-5378	434	18	(	(	PUNCT
easat-5378	434	19	cc	cc	NOUN
easat-5378	434	20	by	by	ADP
easat-5378	434	21	)	)	PUNCT
easat-5378	434	22	license	license	NOUN
easat-5378	434	23	(	(	PUNCT
easat-5378	434	24	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-5378	434	25	)	)	PUNCT
easat-5378	434	26	.	.	PUNCT
easat-5378	435	1	references	reference	NOUN
easat-5378	435	2	[	[	X
easat-5378	435	3	1	1	NUM
easat-5378	435	4	]	]	PUNCT
easat-5378	435	5	a.	a.	NOUN
easat-5378	435	6	s.	s.	PROPN
easat-5378	435	7	garge	garge	PROPN
easat-5378	435	8	and	and	CCONJ
easat-5378	435	9	s.	s.	PROPN
easat-5378	435	10	a.	a.	PROPN
easat-5378	435	11	shirali	shirali	PROPN
easat-5378	435	12	,	,	PUNCT
easat-5378	435	13	"	"	PUNCT
easat-5378	435	14	triangular	triangular	NOUN
easat-5378	435	15	numbers	number	NOUN
easat-5378	435	16	,	,	PUNCT
easat-5378	435	17	"	"	PUNCT
easat-5378	435	18	resonance	resonance	NOUN
easat-5378	435	19	,	,	PUNCT
easat-5378	435	20	vol	vol	NOUN
easat-5378	435	21	.	.	PROPN
easat-5378	436	1	17	17	NUM
easat-5378	436	2	,	,	PUNCT
easat-5378	436	3	pp	pp	ADJ
easat-5378	436	4	.	.	PUNCT
easat-5378	437	1	672	672	NUM
easat-5378	437	2	-	-	SYM
easat-5378	437	3	681	681	NUM
easat-5378	437	4	,	,	PUNCT
easat-5378	437	5	2012	2012	NUM
easat-5378	437	6	.	.	PUNCT
easat-5378	438	1	https://doi.org/10.1007/s12045-012-0074-z	https://doi.org/10.1007/s12045-012-0074-z	PROPN
easat-5378	438	2	[	[	X
easat-5378	438	3	2	2	X
easat-5378	438	4	]	]	PUNCT
easat-5378	438	5	h.	h.	PROPN
easat-5378	438	6	e.	e.	PROPN
easat-5378	438	7	ross	ross	PROPN
easat-5378	438	8	and	and	CCONJ
easat-5378	438	9	b.	b.	PROPN
easat-5378	438	10	i.	i.	PROPN
easat-5378	438	11	knott	knott	PROPN
easat-5378	438	12	,	,	PUNCT
easat-5378	438	13	"	"	PUNCT
easat-5378	438	14	dicuil	dicuil	PROPN
easat-5378	438	15	(	(	PUNCT
easat-5378	438	16	9th	9th	ADJ
easat-5378	438	17	century	century	NOUN
easat-5378	438	18	)	)	PUNCT
easat-5378	438	19	on	on	ADP
easat-5378	438	20	triangular	triangular	NOUN
easat-5378	438	21	and	and	CCONJ
easat-5378	438	22	square	square	ADJ
easat-5378	438	23	numbers	number	NOUN
easat-5378	438	24	,	,	PUNCT
easat-5378	438	25	"	"	PUNCT
easat-5378	438	26	british	british	ADJ
easat-5378	438	27	journal	journal	NOUN
easat-5378	438	28	for	for	ADP
easat-5378	438	29	the	the	DET
easat-5378	438	30	history	history	NOUN
easat-5378	438	31	of	of	ADP
easat-5378	438	32	mathematics	mathematic	NOUN
easat-5378	438	33	,	,	PUNCT
easat-5378	438	34	vol	vol	NOUN
easat-5378	438	35	.	.	PROPN
easat-5378	438	36	34	34	NUM
easat-5378	438	37	,	,	PUNCT
easat-5378	438	38	no	no	INTJ
easat-5378	438	39	.	.	NOUN
easat-5378	438	40	2	2	NUM
easat-5378	438	41	,	,	PUNCT
easat-5378	438	42	pp	pp	ADJ
easat-5378	438	43	.	.	PUNCT
easat-5378	439	1	79	79	NUM
easat-5378	439	2	-	-	SYM
easat-5378	439	3	94	94	NUM
easat-5378	439	4	,	,	PUNCT
easat-5378	439	5	2019	2019	NUM
easat-5378	439	6	.	.	PUNCT
easat-5378	440	1	https://doi.org/10.1080/26375451.2019.1598687	https://doi.org/10.1080/26375451.2019.1598687	X
easat-5378	441	1	[	[	X
easat-5378	441	2	3	3	X
easat-5378	441	3	]	]	X
easat-5378	441	4	j.	j.	PROPN
easat-5378	441	5	dutka	dutka	PROPN
easat-5378	441	6	,	,	PUNCT
easat-5378	441	7	"	"	PUNCT
easat-5378	441	8	the	the	DET
easat-5378	441	9	early	early	ADJ
easat-5378	441	10	history	history	NOUN
easat-5378	441	11	of	of	ADP
easat-5378	441	12	the	the	DET
easat-5378	441	13	factorial	factorial	ADJ
easat-5378	441	14	function	function	NOUN
