id	sid	tid	token	lemma	pos
cana-1174	1	1	higher	high	ADJ
cana-1174	1	2	dimensional	dimensional	ADJ
cana-1174	1	3	triangular	triangular	NOUN
cana-1174	1	4	and	and	CCONJ
cana-1174	1	5	square	square	ADJ
cana-1174	1	6	numbers	number	NOUN
cana-1174	1	7	communications	communication	NOUN
cana-1174	1	8	on	on	ADP
cana-1174	1	9	applied	apply	VERB
cana-1174	1	10	nonlinear	nonlinear	ADJ
cana-1174	1	11	analysis	analysis	NOUN
cana-1174	1	12	issn	issn	NOUN
cana-1174	1	13	:	:	PUNCT
cana-1174	1	14	1074	1074	NUM
cana-1174	1	15	-	-	PUNCT
cana-1174	1	16	133x	133x	NUM
cana-1174	1	17	vol	vol	NOUN
cana-1174	1	18	31	31	NUM
cana-1174	1	19	no	no	NOUN
cana-1174	1	20	.	.	PUNCT
cana-1174	2	1	6s	6s	NUM
cana-1174	2	2	(	(	PUNCT
cana-1174	2	3	2024	2024	NUM
cana-1174	2	4	)	)	PUNCT
cana-1174	2	5	153	153	NUM
cana-1174	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1174	2	7	higher	high	ADJ
cana-1174	2	8	dimensional	dimensional	ADJ
cana-1174	2	9	triangular	triangular	NOUN
cana-1174	2	10	and	and	CCONJ
cana-1174	2	11	square	square	ADJ
cana-1174	2	12	numbers	number	NOUN
cana-1174	2	13	j.	j.	PROPN
cana-1174	2	14	choi1	choi1	PROPN
cana-1174	2	15	,	,	PUNCT
cana-1174	2	16	dr	dr	PROPN
cana-1174	2	17	.	.	PROPN
cana-1174	2	18	e.	e.	PROPN
cana-1174	2	19	lee2	lee2	PROPN
cana-1174	3	1	1chadwick	1chadwick	PROPN
cana-1174	3	2	international	international	ADJ
cana-1174	3	3	school	school	NOUN
cana-1174	3	4	,	,	PUNCT
cana-1174	3	5	incheon	incheon	PROPN
cana-1174	3	6	,	,	PUNCT
cana-1174	3	7	south	south	PROPN
cana-1174	3	8	korea	korea	PROPN
cana-1174	3	9	2phd	2phd	PROPN
cana-1174	3	10	,	,	PUNCT
cana-1174	3	11	korea	korea	PROPN
cana-1174	3	12	university	university	PROPN
cana-1174	3	13	,	,	PUNCT
cana-1174	3	14	cfa	cfa	PROPN
cana-1174	3	15	,	,	PUNCT
cana-1174	3	16	financial	financial	ADJ
cana-1174	3	17	services	service	NOUN
cana-1174	3	18	industry	industry	NOUN
cana-1174	3	19	,	,	PUNCT
cana-1174	3	20	south	south	PROPN
cana-1174	3	21	korea	korea	PROPN
cana-1174	3	22	article	article	NOUN
cana-1174	3	23	history	history	NOUN
cana-1174	3	24	:	:	PUNCT
cana-1174	3	25	received	receive	VERB
cana-1174	3	26	:	:	PUNCT
cana-1174	3	27	25	25	NUM
cana-1174	3	28	-	-	PUNCT
cana-1174	3	29	05	05	NUM
cana-1174	3	30	-	-	PUNCT
cana-1174	3	31	2024	2024	NUM
cana-1174	3	32	revised	revise	VERB
cana-1174	3	33	:	:	PUNCT
cana-1174	3	34	17	17	NUM
cana-1174	3	35	-	-	SYM
cana-1174	3	36	07	07	NUM
cana-1174	3	37	-	-	PUNCT
cana-1174	3	38	2024	2024	NUM
cana-1174	3	39	accepted	accept	VERB
cana-1174	3	40	:	:	PUNCT
cana-1174	3	41	29	29	NUM
cana-1174	3	42	-	-	SYM
cana-1174	3	43	07	07	NUM
cana-1174	3	44	-	-	PUNCT
cana-1174	3	45	2024	2024	NUM
cana-1174	3	46	abstract	abstract	NOUN
cana-1174	3	47	:	:	PUNCT
cana-1174	3	48	triangular	triangular	NOUN
cana-1174	3	49	numbers	number	NOUN
cana-1174	3	50	and	and	CCONJ
cana-1174	3	51	square	square	ADJ
cana-1174	3	52	numbers	number	NOUN
cana-1174	3	53	are	be	AUX
cana-1174	3	54	traditionally	traditionally	ADV
cana-1174	3	55	defined	define	VERB
cana-1174	3	56	in	in	ADP
cana-1174	3	57	two	two	NUM
cana-1174	3	58	-	-	PUNCT
cana-1174	3	59	dimensional	dimensional	ADJ
cana-1174	3	60	space	space	NOUN
cana-1174	3	61	.	.	PUNCT
cana-1174	4	1	in	in	ADP
cana-1174	4	2	this	this	DET
cana-1174	4	3	paper	paper	NOUN
cana-1174	4	4	,	,	PUNCT
cana-1174	4	5	we	we	PRON
cana-1174	4	6	generalize	generalize	VERB
cana-1174	4	7	these	these	DET
cana-1174	4	8	numbers	number	NOUN
cana-1174	4	9	to	to	ADP
cana-1174	4	10	higher	high	ADJ
cana-1174	4	11	dimensions	dimension	NOUN
cana-1174	4	12	and	and	CCONJ
cana-1174	4	13	explore	explore	VERB
cana-1174	4	14	their	their	PRON
cana-1174	4	15	properties	property	NOUN
cana-1174	4	16	using	use	VERB
cana-1174	4	17	a	a	DET
cana-1174	4	18	coordinate	coordinate	NOUN
cana-1174	4	19	system	system	NOUN
cana-1174	4	20	to	to	PART
cana-1174	4	21	conceptualize	conceptualize	VERB
cana-1174	4	22	spaces	space	NOUN
cana-1174	4	23	beyond	beyond	ADP
cana-1174	4	24	three	three	NUM
cana-1174	4	25	dimensions	dimension	NOUN
cana-1174	4	26	.	.	PUNCT
cana-1174	5	1	by	by	ADP
cana-1174	5	2	generalizing	generalize	VERB
cana-1174	5	3	triangular	triangular	NOUN
cana-1174	5	4	numbers	number	NOUN
cana-1174	5	5	,	,	PUNCT
cana-1174	5	6	we	we	PRON
cana-1174	5	7	establish	establish	VERB
cana-1174	5	8	and	and	CCONJ
cana-1174	5	9	prove	prove	VERB
cana-1174	5	10	a	a	DET
cana-1174	5	11	notable	notable	ADJ
cana-1174	5	12	combinatorial	combinatorial	ADJ
cana-1174	5	13	identity	identity	NOUN
cana-1174	5	14	and	and	CCONJ
cana-1174	5	15	recursive	recursive	ADJ
cana-1174	5	16	relationship	relationship	NOUN
cana-1174	5	17	between	between	ADP
cana-1174	5	18	triangular	triangular	NOUN
cana-1174	5	19	numbers	number	NOUN
cana-1174	5	20	of	of	ADP
cana-1174	5	21	different	different	ADJ
cana-1174	5	22	dimensions	dimension	NOUN
cana-1174	5	23	.	.	PUNCT
cana-1174	6	1	key	key	ADJ
cana-1174	6	2	applications	application	NOUN
cana-1174	6	3	discussed	discuss	VERB
cana-1174	6	4	include	include	VERB
cana-1174	6	5	network	network	NOUN
cana-1174	6	6	optimization	optimization	NOUN
cana-1174	6	7	and	and	CCONJ
cana-1174	6	8	geometric	geometric	ADJ
cana-1174	6	9	partitioning	partitioning	NOUN
cana-1174	6	10	.	.	PUNCT
cana-1174	7	1	the	the	DET
cana-1174	7	2	paper	paper	NOUN
cana-1174	7	3	concludes	conclude	VERB
cana-1174	7	4	by	by	ADP
cana-1174	7	5	finding	find	VERB
cana-1174	7	6	the	the	DET
cana-1174	7	7	ratio	ratio	NOUN
cana-1174	7	8	between	between	ADP
cana-1174	7	9	generalized	generalized	ADJ
cana-1174	7	10	triangular	triangular	NOUN
cana-1174	7	11	and	and	CCONJ
cana-1174	7	12	square	square	ADJ
cana-1174	7	13	numbers	number	NOUN
cana-1174	7	14	,	,	PUNCT
cana-1174	7	15	which	which	PRON
cana-1174	7	16	geometrically	geometrically	ADV
cana-1174	7	17	corresponds	correspond	VERB
cana-1174	7	18	to	to	ADP
cana-1174	7	19	the	the	DET
cana-1174	7	20	volume	volume	NOUN
cana-1174	7	21	ratio	ratio	NOUN
cana-1174	7	22	of	of	ADP
cana-1174	7	23	a	a	DET
cana-1174	7	24	tetrahedron	tetrahedron	NOUN
cana-1174	7	25	to	to	ADP
cana-1174	7	26	a	a	DET
cana-1174	7	27	d	d	ADJ
cana-1174	7	28	-	-	ADJ
cana-1174	7	29	dimensional	dimensional	ADJ
cana-1174	7	30	cube	cube	NOUN
cana-1174	7	31	.	.	PUNCT
cana-1174	8	1	keywords	keyword	NOUN
cana-1174	8	2	:	:	PUNCT
cana-1174	8	3	triangular	triangular	NOUN
cana-1174	8	4	numbers	number	NOUN
cana-1174	8	5	,	,	PUNCT
cana-1174	8	6	square	square	ADJ
cana-1174	8	7	numbers	number	NOUN
cana-1174	8	8	,	,	PUNCT
cana-1174	8	9	higher	higher	ADV
cana-1174	8	10	-	-	PUNCT
cana-1174	8	11	dimensional	dimensional	ADJ
cana-1174	8	12	space	space	NOUN
cana-1174	8	13	,	,	PUNCT
cana-1174	8	14	combinatorial	combinatorial	ADJ
cana-1174	8	15	identities	identity	NOUN
cana-1174	8	16	.	.	PUNCT
cana-1174	9	1	1	1	X
cana-1174	9	2	.	.	X
cana-1174	9	3	introduction	introduction	NOUN
cana-1174	9	4	the	the	DET
cana-1174	9	5	study	study	NOUN
cana-1174	9	6	of	of	ADP
cana-1174	9	7	polygonal	polygonal	ADJ
cana-1174	9	8	numbers	number	NOUN
cana-1174	9	9	has	have	VERB
cana-1174	9	10	a	a	DET
cana-1174	9	11	rich	rich	ADJ
cana-1174	9	12	and	and	CCONJ
cana-1174	9	13	ancient	ancient	ADJ
cana-1174	9	14	history	history	NOUN
cana-1174	9	15	,	,	PUNCT
cana-1174	9	16	tracing	trace	VERB
cana-1174	9	17	back	back	ADV
cana-1174	9	18	to	to	ADP
cana-1174	9	19	the	the	DET
cana-1174	9	20	era	era	NOUN
cana-1174	9	21	of	of	ADP
cana-1174	9	22	pythagoras	pythagoras	PROPN
cana-1174	9	23	and	and	CCONJ
cana-1174	9	24	his	his	PRON
cana-1174	9	25	followers	follower	NOUN
cana-1174	9	26	,	,	PUNCT
cana-1174	9	27	based	base	VERB
cana-1174	9	28	on	on	ADP
cana-1174	9	29	egyptian	egyptian	ADJ
cana-1174	9	30	and	and	CCONJ
cana-1174	9	31	babylonian	babylonian	ADJ
cana-1174	9	32	precursors	precursor	NOUN
cana-1174	9	33	.	.	PUNCT
cana-1174	10	1	these	these	DET
cana-1174	10	2	early	early	ADJ
cana-1174	10	3	mathematicians	mathematician	NOUN
cana-1174	10	4	sought	seek	VERB
cana-1174	10	5	to	to	PART
cana-1174	10	6	connect	connect	VERB
cana-1174	10	7	the	the	DET
cana-1174	10	8	realms	realm	NOUN
cana-1174	10	9	of	of	ADP
cana-1174	10	10	geometry	geometry	NOUN
cana-1174	10	11	and	and	CCONJ
cana-1174	10	12	arithmetic	arithmetic	ADJ
cana-1174	10	13	by	by	ADP
cana-1174	10	14	representing	represent	VERB
cana-1174	10	15	numbers	number	NOUN
cana-1174	10	16	through	through	ADP