easat-5378	441	15	,	,	PUNCT
easat-5378	441	16	"	"	PUNCT
easat-5378	441	17	archive	archive	NOUN
easat-5378	441	18	for	for	ADP
easat-5378	441	19	history	history	NOUN
easat-5378	441	20	of	of	ADP
easat-5378	441	21	exact	exact	ADJ
easat-5378	441	22	sciences	science	NOUN
easat-5378	441	23	,	,	PUNCT
easat-5378	441	24	vol	vol	NOUN
easat-5378	441	25	.	.	PROPN
easat-5378	441	26	43	43	NUM
easat-5378	441	27	,	,	PUNCT
easat-5378	441	28	no	no	INTJ
easat-5378	441	29	.	.	NOUN
easat-5378	441	30	3	3	NUM
easat-5378	441	31	,	,	PUNCT
easat-5378	441	32	pp	pp	ADJ
easat-5378	441	33	.	.	PUNCT
easat-5378	442	1	225	225	NUM
easat-5378	442	2	-	-	SYM
easat-5378	442	3	249	249	NUM
easat-5378	442	4	,	,	PUNCT
easat-5378	442	5	1991	1991	NUM
easat-5378	442	6	.	.	PUNCT
easat-5378	443	1	https://doi.org/10.1007/bf00389433	https://doi.org/10.1007/bf00389433	X
easat-5378	444	1	[	[	X
easat-5378	444	2	4	4	NUM
easat-5378	444	3	]	]	X
easat-5378	444	4	a.-m	a.-m	NOUN
easat-5378	444	5	.	.	PUNCT
easat-5378	445	1	legendre	legendre	PROPN
easat-5378	445	2	,	,	PUNCT
easat-5378	445	3	new	new	ADJ
easat-5378	445	4	methods	method	NOUN
easat-5378	445	5	for	for	ADP
easat-5378	445	6	determining	determine	VERB
easat-5378	445	7	the	the	DET
easat-5378	445	8	orbits	orbit	NOUN
easat-5378	445	9	of	of	ADP
easat-5378	445	10	comets	comet	NOUN
easat-5378	445	11	.	.	PUNCT
easat-5378	446	1	paris	paris	PROPN
easat-5378	446	2	:	:	PUNCT
easat-5378	446	3	f.	f.	PROPN
easat-5378	446	4	didot	didot	PROPN
easat-5378	446	5	,	,	PUNCT
easat-5378	446	6	1805	1805	NUM
easat-5378	446	7	.	.	PUNCT
easat-5378	447	1	[	[	X
easat-5378	447	2	5	5	NUM
easat-5378	447	3	]	]	PUNCT
easat-5378	447	4	c.	c.	PROPN
easat-5378	447	5	f.	f.	PROPN
easat-5378	447	6	gauss	gauss	PROPN
easat-5378	447	7	,	,	PUNCT
easat-5378	447	8	disquisitiones	disquisitione	VERB
easat-5378	447	9	arithmeticae	arithmeticae	PROPN
easat-5378	447	10	.	.	PUNCT
easat-5378	448	1	leipzig	leipzig	PROPN
easat-5378	448	2	:	:	PUNCT
easat-5378	448	3	gerhard	gerhard	PROPN
easat-5378	448	4	fleischer	fleischer	PROPN
easat-5378	448	5	,	,	PUNCT
easat-5378	448	6	1801	1801	NUM
easat-5378	448	7	.	.	PUNCT
easat-5378	449	1	[	[	X
easat-5378	449	2	6	6	NUM
easat-5378	449	3	]	]	PUNCT
easat-5378	449	4	r.	r.	PROPN
easat-5378	449	5	c.	c.	PROPN
easat-5378	449	6	castillo	castillo	PROPN
easat-5378	449	7	,	,	PUNCT
easat-5378	449	8	"	"	PUNCT
easat-5378	449	9	a	a	DET
easat-5378	449	10	survey	survey	NOUN
easat-5378	449	11	on	on	ADP
easat-5378	449	12	triangular	triangular	ADJ
easat-5378	449	13	number	number	NOUN
easat-5378	449	14	,	,	PUNCT
easat-5378	449	15	factorial	factorial	NOUN
easat-5378	449	16	and	and	CCONJ
easat-5378	449	17	some	some	DET
easat-5378	449	18	associated	associated	ADJ
easat-5378	449	19	numbers	number	NOUN
easat-5378	449	20	,	,	PUNCT
easat-5378	449	21	"	"	PUNCT
easat-5378	449	22	indian	indian	ADJ
easat-5378	449	23	journal	journal	NOUN
easat-5378	449	24	of	of	ADP
easat-5378	449	25	science	science	NOUN
easat-5378	449	26	and	and	CCONJ
easat-5378	449	27	technology	technology	NOUN
easat-5378	449	28	,	,	PUNCT
easat-5378	449	29	vol	vol	NOUN
easat-5378	449	30	.	.	PROPN
easat-5378	449	31	9	9	NUM
easat-5378	449	32	,	,	PUNCT
easat-5378	449	33	no	no	INTJ
easat-5378	449	34	.	.	NOUN
easat-5378	449	35	41	41	NUM
easat-5378	449	36	,	,	PUNCT
easat-5378	449	37	pp	pp	ADJ
easat-5378	449	38	.	.	PUNCT
easat-5378	450	1	1	1	NUM
easat-5378	450	2	-	-	SYM
easat-5378	450	3	7	7	NUM
easat-5378	450	4	,	,	PUNCT
easat-5378	450	5	2016	2016	NUM
easat-5378	450	6	.	.	PUNCT
easat-5378	451	1	https://doi.org/10.17485/ijst/2016/v9i41/85182	https://doi.org/10.17485/ijst/2016/v9i41/85182	NOUN
easat-5378	452	1	[	[	X
easat-5378	452	2	7	7	NUM
easat-5378	452	3	]	]	PUNCT
easat-5378	452	4	a.	a.	PROPN
easat-5378	452	5	b.	b.	PROPN
easat-5378	452	6	ellis	ellis	PROPN
easat-5378	452	7	,	,	PUNCT
easat-5378	452	8	e.	e.	PROPN
easat-5378	452	9	lockwood	lockwood	PROPN
easat-5378	452	10	,	,	PUNCT
easat-5378	452	11	e.	e.	PROPN
easat-5378	452	12	tillema	tillema	PROPN
easat-5378	452	13	,	,	PUNCT
easat-5378	452	14	and	and	CCONJ
easat-5378	452	15	k.	k.	PROPN
easat-5378	452	16	moore	moore	PROPN
easat-5378	452	17	,	,	PUNCT
easat-5378	452	18	"	"	PUNCT
easat-5378	452	19	generalization	generalization	NOUN
easat-5378	452	20	across	across	ADP
easat-5378	452	21	multiple	multiple	ADJ
easat-5378	452	22	mathematical	mathematical	ADJ
easat-5378	452	23	domains	domain	NOUN
easat-5378	452	24	:	:	PUNCT
easat-5378	452	25	relating	relate	VERB
easat-5378	452	26	,	,	PUNCT
easat-5378	452	27	forming	form	VERB
easat-5378	452	28	,	,	PUNCT
easat-5378	452	29	and	and	CCONJ
easat-5378	452	30	extending	extend	VERB
easat-5378	452	31	,	,	PUNCT
easat-5378	452	32	"	"	PUNCT
easat-5378	452	33	cognition	cognition	NOUN
easat-5378	452	34	and	and	CCONJ
easat-5378	452	35	instruction	instruction	NOUN
easat-5378	452	36	,	,	PUNCT
easat-5378	452	37	vol	vol	NOUN
easat-5378	452	38	.	.	PROPN
easat-5378	452	39	40	40	NUM
easat-5378	452	40	,	,	PUNCT
easat-5378	452	41	no	no	INTJ
easat-5378	452	42	.	.	NOUN
easat-5378	452	43	3	3	NUM
easat-5378	452	44	,	,	PUNCT
easat-5378	452	45	pp	pp	ADJ
easat-5378	452	46	.	.	PUNCT
easat-5378	453	1	351	351	NUM
easat-5378	453	2	-	-	SYM
easat-5378	453	3	384	384	NUM
easat-5378	453	4	,	,	PUNCT
easat-5378	453	5	2022	2022	NUM
easat-5378	453	6	.	.	PUNCT
easat-5378	454	1	https://doi.org/10.1080/07370008.2021.2000989	https://doi.org/10.1080/07370008.2021.2000989	X
easat-5378	455	1	[	[	X
easat-5378	455	2	8	8	NUM
easat-5378	455	3	]	]	X
easat-5378	455	4	d.	d.	PROPN
easat-5378	455	5	w.	w.	PROPN
easat-5378	455	6	carraher	carraher	PROPN
easat-5378	455	7	,	,	PUNCT
easat-5378	455	8	m.	m.	NOUN
easat-5378	455	9	v.	v.	PROPN
easat-5378	455	10	martinez	martinez	PROPN
easat-5378	455	11	,	,	PUNCT
easat-5378	455	12	and	and	CCONJ
easat-5378	455	13	a.	a.	PROPN
easat-5378	455	14	d.	d.	PROPN
easat-5378	455	15	schliemann	schliemann	PROPN
easat-5378	455	16	,	,	PUNCT
easat-5378	455	17	"	"	PUNCT
easat-5378	455	18	early	early	ADJ
easat-5378	455	19	algebra	algebra	NOUN
easat-5378	455	20	and	and	CCONJ
easat-5378	455	21	mathematical	mathematical	ADJ
easat-5378	455	22	generalization	generalization	NOUN
easat-5378	455	23	,	,	PUNCT
easat-5378	455	24	"	"	PUNCT
easat-5378	455	25	zdm	zdm	NOUN
easat-5378	455	26	mathematics	mathematic	NOUN
easat-5378	455	27	education	education	NOUN
easat-5378	455	28	,	,	PUNCT
easat-5378	455	29	vol	vol	NOUN
easat-5378	455	30	.	.	PROPN
easat-5378	455	31	40	40	NUM
easat-5378	455	32	,	,	PUNCT
easat-5378	455	33	no	no	INTJ
easat-5378	455	34	.	.	NOUN
easat-5378	455	35	1	1	NUM
easat-5378	455	36	,	,	PUNCT
easat-5378	455	37	pp	pp	ADJ
easat-5378	455	38	.	.	PUNCT
easat-5378	456	1	3	3	NUM
easat-5378	456	2	-	-	SYM
easat-5378	456	3	22	22	NUM
easat-5378	456	4	,	,	PUNCT
easat-5378	456	5	2008	2008	NUM