cana-1174	10	17	distinct	distinct	ADJ
cana-1174	10	18	geometric	geometric	ADJ
cana-1174	10	19	patterns	pattern	NOUN
cana-1174	10	20	.	.	PUNCT
cana-1174	11	1	a	a	DET
cana-1174	11	2	figurate	figurate	ADJ
cana-1174	11	3	number	number	NOUN
cana-1174	11	4	is	be	AUX
cana-1174	11	5	a	a	DET
cana-1174	11	6	number	number	NOUN
cana-1174	11	7	that	that	PRON
cana-1174	11	8	can	can	AUX
cana-1174	11	9	be	be	AUX
cana-1174	11	10	represented	represent	VERB
cana-1174	11	11	by	by	ADP
cana-1174	11	12	a	a	DET
cana-1174	11	13	regular	regular	ADJ
cana-1174	11	14	and	and	CCONJ
cana-1174	11	15	discrete	discrete	ADJ
cana-1174	11	16	geometric	geometric	ADJ
cana-1174	11	17	arrangement	arrangement	NOUN
cana-1174	11	18	of	of	ADP
cana-1174	11	19	equally	equally	ADV
cana-1174	11	20	spaced	space	VERB
cana-1174	11	21	points	point	NOUN
cana-1174	11	22	.	.	PUNCT
cana-1174	12	1	in	in	ADP
cana-1174	12	2	two	two	NUM
cana-1174	12	3	dimensions	dimension	NOUN
cana-1174	12	4	,	,	PUNCT
cana-1174	12	5	these	these	PRON
cana-1174	12	6	are	be	AUX
cana-1174	12	7	known	know	VERB
cana-1174	12	8	as	as	ADP
cana-1174	12	9	polygonal	polygonal	ADJ
cana-1174	12	10	numbers	number	NOUN
cana-1174	12	11	,	,	PUNCT
cana-1174	12	12	which	which	PRON
cana-1174	12	13	include	include	VERB
cana-1174	12	14	the	the	DET
cana-1174	12	15	well	well	ADV
cana-1174	12	16	-	-	PUNCT
cana-1174	12	17	known	know	VERB
cana-1174	12	18	triangular	triangular	NOUN
cana-1174	12	19	,	,	PUNCT
cana-1174	12	20	square	square	ADJ
cana-1174	12	21	,	,	PUNCT
cana-1174	12	22	pentagonal	pentagonal	ADJ
cana-1174	12	23	,	,	PUNCT
cana-1174	12	24	hexagonal	hexagonal	ADJ
cana-1174	12	25	,	,	PUNCT
cana-1174	12	26	and	and	CCONJ
cana-1174	12	27	heptagonal	heptagonal	ADJ
cana-1174	12	28	numbers	number	NOUN
cana-1174	12	29	.	.	PUNCT
cana-1174	13	1	this	this	DET
cana-1174	13	2	paper	paper	NOUN
cana-1174	13	3	is	be	AUX
cana-1174	13	4	organized	organize	VERB
cana-1174	13	5	as	as	SCONJ
cana-1174	13	6	follows	follow	VERB
cana-1174	13	7	.	.	PUNCT
cana-1174	14	1	section	section	NOUN
cana-1174	14	2	ii	ii	PROPN
cana-1174	14	3	introduces	introduce	VERB
cana-1174	14	4	the	the	DET
cana-1174	14	5	definitions	definition	NOUN
cana-1174	14	6	of	of	ADP
cana-1174	14	7	triangular	triangular	NOUN
cana-1174	14	8	and	and	CCONJ
cana-1174	14	9	square	square	ADJ
cana-1174	14	10	numbers	number	NOUN
cana-1174	14	11	using	use	VERB
cana-1174	14	12	geometric	geometric	ADJ
cana-1174	14	13	and	and	CCONJ
cana-1174	14	14	algebraic	algebraic	ADJ
cana-1174	14	15	representations	representation	NOUN
cana-1174	14	16	.	.	PUNCT
cana-1174	15	1	in	in	ADP
cana-1174	15	2	section	section	PROPN
cana-1174	15	3	iii	iii	PROPN
cana-1174	15	4	,	,	PUNCT
cana-1174	15	5	the	the	DET
cana-1174	15	6	generalization	generalization	NOUN
cana-1174	15	7	of	of	ADP
cana-1174	15	8	square	square	ADJ
cana-1174	15	9	numbers	number	NOUN
cana-1174	15	10	using	use	VERB
cana-1174	15	11	the	the	DET
cana-1174	15	12	coordinate	coordinate	NOUN
cana-1174	15	13	expression	expression	NOUN
cana-1174	15	14	is	be	AUX
cana-1174	15	15	discussed	discuss	VERB
cana-1174	15	16	.	.	PUNCT
cana-1174	16	1	after	after	ADP
cana-1174	16	2	this	this	PRON
cana-1174	16	3	,	,	PUNCT
cana-1174	16	4	the	the	DET
cana-1174	16	5	generalization	generalization	NOUN
cana-1174	16	6	of	of	ADP
cana-1174	16	7	triangular	triangular	NOUN
cana-1174	16	8	numbers	number	NOUN
cana-1174	16	9	to	to	ADP
cana-1174	16	10	arbitrary	arbitrary	ADJ
cana-1174	16	11	dimensions	dimension	NOUN
cana-1174	16	12	is	be	AUX
cana-1174	16	13	elaborated	elaborate	VERB
cana-1174	16	14	in	in	ADP
cana-1174	16	15	section	section	NOUN
cana-1174	16	16	iv	iv	NUM
cana-1174	16	17	.	.	PUNCT
cana-1174	17	1	here	here	ADV
cana-1174	17	2	,	,	PUNCT
cana-1174	17	3	properties	property	NOUN
cana-1174	17	4	of	of	ADP
cana-1174	17	5	generalized	generalized	ADJ
cana-1174	17	6	triangular	triangular	NOUN
cana-1174	17	7	numbers	number	NOUN
cana-1174	17	8	are	be	AUX
cana-1174	17	9	also	also	ADV
cana-1174	17	10	introduced	introduce	VERB
cana-1174	17	11	.	.	PUNCT
cana-1174	18	1	finally	finally	ADV
cana-1174	18	2	,	,	PUNCT
cana-1174	18	3	generalizations	generalization	NOUN
cana-1174	18	4	of	of	ADP
cana-1174	18	5	triangular	triangular	NOUN
cana-1174	18	6	and	and	CCONJ
cana-1174	18	7	square	square	ADJ
cana-1174	18	8	numbers	number	NOUN
cana-1174	18	9	are	be	AUX
cana-1174	18	10	compared	compare	VERB
cana-1174	18	11	in	in	ADP
cana-1174	18	12	section	section	NOUN
cana-1174	18	13	v.	v.	CCONJ
cana-1174	18	14	we	we	PRON
cana-1174	18	15	will	will	AUX
cana-1174	18	16	discuss	discuss	VERB
cana-1174	18	17	the	the	DET
cana-1174	18	18	applications	application	NOUN
cana-1174	18	19	and	and	CCONJ
cana-1174	18	20	value	value	NOUN
cana-1174	18	21	of	of	ADP
cana-1174	18	22	triangular	triangular	NOUN
cana-1174	18	23	numbers	number	NOUN
cana-1174	18	24	.	.	PUNCT
cana-1174	19	1	2	2	X
cana-1174	19	2	.	.	X
cana-1174	19	3	original	original	ADJ
cana-1174	19	4	triangular	triangular	NOUN
cana-1174	19	5	and	and	CCONJ
cana-1174	19	6	square	square	ADJ
cana-1174	19	7	numbers	number	NOUN
cana-1174	19	8	triangular	triangular	NOUN
cana-1174	19	9	and	and	CCONJ
cana-1174	19	10	square	square	ADJ
cana-1174	19	11	numbers	number	NOUN
cana-1174	19	12	are	be	AUX
cana-1174	19	13	defined	define	VERB
cana-1174	19	14	geometrically	geometrically	ADV
cana-1174	19	15	.	.	PUNCT
cana-1174	20	1	from	from	ADP
cana-1174	20	2	stones	stone	NOUN
cana-1174	20	3	,	,	PUNCT
cana-1174	20	4	we	we	PRON
cana-1174	20	5	can	can	AUX
cana-1174	20	6	construct	construct	VERB
cana-1174	20	7	𝑛	𝑛	DET
cana-1174	20	8	×	×	NOUN
cana-1174	20	9	𝑛	𝑛	DET
cana-1174	20	10	grid	grid	NOUN
cana-1174	20	11	.	.	PUNCT
cana-1174	21	1	since	since	SCONJ
cana-1174	21	2	this	this	DET
cana-1174	21	3	structure	structure	NOUN
cana-1174	21	4	looks	look	VERB
cana-1174	21	5	like	like	ADP
cana-1174	21	6	a	a	DET
cana-1174	21	7	square	square	NOUN
cana-1174	21	8	,	,	PUNCT
cana-1174	21	9	we	we	PRON
cana-1174	21	10	call	call	VERB
cana-1174	21	11	n	n	PRON
cana-1174	21	12	2	2	NUM
cana-1174	21	13	is	be	AUX
cana-1174	21	14	the	the	PRON
cana-1174	21	15	we	we	PRON
cana-1174	21	16	denote	denote	VERB
cana-1174	21	17	these	these	DET
cana-1174	21	18	square	square	ADJ
cana-1174	21	19	numbers	number	NOUN
cana-1174	21	20	as	as	ADP
cana-1174	21	21	𝑛-th	𝑛-th	ADJ
cana-1174	21	22	square	square	ADJ
cana-1174	21	23	number	number	NOUN
cana-1174	21	24	.	.	PUNCT
cana-1174	22	1	in	in	ADP
cana-1174	22	2	this	this	DET
cana-1174	22	3	paper	paper	NOUN
cana-1174	22	4	,	,	PUNCT
cana-1174	22	5	for	for	ADP
cana-1174	22	6	any	any	DET
cana-1174	22	7	natural	natural	ADJ
cana-1174	22	8	numbers	number	NOUN
cana-1174	22	9	𝑛.	𝑛.	NOUN
cana-1174	22	10	𝑆(𝑛	𝑆(𝑛	NUM
cana-1174	22	11	)	)	PUNCT
cana-1174	22	12	=	=	VERB
cana-1174	22	13	𝑛2	𝑛2	NOUN
cana-1174	22	14	communications	communication	NOUN
cana-1174	22	15	on	on	ADP
cana-1174	22	16	applied	apply	VERB
cana-1174	22	17	nonlinear	nonlinear	ADJ
cana-1174	22	18	analysis	analysis	NOUN
cana-1174	22	19	issn	issn	NOUN
cana-1174	22	20	:	:	PUNCT
cana-1174	22	21	1074	1074	NUM
cana-1174	22	22	-	-	PUNCT
cana-1174	22	23	133x	133x	NUM
cana-1174	22	24	vol	vol	NOUN
cana-1174	22	25	31	31	NUM
cana-1174	22	26	no	no	NOUN
cana-1174	22	27	.	.	PUNCT
cana-1174	23	1	6s	6s	NUM
cana-1174	23	2	(	(	PUNCT
cana-1174	23	3	2024	2024	NUM
cana-1174	23	4	)	)	PUNCT
cana-1174	23	5	154	154	NUM
cana-1174	23	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1174	23	7	𝑛	𝑛	NOUN
cana-1174	23	8	then	then	ADV
cana-1174	23	9	,	,	PUNCT
cana-1174	23	10	we	we	PRON
cana-1174	23	11	know	know	VERB
cana-1174	23	12	the	the	DET
cana-1174	23	13	𝑛-th	𝑛-th	ADJ
cana-1174	23	14	triangular	triangular	NOUN
cana-1174	23	15	number	number	NOUN
cana-1174	23	16	is	be	AUX
cana-1174	23	17	1	1	NUM
cana-1174	23	18	+	+	CCONJ
cana-1174	23	19	2	2	NUM
cana-1174	23	20	+	+	NOUN
cana-1174	23	21	...	...	PUNCT
cana-1174	24	1	+	+	CCONJ
cana-1174	24	2	𝑛	𝑛	PROPN
cana-1174	24	3	=	=	SYM
cana-1174	24	4	𝑛(𝑛+1	𝑛(𝑛+1	NUM
cana-1174	24	5	)	)	PUNCT
cana-1174	24	6	.	.	PUNCT
cana-1174	25	1	we	we	PRON
cana-1174	25	2	also	also	ADV
cana-1174	25	3	denote	denote	VERB
cana-1174	25	4	these	these	DET
cana-1174	25	5	2	2	NUM
cana-1174	25	6	triangular	triangular	NOUN
cana-1174	25	7	numbers	number	NOUN
cana-1174	25	8	as	as	ADP
cana-1174	25	9	for	for	ADP
cana-1174	25	10	any	any	DET
cana-1174	25	11	natural	natural	ADJ
cana-1174	25	12	numbers	number	NOUN
cana-1174	25	13	𝑛.	𝑛.	VERB
cana-1174	25	14	𝑇(𝑛	𝑇(𝑛	NOUN
cana-1174	25	15	)	)	PUNCT
cana-1174	25	16	=	=	SYM
cana-1174	26	1	𝑛(𝑛+1	𝑛(𝑛+1	X
cana-1174	26	2	)	)	PUNCT
cana-1174	26	3	2	2	NUM
cana-1174	26	4	in	in	ADP
cana-1174	26	5	this	this	DET
cana-1174	26	6	paper	paper	NOUN
cana-1174	26	7	,	,	PUNCT
cana-1174	26	8	we	we	PRON
cana-1174	26	9	will	will	AUX
cana-1174	26	10	generalize	generalize	VERB
cana-1174	26	11	these	these	DET
cana-1174	26	12	numbers	number	NOUN
cana-1174	26	13	to	to	ADP
cana-1174	26	14	arbitrary	arbitrary	ADJ
cana-1174	26	15	higher	higher	ADV
cana-1174	26	16	-	-	PUNCT
cana-1174	26	17	dimensional	dimensional	ADJ
cana-1174	26	18	spaces	space	NOUN
cana-1174	26	19	.	.	PUNCT
cana-1174	27	1	therefore	therefore	ADV
cana-1174	27	2	,	,	PUNCT
cana-1174	27	3	the	the	DET
cana-1174	27	4	definitions	definition	NOUN
cana-1174	27	5	of	of	ADP
cana-1174	27	6	the	the	DET
cana-1174	27	7	above	above	ADJ
cana-1174	27	8	numbers	number	NOUN
cana-1174	27	9	should	should	AUX
cana-1174	27	10	be	be	AUX
cana-1174	27	11	written	write	VERB
cana-1174	27	12	in	in	ADP
cana-1174	27	13	different	different	ADJ
cana-1174	27	14	ways	way	NOUN
cana-1174	27	15	.	.	PUNCT
cana-1174	28	1	since	since	SCONJ
cana-1174	28	2	we	we	PRON
cana-1174	28	3	can	can	AUX
cana-1174	28	4	not	not	PART
cana-1174	28	5	imagine	imagine	VERB
cana-1174	28	6	a	a	DET
cana-1174	28	7	space	space	NOUN
cana-1174	28	8	with	with	ADP
cana-1174	28	9	dimensions	dimension	NOUN
cana-1174	28	10	greater	great	ADJ
cana-1174	28	11	than	than	ADP
cana-1174	28	12	three	three	NUM
cana-1174	28	13	,	,	PUNCT
cana-1174	28	14	we	we	PRON
cana-1174	28	15	introduce	introduce	VERB
cana-1174	28	16	coordinate	coordinate	NOUN
cana-1174	28	17	systems	system	NOUN
cana-1174	28	18	to	to	PART
cana-1174	28	19	define	define	VERB
cana-1174	28	20	triangular	triangular	NOUN
cana-1174	28	21	and	and	CCONJ
cana-1174	28	22	square	square	ADJ
cana-1174	28	23	numbers	number	NOUN
cana-1174	28	24	.	.	PUNCT
cana-1174	29	1	since	since	SCONJ
cana-1174	29	2	the	the	DET
cana-1174	29	3	square	square	ADJ
cana-1174	29	4	number	number	NOUN
cana-1174	29	5	case	case	NOUN
cana-1174	29	6	is	be	AUX
cana-1174	29	7	easier	easy	ADJ
cana-1174	29	8	,	,	PUNCT
cana-1174	29	9	we	we	PRON
cana-1174	29	10	start	start	VERB
cana-1174	29	11	with	with	ADP
cana-1174	29	12	this	this	DET
cana-1174	29	13	case	case	NOUN
cana-1174	29	14	.	.	PUNCT
cana-1174	30	1	we	we	PRON
cana-1174	30	2	can	can	AUX
cana-1174	30	3	describe	describe	VERB
cana-1174	30	4	the	the	DET
cana-1174	30	5	𝑛-th	𝑛-th	ADJ
cana-1174	30	6	square	square	ADJ
cana-1174	30	7	number	number	NOUN
cana-1174	30	8	as	as	SCONJ
cana-1174	30	9	follows	follow	VERB
cana-1174	30	10	:	:	PUNCT
cana-1174	30	11	𝑆(𝑛	𝑆(𝑛	NUM
cana-1174	30	12	)	)	PUNCT
cana-1174	30	13	=	=	SYM
cana-1174	30	14	|{(𝑎	|{(𝑎	NOUN
cana-1174	30	15	,	,	PUNCT
cana-1174	30	16	𝑏	𝑏	NOUN
cana-1174	30	17	)	)	PUNCT
cana-1174	30	18	:	:	PUNCT
cana-1174	30	19	0	0	NUM
cana-1174	30	20	≤	≤	NUM
cana-1174	30	21	𝑎	𝑎	ADJ
cana-1174	30	22	,	,	PUNCT
cana-1174	30	23	𝑏	𝑏	PROPN
cana-1174	30	24	≤	≤	NUM
cana-1174	30	25	𝑛	𝑛	PRON
cana-1174	30	26	−	−	PROPN
cana-1174	30	27	1}|	1}|	NUM
cana-1174	30	28	.	.	PUNCT
cana-1174	31	1	here	here	ADV
cana-1174	31	2	,	,	PUNCT
cana-1174	31	3	|𝐴|	|𝐴|	VERB
cana-1174	31	4	is	be	AUX
cana-1174	31	5	the	the	DET
cana-1174	31	6	number	number	NOUN
cana-1174	31	7	of	of	ADP
cana-1174	31	8	the	the	DET
cana-1174	31	9	elements	element	NOUN
cana-1174	31	10	of	of	ADP
cana-1174	31	11	the	the	DET
cana-1174	31	12	set	set	NOUN
cana-1174	31	13	𝐴.	𝐴.	PROPN
cana-1174	31	14	we	we	PRON
cana-1174	31	15	know	know	VERB
cana-1174	31	16	the	the	DET
cana-1174	31	17	set	set	NOUN
cana-1174	31	18	|{(𝑎	|{(𝑎	NOUN
cana-1174	31	19	,	,	PUNCT
cana-1174	31	20	𝑏	𝑏	NOUN
cana-1174	31	21	)	)	PUNCT
cana-1174	31	22	:	:	PUNCT
cana-1174	32	1	0	0	NUM
cana-1174	32	2	≤	≤	NUM
cana-1174	32	3	𝑎	𝑎	ADJ
cana-1174	32	4	,	,	PUNCT
cana-1174	32	5	𝑏	𝑏	PROPN
cana-1174	32	6	≤	≤	NOUN
cana-1174	32	7	𝑛	𝑛	PRON
cana-1174	32	8	−	−	PROPN
cana-1174	32	9	1}|	1}|	NUM
cana-1174	32	10	forms	form	NOUN
cana-1174	32	11	a	a	DET
cana-1174	32	12	square	square	NOUN
cana-1174	32	13	and	and	CCONJ
cana-1174	32	14	each	each	DET
cana-1174	32	15	side	side	NOUN
cana-1174	32	16	contains	contain	VERB
cana-1174	32	17	𝑛	𝑛	DET
cana-1174	32	18	points	point	NOUN
cana-1174	32	19	.	.	PUNCT
cana-1174	33	1	since	since	SCONJ
cana-1174	33	2	the	the	DET
cana-1174	33	3	𝑥	𝑥	PROPN
cana-1174	33	4	coordinate	coordinate	NOUN
cana-1174	33	5	can	can	AUX
cana-1174	33	6	be	be	AUX
cana-1174	33	7	from	from	ADP
cana-1174	33	8	0	0	NUM
cana-1174	33	9	to	to	ADP
cana-1174	33	10	number	number	NOUN
cana-1174	33	11	of	of	ADP
cana-1174	33	12	points	point	NOUN
cana-1174	33	13	is	be	AUX
cana-1174	33	14	2	2	NUM
cana-1174	33	15	.	.	PUNCT
cana-1174	34	1	𝑛	𝑛	DET
cana-1174	34	2	−	−	NOUN
cana-1174	34	3	1	1	NUM
cana-1174	34	4	and	and	CCONJ
cana-1174	34	5	the	the	DET
cana-1174	34	6	𝑦	𝑦	NOUN
cana-1174	34	7	coordinate	coordinate	NOUN
cana-1174	34	8	can	can	AUX
cana-1174	34	9	be	be	AUX
cana-1174	34	10	from	from	ADP
cana-1174	34	11	0	0	NUM
cana-1174	34	12	to	to	ADP
cana-1174	34	13	𝑛	𝑛	PROPN
cana-1174	34	14	−	−	NUM
cana-1174	34	15	1	1	NUM
cana-1174	34	16	,	,	PUNCT
cana-1174	34	17	the	the	DET
cana-1174	34	18	total	total	NOUN
cana-1174	34	19	now	now	ADV
cana-1174	34	20	,	,	PUNCT
cana-1174	34	21	let	let	VERB
cana-1174	34	22	us	we	PRON
cana-1174	34	23	express	express	VERB
cana-1174	34	24	the	the	DET
cana-1174	34	25	triangular	triangular	NOUN
cana-1174	34	26	numbers	number	NOUN
cana-1174	34	27	using	use	VERB
cana-1174	34	28	the	the	DET
cana-1174	34	29	coordinate	coordinate	NOUN
cana-1174	34	30	expression	expression	NOUN
cana-1174	34	31	.	.	PUNCT
cana-1174	35	1	in	in	ADP
cana-1174	35	2	figure	figure	NOUN
cana-1174	35	3	1	1	NUM
cana-1174	35	4	,	,	PUNCT
cana-1174	35	5	all	all	DET
cana-1174	35	6	triangles	triangle	NOUN
cana-1174	35	7	form	form	VERB
cana-1174	35	8	the	the	DET
cana-1174	35	9	equilateral	equilateral	ADJ
cana-1174	35	10	triangles	triangle	NOUN
cana-1174	35	11	.	.	PUNCT
cana-1174	36	1	however	however	ADV
cana-1174	36	2	,	,	PUNCT
cana-1174	36	3	we	we	PRON
cana-1174	36	4	can	can	AUX
cana-1174	36	5	transform	transform	VERB
cana-1174	36	6	these	these	DET
cana-1174	36	7	points	point	NOUN
cana-1174	36	8	to	to	PART
cana-1174	36	9	form	form	VERB
cana-1174	36	10	an	an	DET
cana-1174	36	11	isosceles	isoscele	NOUN
cana-1174	36	12	right	right	ADJ
cana-1174	36	13	triangle	triangle	NOUN
cana-1174	36	14	as	as	SCONJ
cana-1174	36	15	follows	follow	VERB
cana-1174	36	16	(	(	PUNCT
cana-1174	36	17	figure	figure	NOUN
cana-1174	36	18	2	2	NUM
cana-1174	36	19	)	)	PUNCT
cana-1174	36	20	.	.	PUNCT
cana-1174	37	1	figure	figure	NOUN
cana-1174	37	2	1	1	NUM
cana-1174	37	3	.	.	PUNCT
cana-1174	37	4	triangular	triangular	NOUN
cana-1174	37	5	numbers	number	NOUN
cana-1174	37	6	in	in	ADP
cana-1174	37	7	equilateral	equilateral	ADJ
cana-1174	37	8	triangles	triangle	NOUN
cana-1174	37	9	figure	figure	NOUN
cana-1174	37	10	2	2	NUM
cana-1174	37	11	.	.	PUNCT
cana-1174	38	1	transformation	transformation	NOUN
cana-1174	38	2	of	of	ADP
cana-1174	38	3	the	the	DET
cana-1174	38	4	triangles	triangle	NOUN
cana-1174	38	5	in	in	ADP
cana-1174	38	6	isosceles	isoscele	NOUN
cana-1174	38	7	right	right	ADJ
cana-1174	38	8	triangles	triangle	NOUN
cana-1174	38	9	from	from	ADP
cana-1174	38	10	this	this	DET
cana-1174	38	11	structure	structure	NOUN
cana-1174	38	12	,	,	PUNCT
cana-1174	38	13	we	we	PRON
cana-1174	38	14	can	can	AUX
cana-1174	38	15	express	express	VERB
cana-1174	38	16	the	the	DET
cana-1174	38	17	triangular	triangular	NOUN
cana-1174	38	18	number	number	NOUN
cana-1174	38	19	as	as	SCONJ
cana-1174	38	20	follows	follow	VERB
cana-1174	38	21	:	:	PUNCT
cana-1174	38	22	𝑇(𝑛	𝑇(𝑛	X
cana-1174	38	23	)	)	PUNCT
cana-1174	38	24	=	=	SYM
cana-1174	38	25	|{(𝑎	|{(𝑎	NOUN
cana-1174	38	26	,	,	PUNCT
cana-1174	38	27	𝑏	𝑏	NOUN
cana-1174	38	28	)	)	PUNCT
cana-1174	38	29	:	:	PUNCT
cana-1174	38	30	0	0	NUM
cana-1174	38	31	≤	≤	NUM
cana-1174	38	32	𝑎	𝑎	ADJ
cana-1174	38	33	,	,	PUNCT
cana-1174	38	34	𝑏	𝑏	NOUN
cana-1174	38	35	,	,	PUNCT
cana-1174	38	36	0	0	NUM
cana-1174	38	37	≤	≤	NOUN
cana-1174	39	1	𝑎	𝑎	X
cana-1174	39	2	+	+	NOUN
cana-1174	39	3	𝑏	𝑏	NOUN