easat-5378	456	6	.	.	PUNCT
easat-5378	457	1	https://doi.org/10.1007/s11858-007-0067-7	https://doi.org/10.1007/s11858-007-0067-7	PUNCT
easat-5378	458	1	[	[	X
easat-5378	458	2	9	9	NUM
easat-5378	458	3	]	]	PUNCT
easat-5378	458	4	m.	m.	NOUN
easat-5378	458	5	bhargava	bhargava	PROPN
easat-5378	458	6	,	,	PUNCT
easat-5378	458	7	"	"	PUNCT
easat-5378	458	8	the	the	DET
easat-5378	458	9	factorial	factorial	ADJ
easat-5378	458	10	function	function	NOUN
easat-5378	458	11	and	and	CCONJ
easat-5378	458	12	generalizations	generalization	NOUN
easat-5378	458	13	,	,	PUNCT
easat-5378	458	14	"	"	PUNCT
easat-5378	458	15	the	the	DET
easat-5378	458	16	american	american	PROPN
easat-5378	458	17	mathematical	mathematical	PROPN
easat-5378	458	18	monthly	monthly	ADJ
easat-5378	458	19	,	,	PUNCT
easat-5378	458	20	vol	vol	NOUN
easat-5378	458	21	.	.	PUNCT
easat-5378	458	22	107	107	NUM
easat-5378	458	23	,	,	PUNCT
easat-5378	458	24	no	no	INTJ
easat-5378	458	25	.	.	NOUN
easat-5378	458	26	9	9	NUM
easat-5378	458	27	,	,	PUNCT
easat-5378	458	28	pp	pp	ADJ
easat-5378	458	29	.	.	PUNCT
easat-5378	459	1	783	783	NUM
easat-5378	459	2	-	-	SYM
easat-5378	459	3	799	799	NUM
easat-5378	459	4	,	,	PUNCT
easat-5378	459	5	2000	2000	NUM
easat-5378	459	6	.	.	PUNCT
easat-5378	460	1	https://doi.org/10.1080/00029890.2000.12005273	https://doi.org/10.1080/00029890.2000.12005273	PROPN
easat-5378	461	1	[	[	X
easat-5378	461	2	10	10	NUM
easat-5378	461	3	]	]	X
easat-5378	461	4	j.	j.	PROPN
easat-5378	461	5	choi	choi	PROPN
easat-5378	461	6	and	and	CCONJ
easat-5378	461	7	e.	e.	PROPN
easat-5378	461	8	lee	lee	PROPN
easat-5378	461	9	,	,	PUNCT
easat-5378	461	10	"	"	PUNCT
easat-5378	461	11	higher	high	ADJ
easat-5378	461	12	dimensional	dimensional	ADJ
easat-5378	461	13	triangular	triangular	NOUN
easat-5378	461	14	and	and	CCONJ
easat-5378	461	15	square	square	ADJ
easat-5378	461	16	numbers	number	NOUN
easat-5378	461	17	,	,	PUNCT
easat-5378	461	18	"	"	PUNCT
easat-5378	461	19	communications	communication	NOUN
easat-5378	461	20	on	on	ADP
easat-5378	461	21	applied	apply	VERB
easat-5378	461	22	nonlinear	nonlinear	ADJ
easat-5378	461	23	analysis	analysis	NOUN
easat-5378	461	24	,	,	PUNCT
easat-5378	461	25	vol	vol	NOUN
easat-5378	461	26	.	.	PROPN
easat-5378	461	27	31	31	NUM
easat-5378	461	28	,	,	PUNCT
easat-5378	461	29	no	no	INTJ
easat-5378	461	30	.	.	NOUN
easat-5378	461	31	6	6	NUM
easat-5378	461	32	,	,	PUNCT
easat-5378	461	33	pp	pp	ADJ
easat-5378	461	34	.	.	PUNCT
easat-5378	461	35	153	153	NUM
easat-5378	461	36	-	-	SYM
easat-5378	461	37	159	159	NUM
easat-5378	461	38	,	,	PUNCT
easat-5378	461	39	2024	2024	NUM
easat-5378	461	40	.	.	PUNCT
easat-5378	462	1	https://doi.org/10.52783/cana.v31.1174	https://doi.org/10.52783/cana.v31.1174	PROPN
easat-5378	463	1	[	[	X
easat-5378	463	2	11	11	NUM
easat-5378	463	3	]	]	X
easat-5378	463	4	r.	r.	NOUN
easat-5378	463	5	flórez	flórez	PROPN
easat-5378	463	6	and	and	CCONJ
easat-5378	463	7	l.	l.	PROPN
easat-5378	463	8	junes	junes	PROPN
easat-5378	463	9	,	,	PUNCT
easat-5378	463	10	"	"	PUNCT
easat-5378	463	11	a	a	DET
easat-5378	463	12	relation	relation	NOUN
easat-5378	463	13	between	between	ADP
easat-5378	463	14	triangular	triangular	NOUN
easat-5378	463	15	numbers	number	NOUN
easat-5378	463	16	and	and	CCONJ
easat-5378	463	17	prime	prime	ADJ
easat-5378	463	18	numbers	number	NOUN
easat-5378	463	19	,	,	PUNCT
easat-5378	463	20	"	"	PUNCT