cana-1174	39	4	≤	≤	NOUN
cana-1174	39	5	𝑛	𝑛	PRON
cana-1174	39	6	−	−	PROPN
cana-1174	39	7	1}|	1}|	NUM
cana-1174	39	8	.	.	PUNCT
cana-1174	40	1	there	there	PRON
cana-1174	40	2	was	be	VERB
cana-1174	40	3	no	no	DET
cana-1174	40	4	way	way	NOUN
cana-1174	40	5	to	to	PART
cana-1174	40	6	generalize	generalize	VERB
cana-1174	40	7	the	the	DET
cana-1174	40	8	concept	concept	NOUN
cana-1174	40	9	of	of	ADP
cana-1174	40	10	equilateral	equilateral	ADJ
cana-1174	40	11	triangles	triangle	NOUN
cana-1174	40	12	to	to	ADP
cana-1174	40	13	a	a	DET
cana-1174	40	14	general	general	ADJ
cana-1174	40	15	dimension	dimension	NOUN
cana-1174	40	16	;	;	PUNCT
cana-1174	40	17	however	however	ADV
cana-1174	40	18	,	,	PUNCT
cana-1174	40	19	from	from	ADP
cana-1174	40	20	the	the	DET
cana-1174	40	21	above	above	ADJ
cana-1174	40	22	expression	expression	NOUN
cana-1174	40	23	,	,	PUNCT
cana-1174	40	24	we	we	PRON
cana-1174	40	25	can	can	AUX
cana-1174	40	26	generalize	generalize	VERB
cana-1174	40	27	the	the	DET
cana-1174	40	28	triangular	triangular	NOUN
cana-1174	40	29	number	number	NOUN
cana-1174	40	30	to	to	ADP
cana-1174	40	31	arbitrary	arbitrary	ADJ
cana-1174	40	32	dimensions	dimension	NOUN
cana-1174	40	33	.	.	PUNCT
cana-1174	41	1	3	3	X
cana-1174	41	2	.	.	X
cana-1174	41	3	generalization	generalization	NOUN
cana-1174	41	4	of	of	ADP
cana-1174	41	5	square	square	ADJ
cana-1174	41	6	numbers	number	NOUN
cana-1174	41	7	since	since	SCONJ
cana-1174	41	8	generalizing	generalize	VERB
cana-1174	41	9	square	square	ADJ
cana-1174	41	10	numbers	number	NOUN
cana-1174	41	11	is	be	AUX
cana-1174	41	12	easier	easy	ADJ
cana-1174	41	13	than	than	ADP
cana-1174	41	14	triangular	triangular	NOUN
cana-1174	41	15	numbers	number	NOUN
cana-1174	41	16	,	,	PUNCT
cana-1174	41	17	we	we	PRON
cana-1174	41	18	introduce	introduce	VERB
cana-1174	41	19	this	this	DET
cana-1174	41	20	part	part	NOUN
cana-1174	41	21	first	first	ADV
cana-1174	41	22	.	.	PUNCT
cana-1174	42	1	for	for	ADP
cana-1174	42	2	the	the	DET
cana-1174	42	3	two	two	NUM
cana-1174	42	4	-	-	PUNCT
cana-1174	42	5	dimensional	dimensional	ADJ
cana-1174	42	6	space	space	NOUN
cana-1174	42	7	,	,	PUNCT
cana-1174	42	8	we	we	PRON
cana-1174	42	9	used	use	VERB
cana-1174	42	10	the	the	DET
cana-1174	42	11	coordinate	coordinate	NOUN
cana-1174	42	12	plane	plane	NOUN
cana-1174	42	13	,	,	PUNCT
cana-1174	42	14	and	and	CCONJ
cana-1174	42	15	the	the	DET
cana-1174	42	16	square	square	ADJ
cana-1174	42	17	number	number	NOUN
cana-1174	42	18	was	be	AUX
cana-1174	42	19	expressed	express	VERB
cana-1174	42	20	in	in	ADP
cana-1174	42	21	the	the	DET
cana-1174	42	22	communications	communication	NOUN
cana-1174	42	23	on	on	ADP
cana-1174	42	24	applied	apply	VERB
cana-1174	42	25	nonlinear	nonlinear	ADJ
cana-1174	42	26	analysis	analysis	NOUN
cana-1174	42	27	issn	issn	NOUN
cana-1174	42	28	:	:	PUNCT
cana-1174	42	29	1074	1074	NUM
cana-1174	42	30	-	-	PUNCT
cana-1174	42	31	133x	133x	NUM
cana-1174	42	32	vol	vol	NOUN
cana-1174	42	33	31	31	NUM
cana-1174	42	34	no	no	NOUN
cana-1174	42	35	.	.	PUNCT
cana-1174	43	1	6s	6s	NUM
cana-1174	43	2	(	(	PUNCT
cana-1174	43	3	2024	2024	NUM
cana-1174	43	4	)	)	PUNCT
cana-1174	43	5	155	155	NUM
cana-1174	43	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1174	43	7	𝑑	𝑑	NOUN
cana-1174	43	8	𝑛	𝑛	ADP
cana-1174	43	9	following	follow	VERB
cana-1174	43	10	form	form	NOUN
cana-1174	43	11	:	:	PUNCT
cana-1174	43	12	𝑆(𝑛	𝑆(𝑛	NUM
cana-1174	43	13	)	)	PUNCT
cana-1174	43	14	=	=	SYM
cana-1174	43	15	|{(𝑎	|{(𝑎	NOUN
cana-1174	43	16	,	,	PUNCT
cana-1174	43	17	𝑏	𝑏	NOUN
cana-1174	43	18	)	)	PUNCT
cana-1174	43	19	:	:	PUNCT
cana-1174	43	20	0	0	NUM
cana-1174	43	21	≤	≤	NUM
cana-1174	43	22	𝑎	𝑎	ADJ
cana-1174	43	23	,	,	PUNCT
cana-1174	43	24	𝑏	𝑏	PROPN
cana-1174	43	25	≤	≤	NUM
cana-1174	43	26	𝑛	𝑛	PRON
cana-1174	43	27	−	−	PROPN
cana-1174	43	28	1}|	1}|	NUM
cana-1174	43	29	.	.	PUNCT
cana-1174	44	1	since	since	SCONJ
cana-1174	44	2	this	this	DET
cana-1174	44	3	square	square	ADJ
cana-1174	44	4	number	number	NOUN
cana-1174	44	5	is	be	AUX
cana-1174	44	6	defined	define	VERB
cana-1174	44	7	in	in	ADP
cana-1174	44	8	the	the	DET
cana-1174	44	9	two	two	NUM
cana-1174	44	10	-	-	PUNCT
cana-1174	44	11	dimensional	dimensional	ADJ
cana-1174	44	12	space	space	NOUN
cana-1174	44	13	,	,	PUNCT
cana-1174	44	14	to	to	PART
cana-1174	44	15	emphasize	emphasize	VERB
cana-1174	44	16	the	the	DET
cana-1174	44	17	dimension	dimension	NOUN
cana-1174	44	18	from	from	ADP
cana-1174	44	19	now	now	ADV
cana-1174	44	20	on	on	ADV
cana-1174	44	21	we	we	PRON
cana-1174	44	22	denote	denote	VERB
cana-1174	44	23	𝑆	𝑆	PROPN
cana-1174	44	24	(	(	PUNCT
cana-1174	44	25	𝑛	𝑛	NOUN
cana-1174	44	26	)	)	PUNCT
cana-1174	44	27	=	=	SYM
cana-1174	44	28	|{(𝑎	|{(𝑎	NOUN
cana-1174	44	29	,	,	PUNCT
cana-1174	44	30	𝑏	𝑏	NOUN
cana-1174	44	31	)	)	PUNCT
cana-1174	44	32	:	:	PUNCT
cana-1174	44	33	0	0	NUM
cana-1174	44	34	≤	≤	NUM
cana-1174	44	35	𝑎	𝑎	ADJ
cana-1174	44	36	,	,	PUNCT
cana-1174	44	37	𝑏	𝑏	PROPN
cana-1174	44	38	≤	≤	NUM
cana-1174	44	39	𝑛	𝑛	PRON
cana-1174	44	40	−	−	PROPN
cana-1174	44	41	1}|	1}|	NUM
cana-1174	44	42	.	.	NOUN
cana-1174	44	43	2	2	NUM
cana-1174	44	44	we	we	PRON
cana-1174	44	45	will	will	AUX
cana-1174	44	46	denote	denote	VERB
cana-1174	44	47	the	the	DET
cana-1174	44	48	square	square	ADJ
cana-1174	44	49	number	number	NOUN
cana-1174	44	50	defined	define	VERB
cana-1174	44	51	in	in	ADP
cana-1174	44	52	the	the	DET
cana-1174	44	53	𝑑-dimensional	𝑑-dimensional	PROPN
cana-1174	44	54	space	space	NOUN
cana-1174	44	55	𝑑	𝑑	PROPN
cana-1174	44	56	≥	≥	NUM
cana-1174	44	57	2	2	NUM
cana-1174	44	58	as	as	ADP
cana-1174	44	59	𝑆	𝑆	PROPN
cana-1174	44	60	(	(	PUNCT
cana-1174	44	61	𝑛	𝑛	NOUN
cana-1174	44	62	)	)	PUNCT
cana-1174	44	63	.	.	PUNCT
cana-1174	45	1	the	the	DET
cana-1174	45	2	𝑑	𝑑	PROPN
cana-1174	45	3	definition	definition	NOUN
cana-1174	45	4	of	of	ADP
cana-1174	45	5	this	this	DET
cana-1174	45	6	number	number	NOUN
cana-1174	45	7	can	can	AUX
cana-1174	45	8	be	be	AUX
cana-1174	45	9	generalized	generalize	VERB
cana-1174	45	10	from	from	ADP
cana-1174	45	11	𝑆	𝑆	PROPN
cana-1174	45	12	2	2	NUM
cana-1174	45	13	(	(	PUNCT
cana-1174	45	14	𝑛	𝑛	NOUN
cana-1174	45	15	)	)	PUNCT
cana-1174	45	16	.	.	PUNCT
cana-1174	46	1	if	if	SCONJ
cana-1174	46	2	we	we	PRON
cana-1174	46	3	only	only	ADV
cana-1174	46	4	increase	increase	VERB
cana-1174	46	5	the	the	DET
cana-1174	46	6	dimension	dimension	NOUN
cana-1174	46	7	of	of	ADP
cana-1174	46	8	the	the	DET
cana-1174	46	9	coordinate	coordinate	NOUN
cana-1174	46	10	system	system	NOUN
cana-1174	46	11	,	,	PUNCT
cana-1174	46	12	then	then	ADV
cana-1174	46	13	we	we	PRON
cana-1174	46	14	can	can	AUX
cana-1174	46	15	define	define	VERB
cana-1174	46	16	𝑆	𝑆	PROPN
cana-1174	46	17	𝑑	𝑑	PROPN
cana-1174	46	18	(	(	PUNCT
cana-1174	46	19	𝑛	𝑛	NOUN
cana-1174	46	20	)	)	PUNCT
cana-1174	46	21	for	for	ADP
cana-1174	46	22	𝑑	𝑑	PRON
cana-1174	46	23	≥	≥	NUM
cana-1174	46	24	2	2	NUM
cana-1174	46	25	as	as	SCONJ
cana-1174	46	26	follows	follow	VERB
cana-1174	46	27	:	:	PUNCT
cana-1174	46	28	𝑆	𝑆	PROPN
cana-1174	46	29	𝑑	𝑑	PROPN
cana-1174	46	30	(	(	PUNCT
cana-1174	46	31	𝑛	𝑛	NOUN
cana-1174	46	32	)	)	PUNCT
cana-1174	46	33	=	=	SYM
cana-1174	47	1	|{(𝑥	|{(𝑥	ADJ
cana-1174	47	2	1	1	NUM
cana-1174	47	3	,	,	PUNCT
cana-1174	47	4	𝑥	𝑥	PROPN
cana-1174	47	5	2	2	NUM
cana-1174	47	6	,	,	PUNCT
cana-1174	47	7	...	...	PUNCT
cana-1174	47	8	,	,	PUNCT
cana-1174	47	9	𝑥	𝑥	PROPN
cana-1174	47	10	𝑑	𝑑	NOUN
cana-1174	47	11	)	)	PUNCT
cana-1174	47	12	:	:	PUNCT
cana-1174	47	13	0	0	NUM
cana-1174	47	14	≤	≤	NUM
cana-1174	47	15	𝑥	𝑥	DET
cana-1174	47	16	1	1	NUM
cana-1174	47	17	,	,	PUNCT
cana-1174	47	18	𝑥	𝑥	PROPN
cana-1174	47	19	2	2	NUM
cana-1174	47	20	,	,	PUNCT
cana-1174	47	21	...	...	PUNCT
cana-1174	47	22	,	,	PUNCT
cana-1174	47	23	𝑥	𝑥	PROPN
cana-1174	48	1	𝑑	𝑑	PROPN
cana-1174	48	2	≤	≤	NUM
cana-1174	48	3	𝑛	𝑛	PRON
cana-1174	48	4	−	−	PROPN
cana-1174	48	5	1}|	1}|	NUM
cana-1174	48	6	.	.	PUNCT
cana-1174	49	1	we	we	PRON
cana-1174	49	2	know	know	VERB
cana-1174	49	3	each	each	DET
cana-1174	49	4	𝑥	𝑥	NOUN
cana-1174	49	5	can	can	AUX
cana-1174	49	6	be	be	AUX
cana-1174	49	7	from	from	ADP
cana-1174	49	8	0	0	NUM
cana-1174	49	9	to	to	ADP
cana-1174	49	10	𝑛	𝑛	PROPN
cana-1174	49	11	−	−	NUM
cana-1174	49	12	1	1	NUM
cana-1174	50	1	and	and	CCONJ
cana-1174	50	2	they	they	PRON
cana-1174	50	3	can	can	AUX
cana-1174	50	4	be	be	AUX