easat-5378	463	21	integers	integer	NOUN
easat-5378	463	22	,	,	PUNCT
easat-5378	463	23	vol	vol	NOUN
easat-5378	463	24	.	.	PROPN
easat-5378	463	25	12	12	NUM
easat-5378	463	26	,	,	PUNCT
easat-5378	463	27	no	no	INTJ
easat-5378	463	28	.	.	NOUN
easat-5378	463	29	1	1	NUM
easat-5378	463	30	,	,	PUNCT
easat-5378	463	31	pp	pp	ADJ
easat-5378	463	32	.	.	PUNCT
easat-5378	463	33	8396	8396	NUM
easat-5378	463	34	,	,	PUNCT
easat-5378	463	35	2012	2012	NUM
easat-5378	463	36	.	.	PUNCT
easat-5378	464	1	https://doi.org/10.1515/integ.2011.086	https://doi.org/10.1515/integ.2011.086	PROPN
easat-5378	464	2	[	[	X
easat-5378	464	3	12	12	NUM
easat-5378	464	4	]	]	X
easat-5378	464	5	r.	r.	PROPN
easat-5378	464	6	c.	c.	PROPN
easat-5378	464	7	castillo	castillo	PROPN
easat-5378	464	8	,	,	PUNCT
easat-5378	464	9	"	"	PUNCT
easat-5378	464	10	on	on	ADP
easat-5378	464	11	the	the	DET
easat-5378	464	12	sum	sum	NOUN
easat-5378	464	13	of	of	ADP
easat-5378	464	14	corresponding	correspond	VERB
easat-5378	464	15	factorials	factorial	NOUN
easat-5378	464	16	and	and	CCONJ
easat-5378	464	17	triangular	triangular	NOUN
easat-5378	464	18	numbers	number	NOUN
easat-5378	464	19	:	:	PUNCT
easat-5378	464	20	some	some	DET
easat-5378	464	21	preliminary	preliminary	ADJ
easat-5378	464	22	results	result	NOUN
easat-5378	464	23	,	,	PUNCT
easat-5378	464	24	"	"	PUNCT
easat-5378	464	25	asia	asia	PROPN
easat-5378	464	26	pacific	pacific	PROPN
easat-5378	464	27	journal	journal	PROPN
easat-5378	464	28	of	of	ADP
easat-5378	464	29	multidisciplinary	multidisciplinary	ADJ
easat-5378	464	30	research	research	NOUN
easat-5378	464	31	,	,	PUNCT
easat-5378	464	32	vol	vol	NOUN
easat-5378	464	33	.	.	PROPN
easat-5378	465	1	3	3	NUM
easat-5378	465	2	,	,	PUNCT
easat-5378	465	3	no	no	INTJ
easat-5378	465	4	.	.	NOUN
easat-5378	465	5	4	4	NUM
easat-5378	465	6	,	,	PUNCT
easat-5378	465	7	pp	pp	ADJ
easat-5378	465	8	.	.	PUNCT
easat-5378	466	1	5	5	NUM
easat-5378	466	2	-	-	SYM
easat-5378	466	3	11	11	NUM
easat-5378	466	4	,	,	PUNCT
easat-5378	466	5	2015	2015	NUM
easat-5378	466	6	.	.	PUNCT
easat-5378	467	1	[	[	X
easat-5378	467	2	13	13	NUM
easat-5378	467	3	]	]	X
easat-5378	467	4	oeis	oeis	PROPN
easat-5378	467	5	foundation	foundation	PROPN
easat-5378	467	6	inc	inc	PROPN
easat-5378	467	7	,	,	PUNCT
easat-5378	467	8	"	"	PUNCT
easat-5378	467	9	entry	entry	NOUN
easat-5378	467	10	a101292	a101292	NOUN
easat-5378	467	11	in	in	ADP
easat-5378	467	12	the	the	DET
easat-5378	467	13	online	online	ADJ
easat-5378	467	14	encyclopedia	encyclopedia	NOUN
easat-5378	467	15	of	of	ADP
easat-5378	467	16	integer	integer	NOUN
easat-5378	467	17	sequences	sequence	NOUN
easat-5378	467	18	,	,	PUNCT
easat-5378	467	19	"	"	PUNCT
easat-5378	467	20	retrieved	retrieve	VERB
easat-5378	467	21	:	:	PUNCT
easat-5378	467	22	https://oeis.org/a101292	https://oeis.org/a101292	NOUN
easat-5378	467	23	.	.	PUNCT
easat-5378	468	1	[	[	X
easat-5378	468	2	accessed	access	VERB
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easat-5378	468	4	12	12	NUM
easat-5378	468	5	,	,	PUNCT
easat-5378	468	6	2025	2025	NUM
easat-5378	468	7	]	]	PUNCT
easat-5378	468	8	,	,	PUNCT
easat-5378	468	9	2024	2024	NUM
easat-5378	468	10	.	.	PUNCT
easat-5378	469	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-5378	469	2	https://doi.org/10.1007/s12045-012-0074-z	https://doi.org/10.1007/s12045-012-0074-z	PROPN