cana-1174	50	5	independently	independently	ADV
cana-1174	50	6	determined	determine	VERB
cana-1174	50	7	.	.	PUNCT
cana-1174	51	1	for	for	ADP
cana-1174	51	2	this	this	DET
cana-1174	51	3	𝑖	𝑖	SYM
cana-1174	51	4	reason	reason	NOUN
cana-1174	51	5	,	,	PUNCT
cana-1174	51	6	we	we	PRON
cana-1174	51	7	have	have	VERB
cana-1174	51	8	𝑆	𝑆	PROPN
cana-1174	51	9	(	(	PUNCT
cana-1174	51	10	𝑛	𝑛	NOUN
cana-1174	51	11	)	)	PUNCT
cana-1174	51	12	=	=	SYM
cana-1174	51	13	𝑛	𝑛	PROPN
cana-1174	51	14	.	.	PUNCT
cana-1174	52	1	we	we	PRON
cana-1174	52	2	can	can	AUX
cana-1174	52	3	explain	explain	VERB
cana-1174	52	4	this	this	DET
cana-1174	52	5	number	number	NOUN
cana-1174	52	6	both	both	CCONJ
cana-1174	52	7	algebraically	algebraically	ADV
cana-1174	52	8	and	and	CCONJ
cana-1174	52	9	geometrically	geometrically	ADV
cana-1174	52	10	.	.	PUNCT
cana-1174	53	1	𝑑	𝑑	PRON
cana-1174	53	2	for	for	ADP
cana-1174	53	3	instance	instance	NOUN
cana-1174	53	4	,	,	PUNCT
cana-1174	53	5	when	when	SCONJ
cana-1174	53	6	𝑑	𝑑	PROPN
cana-1174	53	7	=	=	SYM
cana-1174	53	8	3	3	NUM
cana-1174	53	9	,	,	PUNCT
cana-1174	53	10	then	then	ADV
cana-1174	53	11	3	3	NUM
cana-1174	53	12	points	point	NOUN
cana-1174	53	13	form	form	VERB
cana-1174	53	14	a	a	DET
cana-1174	53	15	cube	cube	NOUN
cana-1174	53	16	in	in	ADP
cana-1174	53	17	three	three	NUM
cana-1174	53	18	-	-	PUNCT
cana-1174	53	19	dimensional	dimensional	ADJ
cana-1174	53	20	space	space	NOUN
cana-1174	53	21	.	.	PUNCT
cana-1174	54	1	also	also	ADV
cana-1174	54	2	,	,	PUNCT
cana-1174	54	3	when	when	SCONJ
cana-1174	54	4	𝑑	𝑑	PROPN
cana-1174	54	5	=	=	SYM
cana-1174	54	6	1	1	NUM
cana-1174	54	7	,	,	PUNCT
cana-1174	54	8	then	then	ADV
cana-1174	54	9	we	we	PRON
cana-1174	54	10	just	just	ADV
cana-1174	54	11	get	get	VERB
cana-1174	54	12	natural	natural	ADJ
cana-1174	54	13	numbers	number	NOUN
cana-1174	54	14	.	.	PUNCT
cana-1174	55	1	𝑆	𝑆	PROPN
cana-1174	55	2	(	(	PUNCT
cana-1174	55	3	𝑛	𝑛	NOUN
cana-1174	55	4	)	)	PUNCT
cana-1174	55	5	=	=	SYM
cana-1174	55	6	1	1	NUM
cana-1174	55	7	𝑑	𝑑	NOUN
cana-1174	55	8	and	and	CCONJ
cana-1174	55	9	it	it	PRON
cana-1174	55	10	implies	imply	VERB
cana-1174	55	11	one	one	NUM
cana-1174	55	12	-	-	PUNCT
cana-1174	55	13	dimensional	dimensional	ADJ
cana-1174	55	14	square	square	ADJ
cana-1174	55	15	numbers	number	NOUN
cana-1174	55	16	are	be	AUX
cana-1174	55	17	just	just	ADV
cana-1174	55	18	4	4	NUM
cana-1174	55	19	.	.	PUNCT
cana-1174	55	20	generalization	generalization	NOUN
cana-1174	55	21	of	of	ADP
cana-1174	55	22	triangular	triangular	NOUN
cana-1174	55	23	numbers	number	NOUN
cana-1174	55	24	in	in	ADP
cana-1174	55	25	this	this	DET
cana-1174	55	26	section	section	NOUN
cana-1174	55	27	,	,	PUNCT
cana-1174	55	28	we	we	PRON
cana-1174	55	29	study	study	VERB
cana-1174	55	30	the	the	DET
cana-1174	55	31	generalization	generalization	NOUN
cana-1174	55	32	of	of	ADP
cana-1174	55	33	triangular	triangular	NOUN
cana-1174	55	34	numbers	number	NOUN
cana-1174	55	35	to	to	ADP
cana-1174	55	36	higher	high	ADJ
cana-1174	55	37	dimensional	dimensional	ADJ
cana-1174	55	38	space	space	NOUN
cana-1174	55	39	.	.	PUNCT
cana-1174	56	1	recall	recall	VERB
cana-1174	56	2	that	that	SCONJ
cana-1174	56	3	the	the	DET
cana-1174	56	4	triangular	triangular	NOUN
cana-1174	56	5	number	number	NOUN
cana-1174	56	6	was	be	AUX
cana-1174	56	7	expressed	express	VERB
cana-1174	56	8	as	as	SCONJ
cana-1174	56	9	follows	follow	VERB
cana-1174	56	10	in	in	ADP
cana-1174	56	11	section	section	NOUN
cana-1174	56	12	1	1	NUM
cana-1174	56	13	:	:	PUNCT
cana-1174	56	14	𝑇(𝑛	𝑇(𝑛	X
cana-1174	56	15	)	)	PUNCT
cana-1174	56	16	=	=	SYM
cana-1174	57	1	|{(𝑎	|{(𝑎	NOUN
cana-1174	57	2	,	,	PUNCT
cana-1174	57	3	𝑏	𝑏	NOUN
cana-1174	57	4	)	)	PUNCT
cana-1174	57	5	:	:	PUNCT
cana-1174	57	6	0	0	NUM
cana-1174	58	1	≤	≤	NUM
cana-1174	58	2	𝑎	𝑎	ADJ
cana-1174	58	3	,	,	PUNCT
cana-1174	58	4	𝑏	𝑏	NOUN
cana-1174	58	5	,	,	PUNCT
cana-1174	58	6	0	0	NUM
cana-1174	58	7	≤	≤	NOUN
cana-1174	59	1	𝑎	𝑎	X
cana-1174	59	2	+	+	NOUN
cana-1174	59	3	𝑏	𝑏	NOUN
cana-1174	59	4	≤	≤	NOUN
cana-1174	59	5	𝑛	𝑛	PRON
cana-1174	59	6	−	−	PROPN
cana-1174	59	7	1}|	1}|	NUM
cana-1174	59	8	.	.	PUNCT
cana-1174	60	1	to	to	PART
cana-1174	60	2	emphasize	emphasize	VERB
cana-1174	60	3	the	the	DET
cana-1174	60	4	original	original	ADJ
cana-1174	60	5	triangular	triangular	NOUN
cana-1174	60	6	number	number	NOUN
cana-1174	60	7	is	be	AUX
cana-1174	60	8	defined	define	VERB
cana-1174	60	9	on	on	ADP
cana-1174	60	10	the	the	DET
cana-1174	60	11	two	two	NUM
cana-1174	60	12	dimensional	dimensional	ADJ
cana-1174	60	13	space	space	NOUN
cana-1174	60	14	,	,	PUNCT
cana-1174	60	15	we	we	PRON
cana-1174	60	16	again	again	ADV
cana-1174	60	17	denote	denote	VERB
cana-1174	60	18	𝑇	𝑇	PROPN
cana-1174	60	19	(	(	PUNCT
cana-1174	60	20	𝑛	𝑛	NOUN
cana-1174	60	21	)	)	PUNCT
cana-1174	60	22	=	=	SYM
cana-1174	60	23	𝑇(𝑛	𝑇(𝑛	NOUN
cana-1174	60	24	)	)	PUNCT
cana-1174	60	25	.	.	PUNCT
cana-1174	61	1	from	from	ADP
cana-1174	61	2	the	the	DET
cana-1174	61	3	above	above	ADJ
cana-1174	61	4	expression	expression	NOUN
cana-1174	61	5	of	of	ADP
cana-1174	61	6	the	the	DET
cana-1174	61	7	triangular	triangular	NOUN
cana-1174	61	8	number	number	NOUN
cana-1174	61	9	defined	define	VERB
cana-1174	61	10	in	in	ADP
cana-1174	61	11	the	the	DET
cana-1174	61	12	2	2	NUM
cana-1174	61	13	two	two	NUM
cana-1174	61	14	-	-	PUNCT
cana-1174	61	15	dimensional	dimensional	ADJ
cana-1174	61	16	space	space	NOUN
cana-1174	61	17	,	,	PUNCT
cana-1174	61	18	we	we	PRON
cana-1174	61	19	can	can	AUX
cana-1174	61	20	define	define	VERB
cana-1174	61	21	the	the	DET
cana-1174	61	22	𝑑-dimensional	𝑑-dimensional	ADJ
cana-1174	61	23	triangular	triangular	NOUN
cana-1174	61	24	number	number	NOUN
cana-1174	61	25	as	as	ADP
cana-1174	61	26	𝑇	𝑇	PROPN
cana-1174	61	27	(	(	PUNCT
cana-1174	61	28	𝑛	𝑛	NOUN
cana-1174	61	29	)	)	PUNCT
cana-1174	61	30	=	=	SYM
cana-1174	62	1	|{(𝑥	|{(𝑥	NOUN
cana-1174	62	2	,	,	PUNCT
cana-1174	62	3	𝑥	𝑥	X
cana-1174	62	4	,	,	PUNCT
cana-1174	62	5	...	...	PUNCT
cana-1174	62	6	,	,	PUNCT
cana-1174	62	7	𝑥	𝑥	PROPN
cana-1174	62	8	)	)	PUNCT
cana-1174	62	9	:	:	PUNCT
cana-1174	62	10	0	0	NUM
cana-1174	62	11	≤	≤	NUM
cana-1174	63	1	𝑥	𝑥	PRON
cana-1174	63	2	,	,	PUNCT
cana-1174	63	3	𝑥	𝑥	X
cana-1174	63	4	,	,	PUNCT
cana-1174	63	5	...	...	PUNCT
cana-1174	63	6	,	,	PUNCT
cana-1174	63	7	𝑥	𝑥	X
cana-1174	63	8	,	,	PUNCT
cana-1174	63	9	0	0	NUM
cana-1174	63	10	≤	≤	NUM
cana-1174	63	11	𝑥	𝑥	PROPN
cana-1174	64	1	+	+	CCONJ
cana-1174	64	2	...	...	PUNCT
cana-1174	65	1	+	+	CCONJ
cana-1174	65	2	𝑥	𝑥	X
cana-1174	65	3	≤	≤	NUM
cana-1174	65	4	𝑛	𝑛	PRON
cana-1174	65	5	−	−	PROPN
cana-1174	65	6	1}|	1}|	NUM
cana-1174	65	7	.	.	PUNCT
cana-1174	66	1	𝑑	𝑑	PROPN
cana-1174	66	2	1	1	NUM
cana-1174	66	3	2	2	NUM
cana-1174	66	4	𝑑	𝑑	PROPN
cana-1174	66	5	1	1	NUM
cana-1174	66	6	2	2	NUM
cana-1174	66	7	𝑑	𝑑	PROPN
cana-1174	66	8	1	1	NUM
cana-1174	66	9	𝑑	𝑑	NOUN
cana-1174	66	10	when	when	SCONJ
cana-1174	66	11	𝑑	𝑑	PROPN
cana-1174	66	12	=	=	SYM
cana-1174	66	13	3	3	NUM
cana-1174	66	14	,	,	PUNCT
cana-1174	66	15	then	then	ADV
cana-1174	66	16	we	we	PRON
cana-1174	66	17	can	can	AUX
cana-1174	66	18	construct	construct	VERB
cana-1174	66	19	a	a	DET
cana-1174	66	20	tetrahedron	tetrahedron	NOUN
cana-1174	66	21	using	use	VERB
cana-1174	66	22	𝑇	𝑇	PROPN
cana-1174	66	23	(	(	PUNCT
cana-1174	66	24	𝑛	𝑛	NOUN
cana-1174	66	25	)	)	PUNCT
cana-1174	66	26	points	point	NOUN
cana-1174	66	27	.	.	PUNCT
cana-1174	67	1	we	we	PRON
cana-1174	67	2	have	have	VERB
cana-1174	67	3	one	one	NUM
cana-1174	67	4	interesting	interesting	ADJ
cana-1174	67	5	𝑑	𝑑	NOUN
cana-1174	67	6	property	property	NOUN
cana-1174	67	7	.	.	PUNCT
cana-1174	68	1	if	if	SCONJ
cana-1174	68	2	we	we	PRON
cana-1174	68	3	consider	consider	VERB
cana-1174	68	4	𝑇	𝑇	PROPN
cana-1174	68	5	(	(	PUNCT
cana-1174	68	6	𝑛	𝑛	PROPN
cana-1174	68	7	+	+	NOUN
cana-1174	68	8	1	1	NUM
cana-1174	68	9	)	)	PUNCT
cana-1174	68	10	,	,	PUNCT
cana-1174	68	11	then	then	ADV
cana-1174	68	12	we	we	PRON
cana-1174	68	13	have	have	VERB
cana-1174	68	14	𝑑	𝑑	PROPN
cana-1174	68	15	𝑇	𝑇	PROPN
cana-1174	68	16	(	(	PUNCT
cana-1174	68	17	𝑛	𝑛	PROPN
cana-1174	68	18	+	+	NOUN
cana-1174	68	19	1	1	NUM
cana-1174	68	20	)	)	PUNCT
cana-1174	68	21	=	=	SYM
cana-1174	68	22	|{(𝑥	|{(𝑥	NOUN
cana-1174	68	23	,	,	PUNCT
cana-1174	68	24	𝑥	𝑥	X
cana-1174	68	25	,	,	PUNCT
cana-1174	68	26	...	...	PUNCT
cana-1174	68	27	,	,	PUNCT
cana-1174	68	28	𝑥	𝑥	PROPN
cana-1174	68	29	)	)	PUNCT
cana-1174	68	30	:	:	PUNCT
cana-1174	69	1	0	0	NUM
cana-1174	69	2	≤	≤	NUM
cana-1174	69	3	𝑥	𝑥	PRON
cana-1174	69	4	,	,	PUNCT
cana-1174	69	5	𝑥	𝑥	X
cana-1174	69	6	,	,	PUNCT
cana-1174	69	7	...	...	PUNCT
cana-1174	69	8	,	,	PUNCT
cana-1174	69	9	𝑥	𝑥	X
cana-1174	69	10	,	,	PUNCT
cana-1174	69	11	0	0	NUM
cana-1174	69	12	≤	≤	NUM
cana-1174	69	13	𝑥	𝑥	PROPN
cana-1174	70	1	+	+	CCONJ
cana-1174	70	2	...	...	PUNCT
cana-1174	71	1	+	+	CCONJ
cana-1174	71	2	𝑥	𝑥	PROPN
cana-1174	71	3	≤	≤	NUM
cana-1174	71	4	𝑛}|	𝑛}|	PROPN
cana-1174	71	5	.	.	PUNCT
cana-1174	72	1	𝑑	𝑑	PROPN
cana-1174	72	2	1	1	NUM
cana-1174	72	3	2	2	NUM
cana-1174	72	4	𝑑	𝑑	PROPN
cana-1174	72	5	1	1	NUM
cana-1174	72	6	2	2	NUM
cana-1174	72	7	𝑑	𝑑	PROPN
cana-1174	72	8	1	1	NUM
cana-1174	72	9	𝑑	𝑑	NOUN
cana-1174	72	10	the	the	DET
cana-1174	72	11	set	set	NOUN
cana-1174	72	12	on	on	ADP
cana-1174	72	13	the	the	DET
cana-1174	72	14	right	right	ADJ
cana-1174	72	15	-	-	PUNCT
cana-1174	72	16	hand	hand	NOUN
cana-1174	72	17	side	side	NOUN
cana-1174	72	18	can	can	AUX
cana-1174	72	19	be	be	AUX
cana-1174	72	20	decomposed	decompose	VERB
cana-1174	72	21	into	into	ADP
cana-1174	72	22	two	two	NUM
cana-1174	72	23	parts	part	NOUN
cana-1174	72	24	:	:	PUNCT
cana-1174	72	25	{	{	PUNCT
cana-1174	72	26	(	(	PUNCT
cana-1174	72	27	𝑥	𝑥	INTJ
cana-1174	72	28	,	,	PUNCT
cana-1174	72	29	𝑥	𝑥	X
cana-1174	72	30	,	,	PUNCT
cana-1174	72	31	...	...	PUNCT
cana-1174	72	32	,	,	PUNCT
cana-1174	72	33	𝑥	𝑥	PROPN
cana-1174	72	34	)	)	PUNCT
cana-1174	72	35	:	:	PUNCT
cana-1174	72	36	0	0	NUM
cana-1174	72	37	≤	≤	NUM
cana-1174	72	38	𝑥	𝑥	PRON
cana-1174	72	39	,	,	PUNCT
cana-1174	72	40	𝑥	𝑥	X
cana-1174	72	41	,	,	PUNCT
cana-1174	72	42	...	...	PUNCT
cana-1174	72	43	,	,	PUNCT
cana-1174	72	44	𝑥	𝑥	X
cana-1174	72	45	,	,	PUNCT
cana-1174	72	46	0	0	NUM
cana-1174	72	47	≤	≤	NUM
cana-1174	72	48	𝑥	𝑥	PROPN
cana-1174	73	1	+	+	CCONJ
cana-1174	73	2	...	...	PUNCT
cana-1174	74	1	+	+	CCONJ
cana-1174	74	2	𝑥	𝑥	X
cana-1174	74	3	≤	≤	NUM
cana-1174	74	4	𝑛	𝑛	DET
cana-1174	74	5	−	−	PROPN
cana-1174	74	6	1	1	NUM
cana-1174	74	7	}	}	SYM
cana-1174	74	8	1	1	NUM
cana-1174	74	9	2	2	NUM
cana-1174	74	10	𝑑	𝑑	NOUN
cana-1174	74	11	1	1	NUM
cana-1174	74	12	2	2	NUM
cana-1174	74	13	𝑑	𝑑	PROPN
cana-1174	74	14	1	1	NUM
cana-1174	74	15	𝑑	𝑑	NOUN
cana-1174	74	16	and	and	CCONJ
cana-1174	74	17	{	{	PUNCT
cana-1174	74	18	(	(	PUNCT
cana-1174	74	19	𝑥	𝑥	INTJ
cana-1174	74	20	,	,	PUNCT
cana-1174	74	21	𝑥	𝑥	X
cana-1174	74	22	,	,	PUNCT
cana-1174	74	23	...	...	PUNCT
cana-1174	74	24	,	,	PUNCT
cana-1174	74	25	𝑥	𝑥	PROPN
cana-1174	74	26	)	)	PUNCT
cana-1174	74	27	:	:	PUNCT
cana-1174	75	1	0	0	NUM
cana-1174	75	2	≤	≤	NUM
cana-1174	75	3	𝑥	𝑥	PRON
cana-1174	75	4	,	,	PUNCT
cana-1174	75	5	𝑥	𝑥	X
cana-1174	75	6	,	,	PUNCT
cana-1174	75	7	...	...	PUNCT
cana-1174	75	8	,	,	PUNCT
cana-1174	75	9	𝑥	𝑥	X
cana-1174	75	10	,	,	PUNCT
cana-1174	75	11	𝑥	𝑥	PROPN
cana-1174	75	12	+	+	CCONJ
cana-1174	75	13	...	...	PUNCT
cana-1174	76	1	+	+	CCONJ
cana-1174	76	2	𝑥	𝑥	X
cana-1174	76	3	=	=	SYM
cana-1174	76	4	𝑛	𝑛	ADJ
cana-1174	76	5	}	}	SYM
cana-1174	76	6	1	1	NUM
cana-1174	76	7	2	2	NUM
cana-1174	76	8	𝑑	𝑑	NOUN
cana-1174	76	9	1	1	NUM
cana-1174	76	10	2	2	NUM
cana-1174	76	11	𝑑	𝑑	PROPN
cana-1174	76	12	1	1	NUM
cana-1174	76	13	𝑑	𝑑	NOUN
cana-1174	76	14	the	the	DET
cana-1174	76	15	number	number	NOUN
cana-1174	76	16	of	of	ADP
cana-1174	76	17	elements	element	NOUN
cana-1174	76	18	of	of	ADP
cana-1174	76	19	the	the	DET
cana-1174	76	20	above	above	ADJ
cana-1174	76	21	set	set	NOUN
cana-1174	76	22	is	be	AUX
cana-1174	76	23	𝑇	𝑇	PROPN
cana-1174	76	24	(	(	PUNCT
cana-1174	76	25	𝑛	𝑛	NOUN
cana-1174	76	26	)	)	PUNCT
cana-1174	76	27	from	from	ADP
cana-1174	76	28	the	the	DET
cana-1174	76	29	definition	definition	NOUN
cana-1174	76	30	of	of	ADP
cana-1174	76	31	the	the	DET
cana-1174	76	32	𝑑	𝑑	PROPN
cana-1174	76	33	𝑑-dimensional	𝑑-dimensional	PROPN
cana-1174	76	34	communications	communication	NOUN
cana-1174	76	35	on	on	ADP
cana-1174	76	36	applied	apply	VERB
cana-1174	76	37	nonlinear	nonlinear	ADJ
cana-1174	76	38	analysis	analysis	NOUN
cana-1174	76	39	issn	issn	NOUN
cana-1174	76	40	:	:	PUNCT
cana-1174	76	41	1074	1074	NUM
cana-1174	76	42	-	-	PUNCT
cana-1174	76	43	133x	133x	NUM
cana-1174	76	44	vol	vol	NOUN
cana-1174	76	45	31	31	NUM
cana-1174	76	46	no	no	NOUN
cana-1174	76	47	.	.	PUNCT
cana-1174	77	1	6s	6s	NUM
cana-1174	77	2	(	(	PUNCT
cana-1174	77	3	2024	2024	NUM
cana-1174	77	4	)	)	PUNCT
cana-1174	77	5	156	156	NUM
cana-1174	77	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1174	77	7	triangular	triangular	NOUN
cana-1174	77	8	number	number	NOUN
cana-1174	77	9	.	.	PUNCT
cana-1174	78	1	so	so	ADV
cana-1174	78	2	,	,	PUNCT
cana-1174	78	3	we	we	PRON
cana-1174	78	4	have	have	VERB
cana-1174	78	5	the	the	DET
cana-1174	78	6	following	follow	VERB
cana-1174	78	7	identity	identity	NOUN
cana-1174	78	8	:	:	PUNCT
cana-1174	78	9	{	{	PUNCT
cana-1174	78	10	(	(	PUNCT
cana-1174	78	11	𝑥	𝑥	INTJ
cana-1174	78	12	,	,	PUNCT
cana-1174	78	13	𝑥	𝑥	X
cana-1174	78	14	,	,	PUNCT
cana-1174	78	15	...	...	PUNCT
cana-1174	78	16	,	,	PUNCT
cana-1174	78	17	𝑥	𝑥	X
cana-1174	78	18	,	,	PUNCT
cana-1174	78	19	0	0	NUM
cana-1174	78	20	)	)	PUNCT
cana-1174	78	21	:	:	PUNCT
cana-1174	78	22	0	0	NUM
cana-1174	78	23	≤	≤	NUM
cana-1174	78	24	𝑥	𝑥	PRON
cana-1174	78	25	,	,	PUNCT
cana-1174	78	26	𝑥	𝑥	X
cana-1174	78	27	,	,	PUNCT
cana-1174	78	28	...	...	PUNCT
cana-1174	78	29	,	,	PUNCT
cana-1174	78	30	𝑥	𝑥	X
cana-1174	78	31	,	,	PUNCT
cana-1174	78	32	0	0	NUM
cana-1174	78	33	≤	≤	NUM
cana-1174	78	34	𝑥	𝑥	PROPN
cana-1174	79	1	+	+	CCONJ
cana-1174	79	2	...	...	PUNCT
cana-1174	80	1	+	+	CCONJ
cana-1174	80	2	𝑥	𝑥	X
cana-1174	80	3	≤	≤	NUM
cana-1174	80	4	𝑛	𝑛	PROPN
cana-1174	80	5	}	}	PUNCT
cana-1174	80	6	,	,	PUNCT
cana-1174	80	7	1	1	NUM
cana-1174	80	8	2	2	NUM
cana-1174	80	9	𝑑−1	𝑑−1	PROPN
cana-1174	80	10	1	1	NUM
cana-1174	80	11	2	2	NUM
cana-1174	80	12	𝑑−1	𝑑−1	PROPN
cana-1174	80	13	1	1	NUM
cana-1174	80	14	𝑑−1	𝑑−1	PROPN
cana-1174	80	15	{	{	PUNCT
cana-1174	80	16	(	(	PUNCT
cana-1174	80	17	𝑥	𝑥	PROPN
cana-1174	80	18	,	,	PUNCT
cana-1174	80	19	𝑥	𝑥	X
cana-1174	80	20	,	,	PUNCT
cana-1174	80	21	...	...	PUNCT
cana-1174	80	22	,	,	PUNCT
cana-1174	80	23	𝑥	𝑥	X
cana-1174	80	24	,	,	PUNCT
cana-1174	80	25	1	1	NUM
cana-1174	80	26	)	)	PUNCT
cana-1174	80	27	:	:	PUNCT
cana-1174	81	1	0	0	NUM
cana-1174	81	2	≤	≤	NUM
cana-1174	81	3	𝑥	𝑥	PRON
cana-1174	81	4	,	,	PUNCT
cana-1174	81	5	𝑥	𝑥	X
cana-1174	81	6	,	,	PUNCT
cana-1174	81	7	...	...	PUNCT
cana-1174	81	8	,	,	PUNCT
cana-1174	81	9	𝑥	𝑥	X
cana-1174	81	10	,	,	PUNCT
cana-1174	81	11	0	0	NUM
cana-1174	81	12	≤	≤	NUM
cana-1174	81	13	𝑥	𝑥	PROPN
cana-1174	81	14	+	+	CCONJ
cana-1174	81	15	...	...	PUNCT
cana-1174	82	1	+	+	CCONJ
cana-1174	83	1	𝑥	𝑥	X
cana-1174	83	2	≤	≤	NUM
cana-1174	83	3	𝑛	𝑛	PRON
cana-1174	83	4	−	−	PROPN
cana-1174	83	5	1	1	NUM
cana-1174	83	6	}	}	PUNCT
cana-1174	83	7	,	,	PUNCT
cana-1174	83	8	1	1	NUM
cana-1174	83	9	2	2	NUM
cana-1174	83	10	𝑑−1	𝑑−1	PROPN
cana-1174	83	11	1	1	NUM
cana-1174	83	12	2	2	NUM
cana-1174	83	13	𝑑−1	𝑑−1	PROPN
cana-1174	83	14	1	1	NUM
cana-1174	83	15	𝑑−1	𝑑−1	PROPN
cana-1174	83	16	…	…	PUNCT
cana-1174	83	17	{	{	PUNCT
cana-1174	83	18	(	(	PUNCT
cana-1174	83	19	𝑥	𝑥	INTJ
cana-1174	83	20	,	,	PUNCT
cana-1174	83	21	𝑥	𝑥	X
cana-1174	83	22	,	,	PUNCT
cana-1174	83	23	...	...	PUNCT
cana-1174	83	24	,	,	PUNCT
cana-1174	83	25	𝑥	𝑥	X
cana-1174	83	26	,	,	PUNCT
cana-1174	83	27	𝑛	𝑛	PRON
cana-1174	83	28	−	−	PROPN
cana-1174	83	29	1	1	NUM
cana-1174	83	30	)	)	PUNCT
cana-1174	83	31	:	:	PUNCT
cana-1174	83	32	0	0	NUM
cana-1174	83	33	≤	≤	NUM
cana-1174	83	34	𝑥	𝑥	PRON
cana-1174	83	35	,	,	PUNCT
cana-1174	83	36	𝑥	𝑥	X
cana-1174	83	37	,	,	PUNCT
cana-1174	83	38	...	...	PUNCT
cana-1174	83	39	,	,	PUNCT
cana-1174	83	40	𝑥	𝑥	X
cana-1174	83	41	,	,	PUNCT
cana-1174	83	42	0	0	NUM
cana-1174	83	43	≤	≤	NUM
cana-1174	83	44	𝑥	𝑥	PROPN
cana-1174	83	45	+	+	CCONJ
cana-1174	83	46	...	...	PUNCT
cana-1174	84	1	+	+	CCONJ
cana-1174	84	2	𝑥	𝑥	NOUN
cana-1174	84	3	=	=	SYM
cana-1174	84	4	1	1	NUM
cana-1174	84	5	}	}	PUNCT
cana-1174	84	6	,	,	PUNCT
cana-1174	84	7	1	1	NUM
cana-1174	84	8	2	2	NUM
cana-1174	84	9	𝑑−1	𝑑−1	PROPN
cana-1174	84	10	1	1	NUM
cana-1174	84	11	2	2	NUM
cana-1174	84	12	𝑑−1	𝑑−1	PROPN
cana-1174	84	13	1	1	NUM
cana-1174	84	14	𝑑−1	𝑑−1	PROPN
cana-1174	84	15	{	{	PUNCT
cana-1174	84	16	(	(	PUNCT
cana-1174	84	17	𝑥	𝑥	PROPN