easat-5378	469	3	https://doi.org/10.1080/26375451.2019.1598687	https://doi.org/10.1080/26375451.2019.1598687	PROPN
easat-5378	469	4	https://doi.org/10.1007/bf00389433	https://doi.org/10.1007/bf00389433	ADJ
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easat-5378	470	1	https://doi.org/10.52783/cana.v31.1174	https://doi.org/10.52783/cana.v31.1174	PROPN
easat-5378	470	2	https://doi.org/10.1515/integ.2011.086	https://doi.org/10.1515/integ.2011.086	PROPN
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easat-5378	470	5	edelweiss	edelweiss	PROPN
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easat-5378	470	8	and	and	CCONJ
easat-5378	470	9	technology	technology	NOUN
easat-5378	470	10	issn	issn	PROPN
easat-5378	470	11	:	:	PUNCT
easat-5378	470	12	2576	2576	NUM
easat-5378	470	13	-	-	SYM
easat-5378	470	14	8484	8484	NUM
easat-5378	470	15	vol	vol	NOUN
easat-5378	470	16	.	.	PROPN
easat-5378	471	1	9	9	NUM
easat-5378	471	2	,	,	PUNCT
easat-5378	471	3	no	no	INTJ
easat-5378	471	4	.	.	NOUN
easat-5378	472	1	3	3	NUM
easat-5378	472	2	:	:	PUNCT
easat-5378	472	3	891	891	NUM
easat-5378	472	4	-	-	SYM
easat-5378	472	5	904	904	NUM
easat-5378	472	6	,	,	PUNCT
easat-5378	472	7	2025	2025	NUM
easat-5378	472	8	doi	doi	NOUN
easat-5378	472	9	:	:	PUNCT
easat-5378	472	10	10.55214/25768484.v9i3.5378	10.55214/25768484.v9i3.5378	NUM
easat-5378	472	11	©	©	PROPN
easat-5378	472	12	2025	2025	NUM
easat-5378	472	13	by	by	ADP
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easat-5378	472	17	licensee	licensee	PROPN
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easat-5378	472	19	gate	gate	NOUN
easat-5378	473	1	[	[	X
easat-5378	473	2	14	14	NUM
easat-5378	473	3	]	]	X
easat-5378	473	4	c.	c.	PROPN
easat-5378	473	5	a.	a.	PROPN
easat-5378	473	6	g.	g.	PROPN
easat-5378	473	7	ruiz	ruiz	PROPN
easat-5378	473	8	and	and	CCONJ
easat-5378	473	9	f.	f.	PROPN
easat-5378	473	10	luca	luca	PROPN
easat-5378	473	11	,	,	PUNCT
easat-5378	473	12	"	"	PUNCT
easat-5378	473	13	fibonacci	fibonacci	NOUN
easat-5378	473	14	factoriangular	factoriangular	ADJ
easat-5378	473	15	numbers	number	NOUN
easat-5378	473	16	,	,	PUNCT
easat-5378	473	17	"	"	PUNCT
easat-5378	473	18	indagationes	indagatione	NOUN
easat-5378	473	19	mathematicae	mathematicae	PROPN
easat-5378	473	20	,	,	PUNCT
easat-5378	473	21	vol	vol	NOUN
easat-5378	473	22	.	.	PROPN
easat-5378	473	23	28	28	NUM
easat-5378	473	24	,	,	PUNCT
easat-5378	473	25	no	no	INTJ
easat-5378	473	26	.	.	NOUN
easat-5378	473	27	4	4	NUM
easat-5378	473	28	,	,	PUNCT
easat-5378	473	29	pp	pp	ADJ
easat-5378	473	30	.	.	PUNCT
easat-5378	474	1	796	796	NUM
easat-5378	474	2	-	-	SYM
easat-5378	474	3	804	804	NUM
easat-5378	474	4	,	,	PUNCT
easat-5378	474	5	2017	2017	NUM
easat-5378	474	6	.	.	PUNCT
easat-5378	475	1	https://doi.org/10.1016/j.indag.2017.05.002	https://doi.org/10.1016/j.indag.2017.05.002	VERB
easat-5378	476	1	[	[	X
easat-5378	476	2	15	15	NUM
easat-5378	476	3	]	]	X
easat-5378	476	4	f.	f.	PROPN
easat-5378	476	5	luca	luca	PROPN
easat-5378	476	6	,	,	PUNCT
easat-5378	476	7	j.	j.	PROPN
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easat-5378	476	9	,	,	PUNCT
easat-5378	476	10	and	and	CCONJ
easat-5378	476	11	a.	a.	NOUN
easat-5378	476	12	togbé	togbé	PROPN
easat-5378	476	13	,	,	PUNCT
easat-5378	476	14	"	"	PUNCT
easat-5378	476	15	pell	pell	VERB
easat-5378	476	16	factoriangular	factoriangular	ADJ
easat-5378	476	17	numbers	number	NOUN
easat-5378	476	18	,	,	PUNCT
easat-5378	476	19	"	"	PUNCT
easat-5378	476	20	publications	publication	NOUN
easat-5378	476	21	de	de	X
easat-5378	476	22	l'institut	l'institut	X
easat-5378	476	23	mathematique	mathematique	NOUN
easat-5378	476	24	,	,	PUNCT
easat-5378	476	25	vol	vol	NOUN
easat-5378	476	26	.	.	PROPN
easat-5378	477	1	105	105	NUM
easat-5378	477	2	,	,	PUNCT
easat-5378	477	3	no	no	INTJ
easat-5378	477	4	.	.	NOUN
easat-5378	477	5	119	119	NUM
easat-5378	477	6	,	,	PUNCT
easat-5378	477	7	pp	pp	ADJ
easat-5378	477	8	.	.	PUNCT
easat-5378	478	1	93	93	NUM
easat-5378	478	2	-	-	SYM
easat-5378	478	3	100	100	NUM
easat-5378	478	4	,	,	PUNCT
easat-5378	478	5	2019	2019	NUM
easat-5378	478	6	.	.	PUNCT
easat-5378	479	1	https://doi.org/10.2298/pim1919093l	https://doi.org/10.2298/pim1919093l	NOUN
easat-5378	480	1	[	[	X
easat-5378	480	2	16	16	NUM
easat-5378	480	3	]	]	PUNCT
easat-5378	480	4	b.	b.	PROPN
easat-5378	480	5	kafle	kafle	PROPN
easat-5378	480	6	,	,	PUNCT
easat-5378	480	7	f.	f.	PROPN
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easat-5378	480	9	,	,	PUNCT
easat-5378	480	10	and	and	CCONJ
easat-5378	480	11	a.	a.	NOUN
easat-5378	480	12	togbé	togbé	PROPN
easat-5378	480	13	,	,	PUNCT
easat-5378	480	14	"	"	PUNCT
easat-5378	480	15	lucas	lucas	ADJ
easat-5378	480	16	factoriangular	factoriangular	ADJ
easat-5378	480	17	numbers	number	NOUN
easat-5378	480	18	,	,	PUNCT
easat-5378	480	19	"	"	PUNCT
easat-5378	480	20	mathematica	mathematica	PROPN
easat-5378	480	21	bohemica	bohemica	PROPN
easat-5378	480	22	,	,	PUNCT
easat-5378	480	23	vol	vol	NOUN
easat-5378	480	24	.	.	PROPN
easat-5378	480	25	145	145	NUM
easat-5378	480	26	,	,	PUNCT
easat-5378	480	27	no	no	INTJ
easat-5378	480	28	.	.	NOUN
easat-5378	480	29	1	1	NUM
easat-5378	480	30	,	,	PUNCT
easat-5378	480	31	pp	pp	ADJ
easat-5378	480	32	.	.	PUNCT
easat-5378	481	1	33	33	NUM
easat-5378	481	2	-	-	SYM
easat-5378	481	3	43	43	NUM
easat-5378	481	4	,	,	PUNCT
easat-5378	481	5	2020	2020	NUM
easat-5378	481	6	.	.	PUNCT
easat-5378	482	1	https://doi.org/10.21136/mb.2018.0021-18	https://doi.org/10.21136/mb.2018.0021-18	NOUN
easat-5378	482	2	[	[	X
easat-5378	482	3	17	17	NUM
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easat-5378	482	5	s.	s.	PROPN
easat-5378	482	6	g.	g.	PROPN
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easat-5378	482	8	,	,	PUNCT
easat-5378	482	9	j.	j.	PROPN
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easat-5378	482	11	,	,	PUNCT
easat-5378	482	12	and	and	CCONJ
easat-5378	482	13	g.	g.	PROPN
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easat-5378	482	16	,	,	PUNCT
easat-5378	482	17	"	"	PUNCT
easat-5378	482	18	factoriangular	factoriangular	ADJ
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easat-5378	482	20	in	in	ADP
easat-5378	482	21	balancing	balancing	NOUN
easat-5378	482	22	and	and	CCONJ
easat-5378	482	23	lucas	lucas	NOUN
easat-5378	482	24	-	-	PUNCT
easat-5378	482	25	balancing	balance	VERB
easat-5378	482	26	sequence	sequence	NOUN
easat-5378	482	27	,	,	PUNCT
easat-5378	482	28	"	"	PUNCT
easat-5378	482	29	boletín	boletín	X
easat-5378	482	30	de	de	PROPN
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easat-5378	482	36	vol	vol	NOUN
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easat-5378	482	39	,	,	PUNCT
easat-5378	482	40	pp	pp	ADJ
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easat-5378	483	1	865	865	NUM