cana-1174	84	18	,	,	PUNCT
cana-1174	84	19	𝑥	𝑥	X
cana-1174	84	20	,	,	PUNCT
cana-1174	84	21	...	...	PUNCT
cana-1174	84	22	,	,	PUNCT
cana-1174	84	23	𝑥	𝑥	X
cana-1174	84	24	,	,	PUNCT
cana-1174	84	25	𝑛	𝑛	PROPN
cana-1174	84	26	)	)	PUNCT
cana-1174	84	27	:	:	PUNCT
cana-1174	84	28	0	0	NUM
cana-1174	84	29	≤	≤	NUM
cana-1174	84	30	𝑥	𝑥	PRON
cana-1174	84	31	,	,	PUNCT
cana-1174	84	32	𝑥	𝑥	X
cana-1174	84	33	,	,	PUNCT
cana-1174	84	34	...	...	PUNCT
cana-1174	84	35	,	,	PUNCT
cana-1174	84	36	𝑥	𝑥	X
cana-1174	84	37	,	,	PUNCT
cana-1174	84	38	0	0	NUM
cana-1174	84	39	≤	≤	NUM
cana-1174	84	40	𝑥	𝑥	PROPN
cana-1174	84	41	+	+	CCONJ
cana-1174	84	42	...	...	PUNCT
cana-1174	85	1	+	+	CCONJ
cana-1174	85	2	𝑥	𝑥	X
cana-1174	85	3	=	=	SYM
cana-1174	85	4	0	0	NUM
cana-1174	85	5	}	}	PUNCT
cana-1174	85	6	,	,	PUNCT
cana-1174	85	7	1	1	NUM
cana-1174	85	8	2	2	NUM
cana-1174	85	9	𝑑−1	𝑑−1	PROPN
cana-1174	85	10	1	1	NUM
cana-1174	85	11	2	2	NUM
cana-1174	85	12	𝑑−1	𝑑−1	PROPN
cana-1174	85	13	1	1	NUM
cana-1174	85	14	𝑑−1	𝑑−1	PROPN
cana-1174	85	15	so	so	ADV
cana-1174	85	16	,	,	PUNCT
cana-1174	85	17	we	we	PRON
cana-1174	85	18	have	have	VERB
cana-1174	85	19	|{(𝑥	|{(𝑥	NOUN
cana-1174	85	20	,	,	PUNCT
cana-1174	85	21	𝑥	𝑥	X
cana-1174	85	22	,	,	PUNCT
cana-1174	85	23	...	...	PUNCT
cana-1174	85	24	,	,	PUNCT
cana-1174	85	25	𝑥	𝑥	PROPN
cana-1174	85	26	)	)	PUNCT
cana-1174	85	27	:	:	PUNCT
cana-1174	85	28	0	0	NUM
cana-1174	85	29	≤	≤	NUM
cana-1174	85	30	𝑥	𝑥	PRON
cana-1174	85	31	,	,	PUNCT
cana-1174	85	32	𝑥	𝑥	X
cana-1174	85	33	,	,	PUNCT
cana-1174	85	34	...	...	PUNCT
cana-1174	85	35	,	,	PUNCT
cana-1174	85	36	𝑥	𝑥	X
cana-1174	85	37	,	,	PUNCT
cana-1174	85	38	𝑥	𝑥	PROPN
cana-1174	86	1	+	+	CCONJ
cana-1174	86	2	...	...	PUNCT
cana-1174	87	1	+	+	CCONJ
cana-1174	87	2	𝑥	𝑥	X
cana-1174	87	3	=	=	SYM
cana-1174	87	4	𝑛	𝑛	NOUN
cana-1174	87	5	}	}	PUNCT
cana-1174	87	6	,	,	PUNCT
cana-1174	87	7	1	1	NUM
cana-1174	87	8	2	2	NUM
cana-1174	87	9	𝑑	𝑑	NOUN
cana-1174	87	10	1	1	NUM
cana-1174	87	11	2	2	NUM
cana-1174	87	12	𝑑	𝑑	PROPN
cana-1174	87	13	1	1	NUM
cana-1174	87	14	𝑑	𝑑	NOUN
cana-1174	87	15	=	=	ADJ
cana-1174	87	16	|{(𝑥	|{(𝑥	NOUN
cana-1174	87	17	,	,	PUNCT
cana-1174	87	18	𝑥	𝑥	X
cana-1174	87	19	,	,	PUNCT
cana-1174	87	20	...	...	PUNCT
cana-1174	87	21	,	,	PUNCT
cana-1174	87	22	𝑥	𝑥	PROPN
cana-1174	87	23	)	)	PUNCT
cana-1174	87	24	:	:	PUNCT
cana-1174	87	25	0	0	NUM
cana-1174	87	26	≤	≤	NUM
cana-1174	87	27	𝑥	𝑥	PRON
cana-1174	87	28	,	,	PUNCT
cana-1174	87	29	𝑥	𝑥	X
cana-1174	87	30	,	,	PUNCT
cana-1174	87	31	...	...	PUNCT
cana-1174	87	32	,	,	PUNCT
cana-1174	87	33	𝑥	𝑥	X
cana-1174	87	34	,	,	PUNCT
cana-1174	87	35	𝑥	𝑥	PROPN
cana-1174	88	1	+	+	CCONJ
cana-1174	88	2	...	...	PUNCT
cana-1174	89	1	+	+	CCONJ
cana-1174	89	2	𝑥	𝑥	X
cana-1174	89	3	≤	≤	NUM
cana-1174	89	4	𝑛	𝑛	PROPN
cana-1174	89	5	}	}	PUNCT
cana-1174	89	6	,	,	PUNCT
cana-1174	89	7	1	1	NUM
cana-1174	89	8	2	2	NUM
cana-1174	89	9	𝑑−1	𝑑−1	PROPN
cana-1174	89	10	1	1	NUM
cana-1174	89	11	2	2	NUM
cana-1174	89	12	𝑑−1	𝑑−1	PROPN
cana-1174	89	13	1	1	NUM
cana-1174	89	14	𝑑−1	𝑑−1	PROPN
cana-1174	89	15	since	since	SCONJ
cana-1174	89	16	the	the	DET
cana-1174	89	17	right	right	ADJ
cana-1174	89	18	-	-	PUNCT
cana-1174	89	19	hand	hand	NOUN
cana-1174	89	20	side	side	NOUN
cana-1174	89	21	of	of	ADP
cana-1174	89	22	the	the	DET
cana-1174	89	23	above	above	ADJ
cana-1174	89	24	equality	equality	NOUN
cana-1174	89	25	is	be	AUX
cana-1174	89	26	𝑇	𝑇	PROPN
cana-1174	89	27	(	(	PUNCT
cana-1174	89	28	𝑛	𝑛	NOUN
cana-1174	89	29	)	)	PUNCT
cana-1174	89	30	𝑑−1	𝑑−1	PROPN
cana-1174	89	31	from	from	ADP
cana-1174	89	32	the	the	DET
cana-1174	89	33	definition	definition	NOUN
cana-1174	89	34	of	of	ADP
cana-1174	89	35	the	the	DET
cana-1174	89	36	higher	high	ADJ
cana-1174	89	37	dimensional	dimensional	ADJ
cana-1174	89	38	triangular	triangular	NOUN
cana-1174	89	39	number	number	NOUN
cana-1174	89	40	,	,	PUNCT
cana-1174	89	41	we	we	PRON
cana-1174	89	42	eventually	eventually	ADV
cana-1174	89	43	get	get	VERB
cana-1174	89	44	the	the	DET
cana-1174	89	45	relationship	relationship	NOUN
cana-1174	89	46	:	:	PUNCT
cana-1174	89	47	𝑇	𝑇	PROPN
cana-1174	89	48	(	(	PUNCT
cana-1174	89	49	𝑛	𝑛	PROPN
cana-1174	89	50	+	+	NOUN
cana-1174	89	51	1	1	NUM
cana-1174	89	52	)	)	PUNCT
cana-1174	89	53	=	=	SYM
cana-1174	89	54	𝑇	𝑇	PROPN
cana-1174	89	55	(	(	PUNCT
cana-1174	89	56	𝑛	𝑛	NOUN
cana-1174	89	57	)	)	PUNCT
cana-1174	90	1	+	+	CCONJ
cana-1174	90	2	𝑇	𝑇	PROPN
cana-1174	90	3	(	(	PUNCT
cana-1174	90	4	𝑛	𝑛	PROPN
cana-1174	90	5	+	+	NOUN
cana-1174	90	6	1	1	X
cana-1174	90	7	)	)	PUNCT
cana-1174	90	8	𝑑	𝑑	PROPN
cana-1174	90	9	𝑑	𝑑	PROPN
cana-1174	90	10	𝑑−1	𝑑−1	PROPN
cana-1174	90	11	𝑇	𝑇	PROPN
cana-1174	90	12	(	(	PUNCT
cana-1174	90	13	𝑛	𝑛	PROPN
cana-1174	90	14	+	+	NOUN
cana-1174	90	15	1	1	NUM
cana-1174	90	16	)	)	PUNCT
cana-1174	90	17	−	−	PROPN
cana-1174	90	18	𝑇	𝑇	PROPN
cana-1174	90	19	(	(	PUNCT
cana-1174	90	20	𝑛	𝑛	NOUN
cana-1174	90	21	)	)	PUNCT
cana-1174	90	22	=	=	SYM
cana-1174	90	23	|{(𝑥	|{(𝑥	NOUN
cana-1174	90	24	,	,	PUNCT
cana-1174	90	25	𝑥	𝑥	X
cana-1174	90	26	,	,	PUNCT
cana-1174	90	27	...	...	PUNCT
cana-1174	90	28	,	,	PUNCT
cana-1174	90	29	𝑥	𝑥	PROPN
cana-1174	90	30	)	)	PUNCT
cana-1174	90	31	:	:	PUNCT
cana-1174	90	32	0	0	NUM
cana-1174	90	33	≤	≤	NUM
cana-1174	90	34	𝑥	𝑥	PRON
cana-1174	90	35	,	,	PUNCT
cana-1174	90	36	𝑥	𝑥	X
cana-1174	90	37	,	,	PUNCT
cana-1174	90	38	...	...	PUNCT
cana-1174	90	39	,	,	PUNCT
cana-1174	90	40	𝑥	𝑥	X
cana-1174	90	41	,	,	PUNCT
cana-1174	90	42	𝑥	𝑥	PROPN
cana-1174	90	43	+	+	CCONJ
cana-1174	90	44	...	...	PUNCT
cana-1174	91	1	+	+	CCONJ
cana-1174	91	2	𝑥	𝑥	X
cana-1174	91	3	=	=	SYM
cana-1174	91	4	𝑛}|	𝑛}|	PROPN
cana-1174	91	5	.	.	PUNCT
cana-1174	92	1	𝑑	𝑑	AUX
cana-1174	92	2	𝑑	𝑑	PROPN
cana-1174	92	3	1	1	NUM
cana-1174	92	4	2	2	NUM
cana-1174	92	5	𝑑	𝑑	PROPN
cana-1174	92	6	1	1	NUM
cana-1174	92	7	2	2	NUM
cana-1174	92	8	𝑑	𝑑	PROPN
cana-1174	92	9	1	1	NUM
cana-1174	92	10	𝑑	𝑑	NOUN
cana-1174	92	11	to	to	PART
cana-1174	92	12	simplify	simplify	VERB
cana-1174	92	13	the	the	DET
cana-1174	92	14	above	above	ADJ
cana-1174	92	15	identity	identity	NOUN
cana-1174	92	16	,	,	PUNCT
cana-1174	92	17	we	we	PRON
cana-1174	92	18	again	again	ADV
cana-1174	92	19	decompose	decompose	VERB
cana-1174	92	20	the	the	DET
cana-1174	92	21	set	set	NOUN
cana-1174	92	22	on	on	ADP
cana-1174	92	23	the	the	DET
cana-1174	92	24	right	right	ADJ
cana-1174	92	25	-	-	PUNCT
cana-1174	92	26	hand	hand	NOUN
cana-1174	92	27	side	side	NOUN
cana-1174	92	28	.	.	PUNCT
cana-1174	93	1	since	since	SCONJ
cana-1174	93	2	can	can	AUX
cana-1174	93	3	be	be	AUX
cana-1174	93	4	from	from	ADP
cana-1174	93	5	0	0	NUM
cana-1174	93	6	to	to	ADP
cana-1174	93	7	𝑛	𝑛	PROPN
cana-1174	93	8	,	,	PUNCT
cana-1174	93	9	we	we	PRON
cana-1174	93	10	can	can	AUX
cana-1174	93	11	decompose	decompose	VERB
cana-1174	93	12	the	the	DET
cana-1174	93	13	set	set	NOUN
cana-1174	93	14	as	as	SCONJ
cana-1174	93	15	follows	follow	VERB
cana-1174	93	16	:	:	PUNCT
cana-1174	93	17	we	we	PRON
cana-1174	93	18	can	can	AUX
cana-1174	93	19	interpret	interpret	VERB
cana-1174	93	20	the	the	DET
cana-1174	93	21	above	above	ADJ
cana-1174	93	22	relation	relation	NOUN
cana-1174	93	23	geometrically	geometrically	ADV
cana-1174	93	24	.	.	PUNCT
cana-1174	94	1	when	when	SCONJ
cana-1174	94	2	𝑑	𝑑	PROPN
cana-1174	94	3	=	=	SYM
cana-1174	94	4	3	3	NUM
cana-1174	94	5	,	,	PUNCT
cana-1174	94	6	we	we	PRON
cana-1174	94	7	know	know	VERB
cana-1174	94	8	𝑇	𝑇	PROPN
cana-1174	94	9	(	(	PUNCT
cana-1174	94	10	𝑛	𝑛	PROPN
cana-1174	94	11	+	+	NOUN
cana-1174	94	12	1	1	NUM
cana-1174	94	13	)	)	PUNCT
cana-1174	94	14	and	and	CCONJ
cana-1174	94	15	𝑇	𝑇	PROPN
cana-1174	94	16	(	(	PUNCT
cana-1174	94	17	𝑛	𝑛	NOUN
cana-1174	94	18	)	)	PUNCT
cana-1174	94	19	𝑑	𝑑	PROPN
cana-1174	94	20	𝑑	𝑑	NOUN
cana-1174	94	21	are	be	AUX
cana-1174	94	22	tetrahedrons	tetrahedron	NOUN
cana-1174	94	23	with	with	ADP
cana-1174	94	24	side	side	ADJ
cana-1174	94	25	length	length	NOUN
cana-1174	94	26	𝑛	𝑛	PROPN
cana-1174	94	27	+	+	NUM
cana-1174	94	28	1	1	NUM
cana-1174	94	29	and	and	CCONJ
cana-1174	94	30	𝑛	𝑛	NOUN
cana-1174	94	31	,	,	PUNCT
cana-1174	94	32	respectively	respectively	ADV
cana-1174	94	33	.	.	PUNCT
cana-1174	95	1	if	if	SCONJ
cana-1174	95	2	we	we	PRON
cana-1174	95	3	compare	compare	VERB
cana-1174	95	4	these	these	DET
cana-1174	95	5	two	two	NUM
cana-1174	95	6	structures	structure	NOUN
cana-1174	95	7	,	,	PUNCT