easat-5378	483	2	-	-	SYM
easat-5378	483	3	878	878	NUM
easat-5378	483	4	,	,	PUNCT
easat-5378	483	5	2020	2020	NUM
easat-5378	483	6	.	.	PUNCT
easat-5378	484	1	https://doi.org/10.1007/s40590020-00303-1	https://doi.org/10.1007/s40590020-00303-1	PROPN
easat-5378	485	1	[	[	X
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easat-5378	485	4	s.	s.	PROPN
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easat-5378	485	6	and	and	CCONJ
easat-5378	485	7	a.	a.	PROPN
easat-5378	485	8	s.	s.	PROPN
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easat-5378	485	19	,	,	PUNCT
easat-5378	485	20	"	"	PUNCT
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easat-5378	485	27	vol	vol	NOUN
easat-5378	485	28	.	.	PROPN
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easat-5378	485	30	,	,	PUNCT
easat-5378	485	31	no	no	INTJ
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easat-5378	485	33	9	9	NUM
easat-5378	485	34	,	,	PUNCT
easat-5378	485	35	pp	pp	ADJ
easat-5378	485	36	.	.	PUNCT
easat-5378	486	1	448	448	NUM
easat-5378	486	2	-	-	SYM
easat-5378	486	3	451	451	NUM
easat-5378	486	4	,	,	PUNCT
easat-5378	486	5	2018	2018	NUM
easat-5378	486	6	.	.	PUNCT
easat-5378	487	1	https://doi.org/10.21474/ijar01/7694	https://doi.org/10.21474/ijar01/7694	PROPN
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easat-5378	487	13	and	and	CCONJ
easat-5378	487	14	factoriangular	factoriangular	ADJ
easat-5378	487	15	triangles	triangle	NOUN
easat-5378	487	16	,	,	PUNCT
easat-5378	487	17	"	"	PUNCT
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easat-5378	487	19	journal	journal	NOUN
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easat-5378	487	23	and	and	CCONJ
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easat-5378	487	25	,	,	PUNCT
easat-5378	487	26	vol	vol	NOUN
easat-5378	487	27	.	.	PROPN
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easat-5378	487	29	,	,	PUNCT
easat-5378	487	30	no	no	INTJ
easat-5378	487	31	.	.	NOUN
easat-5378	487	32	5	5	NUM
easat-5378	487	33	,	,	PUNCT
easat-5378	487	34	pp	pp	ADJ
easat-5378	487	35	.	.	PUNCT
easat-5378	488	1	413	413	NUM
easat-5378	488	2	-	-	SYM
easat-5378	488	3	415	415	NUM
easat-5378	488	4	,	,	PUNCT
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easat-5378	488	6	.	.	PUNCT
easat-5378	489	1	[	[	X
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easat-5378	489	3	]	]	PUNCT
easat-5378	489	4	r.	r.	PROPN
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easat-5378	489	12	of	of	ADP
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easat-5378	489	15	,	,	PUNCT
easat-5378	489	16	"	"	PUNCT
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easat-5378	489	26	,	,	PUNCT
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easat-5378	489	29	5	5	NUM
easat-5378	489	30	,	,	PUNCT
easat-5378	489	31	pp	pp	ADJ
easat-5378	489	32	.	.	PUNCT
easat-5378	490	1	1	1	NUM
easat-5378	490	2	-	-	SYM
easat-5378	490	3	21	21	NUM
easat-5378	490	4	,	,	PUNCT
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easat-5378	491	1	https://doi.org/10.9734/arjom/2022/v18i530374	https://doi.org/10.9734/arjom/2022/v18i530374	X
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easat-5378	492	16	in	in	ADP
easat-5378	492	17	the	the	DET
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easat-5378	492	28	a	a	DET
easat-5378	492	29	k	k	PROPN
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