cana-1174	95	8	then	then	ADV
cana-1174	95	9	the	the	DET
cana-1174	95	10	only	only	ADJ
cana-1174	95	11	difference	difference	NOUN
cana-1174	95	12	is	be	AUX
cana-1174	95	13	the	the	DET
cana-1174	95	14	triangular	triangular	NOUN
cana-1174	95	15	surface	surface	NOUN
cana-1174	95	16	.	.	PUNCT
cana-1174	96	1	if	if	SCONJ
cana-1174	96	2	we	we	PRON
cana-1174	96	3	stack	stack	VERB
cana-1174	96	4	one	one	NUM
cana-1174	96	5	more	more	ADJ
cana-1174	96	6	layer	layer	NOUN
cana-1174	96	7	of	of	ADP
cana-1174	96	8	points	point	NOUN
cana-1174	96	9	on	on	ADP
cana-1174	96	10	the	the	DET
cana-1174	96	11	triangular	triangular	NOUN
cana-1174	96	12	surface	surface	NOUN
cana-1174	96	13	of	of	ADP
cana-1174	96	14	𝑇	𝑇	PROPN
cana-1174	96	15	(	(	PUNCT
cana-1174	96	16	𝑛	𝑛	NOUN
cana-1174	96	17	)	)	PUNCT
cana-1174	96	18	,	,	PUNCT
cana-1174	96	19	then	then	ADV
cana-1174	96	20	we	we	PRON
cana-1174	96	21	get	get	VERB
cana-1174	96	22	𝑑	𝑑	PROPN
cana-1174	96	23	𝑇	𝑇	PROPN
cana-1174	96	24	(	(	PUNCT
cana-1174	96	25	𝑛	𝑛	PROPN
cana-1174	96	26	+	+	NOUN
cana-1174	96	27	1	1	NUM
cana-1174	96	28	)	)	PUNCT
cana-1174	96	29	.	.	PUNCT
cana-1174	97	1	also	also	ADV
cana-1174	97	2	,	,	PUNCT
cana-1174	97	3	we	we	PRON
cana-1174	97	4	know	know	VERB
cana-1174	97	5	the	the	DET
cana-1174	97	6	dimension	dimension	NOUN
cana-1174	97	7	of	of	ADP
cana-1174	97	8	the	the	DET
cana-1174	97	9	𝑑	𝑑	PROPN
cana-1174	97	10	triangular	triangular	NOUN
cana-1174	97	11	surface	surface	NOUN
cana-1174	97	12	of	of	ADP
cana-1174	97	13	𝑇	𝑇	PROPN
cana-1174	97	14	(	(	PUNCT
cana-1174	97	15	𝑛	𝑛	NOUN
cana-1174	97	16	)	)	PUNCT
cana-1174	97	17	is	be	AUX
cana-1174	97	18	two	two	NUM
cana-1174	97	19	-	-	PUNCT
cana-1174	97	20	dimensional	dimensional	ADJ
cana-1174	97	21	,	,	PUNCT
cana-1174	97	22	so	so	ADV
cana-1174	97	23	𝑇	𝑇	PROPN
cana-1174	97	24	(	(	PUNCT
cana-1174	97	25	𝑛	𝑛	PROPN
cana-1174	97	26	+	+	NOUN
cana-1174	97	27	1	1	NUM
cana-1174	97	28	)	)	PUNCT
cana-1174	97	29	will	will	AUX
cana-1174	97	30	be	be	AUX
cana-1174	97	31	added	add	VERB
cana-1174	97	32	.	.	PUNCT
cana-1174	98	1	𝑑	𝑑	PROPN
cana-1174	98	2	𝑑−1	𝑑−1	PROPN
cana-1174	98	3	now	now	ADV
cana-1174	98	4	,	,	PUNCT
cana-1174	98	5	we	we	PRON
cana-1174	98	6	investigate	investigate	VERB
cana-1174	98	7	the	the	DET
cana-1174	98	8	explicit	explicit	ADJ
cana-1174	98	9	form	form	NOUN
cana-1174	98	10	of	of	ADP
cana-1174	98	11	𝑇	𝑇	PROPN
cana-1174	98	12	(	(	PUNCT
cana-1174	98	13	𝑛	𝑛	NOUN
cana-1174	98	14	)	)	PUNCT
cana-1174	98	15	.	.	PUNCT
cana-1174	99	1	rather	rather	ADV
cana-1174	99	2	than	than	ADP
cana-1174	99	3	using	use	VERB
cana-1174	99	4	the	the	DET
cana-1174	99	5	geometric	geometric	ADJ
cana-1174	99	6	property	property	NOUN
cana-1174	99	7	,	,	PUNCT
cana-1174	99	8	we	we	PRON
cana-1174	99	9	𝑑	𝑑	VERB
cana-1174	99	10	will	will	AUX
cana-1174	99	11	again	again	ADV
cana-1174	99	12	use	use	VERB
cana-1174	99	13	the	the	DET
cana-1174	99	14	coordinate	coordinate	NOUN
cana-1174	99	15	system	system	NOUN
cana-1174	99	16	expression	expression	NOUN
cana-1174	99	17	:	:	PUNCT
cana-1174	99	18	𝑇	𝑇	PROPN
cana-1174	99	19	(	(	PUNCT
cana-1174	99	20	𝑛	𝑛	NOUN
cana-1174	99	21	)	)	PUNCT
cana-1174	99	22	=	=	SYM
cana-1174	99	23	|{(𝑥	|{(𝑥	NOUN
cana-1174	99	24	,	,	PUNCT
cana-1174	99	25	𝑥	𝑥	X
cana-1174	99	26	,	,	PUNCT
cana-1174	99	27	...	...	PUNCT
cana-1174	99	28	,	,	PUNCT
cana-1174	99	29	𝑥	𝑥	PROPN
cana-1174	99	30	)	)	PUNCT
cana-1174	99	31	:	:	PUNCT
cana-1174	99	32	0	0	NUM
cana-1174	99	33	≤	≤	NUM
cana-1174	99	34	𝑥	𝑥	PRON
cana-1174	99	35	,	,	PUNCT
cana-1174	99	36	𝑥	𝑥	X
cana-1174	99	37	,	,	PUNCT
cana-1174	99	38	...	...	PUNCT
cana-1174	99	39	,	,	PUNCT
cana-1174	99	40	𝑥	𝑥	X
cana-1174	99	41	,	,	PUNCT
cana-1174	99	42	0	0	NUM
cana-1174	99	43	≤	≤	NUM
cana-1174	100	1	𝑥	𝑥	PROPN
cana-1174	101	1	+	+	CCONJ
cana-1174	101	2	...	...	PUNCT
cana-1174	102	1	+	+	CCONJ
cana-1174	103	1	𝑥	𝑥	X
cana-1174	103	2	≤	≤	NUM
cana-1174	103	3	𝑛	𝑛	PRON
cana-1174	103	4	−	−	PROPN
cana-1174	103	5	1}|	1}|	NUM
cana-1174	103	6	,	,	PUNCT
cana-1174	103	7	𝑑	𝑑	PROPN
cana-1174	103	8	1	1	NUM
cana-1174	103	9	2	2	NUM
cana-1174	103	10	𝑑	𝑑	PROPN
cana-1174	103	11	1	1	NUM
cana-1174	103	12	2	2	NUM
cana-1174	103	13	𝑑	𝑑	PROPN
cana-1174	103	14	1	1	NUM
cana-1174	103	15	𝑑	𝑑	VERB
cana-1174	103	16	since	since	SCONJ
cana-1174	103	17	using	use	VERB
cana-1174	103	18	the	the	DET
cana-1174	103	19	algebraic	algebraic	ADJ
cana-1174	103	20	properties	property	NOUN
cana-1174	103	21	is	be	AUX
cana-1174	103	22	better	well	ADJ
cana-1174	103	23	for	for	ADP
cana-1174	103	24	higher	high	ADJ
cana-1174	103	25	dimensional	dimensional	ADJ
cana-1174	103	26	space	space	NOUN
cana-1174	103	27	.	.	PUNCT
cana-1174	104	1	the	the	DET
cana-1174	104	2	above	above	ADJ
cana-1174	104	3	expression	expression	NOUN
cana-1174	104	4	can	can	AUX
cana-1174	104	5	be	be	AUX
cana-1174	104	6	rewritten	rewrite	VERB
cana-1174	104	7	as	as	ADP
cana-1174	104	8	𝑇	𝑇	PROPN
cana-1174	104	9	(	(	PUNCT
cana-1174	104	10	𝑛	𝑛	NOUN
cana-1174	104	11	)	)	PUNCT
cana-1174	104	12	=	=	SYM
cana-1174	105	1	|{(𝑥	|{(𝑥	NOUN
cana-1174	105	2	,	,	PUNCT
cana-1174	105	3	𝑥	𝑥	X
cana-1174	105	4	,	,	PUNCT
cana-1174	105	5	...	...	PUNCT
cana-1174	105	6	,	,	PUNCT
cana-1174	105	7	𝑥	𝑥	PROPN
cana-1174	105	8	)	)	PUNCT
cana-1174	105	9	:	:	PUNCT
cana-1174	105	10	0	0	NUM
cana-1174	105	11	≤	≤	NUM
cana-1174	106	1	𝑥	𝑥	PRON
cana-1174	106	2	,	,	PUNCT
cana-1174	106	3	𝑥	𝑥	X
cana-1174	106	4	,	,	PUNCT
cana-1174	106	5	...	...	PUNCT
cana-1174	106	6	,	,	PUNCT
cana-1174	106	7	𝑥	𝑥	X
cana-1174	106	8	,	,	PUNCT
cana-1174	106	9	𝑥	𝑥	PROPN
cana-1174	107	1	+	+	CCONJ
cana-1174	107	2	...	...	PUNCT
cana-1174	108	1	+	+	CCONJ
cana-1174	108	2	𝑥	𝑥	X
cana-1174	108	3	=	=	SYM
cana-1174	108	4	𝑛	𝑛	PRON
cana-1174	108	5	−	−	PROPN
cana-1174	108	6	1}|	1}|	NUM
cana-1174	108	7	𝑑	𝑑	NOUN
cana-1174	108	8	1	1	NUM
cana-1174	108	9	2	2	NUM
cana-1174	108	10	𝑑+1	𝑑+1	NUM
cana-1174	108	11	1	1	NUM
cana-1174	108	12	2	2	NUM
cana-1174	108	13	𝑑+1	𝑑+1	PROPN
cana-1174	108	14	1	1	NUM
cana-1174	108	15	𝑑+1	𝑑+1	NUM
cana-1174	108	16	by	by	ADP
cana-1174	108	17	introducing	introduce	VERB
cana-1174	108	18	one	one	NUM
cana-1174	108	19	more	more	ADV
cana-1174	108	20	variable	variable	ADJ
cana-1174	108	21	𝑥	𝑥	INTJ
cana-1174	108	22	.	.	PUNCT
cana-1174	109	1	if	if	SCONJ
cana-1174	109	2	𝑑+1	𝑑+1	PRON
cana-1174	109	3	𝑥	𝑥	X
cana-1174	109	4	,	,	PUNCT
cana-1174	109	5	𝑥	𝑥	X
cana-1174	109	6	,	,	PUNCT
cana-1174	109	7	...	...	PUNCT
cana-1174	109	8	,	,	PUNCT
cana-1174	109	9	𝑥	𝑥	PRON
cana-1174	109	10	1	1	NUM
cana-1174	109	11	2	2	NUM
cana-1174	109	12	𝑑	𝑑	NOUN
cana-1174	109	13	are	be	AUX
cana-1174	109	14	determined	determine	VERB
cana-1174	109	15	then	then	ADV
cana-1174	109	16	𝑥	𝑥	PROPN
cana-1174	109	17	𝑑+1	𝑑+1	PRON
cana-1174	109	18	will	will	AUX
cana-1174	109	19	be	be	AUX
cana-1174	109	20	determined	determine	VERB
cana-1174	109	21	by	by	ADP
cana-1174	109	22	𝑥	𝑥	PROPN
cana-1174	109	23	=	=	SYM
cana-1174	109	24	𝑛	𝑛	PRON
cana-1174	109	25	−	−	NUM
cana-1174	109	26	1	1	NUM
cana-1174	110	1	−	−	NOUN
cana-1174	110	2	𝑥	𝑥	PRON
cana-1174	110	3	−	−	NOUN
cana-1174	110	4	...	...	PUNCT
cana-1174	111	1	−	−	ADP
cana-1174	111	2	𝑥	𝑥	X
cana-1174	111	3	automatically	automatically	ADV
cana-1174	111	4	.	.	PUNCT
cana-1174	112	1	so	so	ADV
cana-1174	112	2	,	,	PUNCT
cana-1174	112	3	the	the	DET
cana-1174	112	4	number	number	NOUN
cana-1174	112	5	of	of	ADP
cana-1174	112	6	elements	element	NOUN
cana-1174	112	7	of	of	ADP
cana-1174	112	8	two	two	NUM
cana-1174	112	9	𝑑+1	𝑑+1	NUM
cana-1174	112	10	1	1	NUM
cana-1174	112	11	𝑑	𝑑	NOUN
cana-1174	112	12	sets	set	NOUN
cana-1174	112	13	in	in	ADP
cana-1174	112	14	two	two	NUM
cana-1174	112	15	expressions	expression	NOUN
cana-1174	112	16	is	be	AUX
cana-1174	112	17	the	the	DET
cana-1174	112	18	same	same	ADJ
cana-1174	112	19	.	.	PUNCT
cana-1174	113	1	let	let	AUX
cana-1174	113	2	assume	assume	VERB
cana-1174	113	3	that	that	SCONJ
cana-1174	113	4	there	there	PRON
cana-1174	113	5	are	be	VERB
cana-1174	113	6	𝑛	𝑛	DET
cana-1174	113	7	−	−	NUM
cana-1174	113	8	1	1	NUM
cana-1174	113	9	candies	candy	NOUN
cana-1174	113	10	are	be	AUX
cana-1174	113	11	on	on	ADP
cana-1174	113	12	the	the	DET
cana-1174	113	13	line	line	NOUN
cana-1174	113	14	and	and	CCONJ
cana-1174	113	15	we	we	PRON
cana-1174	113	16	would	would	AUX
cana-1174	113	17	like	like	VERB
cana-1174	113	18	to	to	PART
cana-1174	113	19	distribute	distribute	VERB
cana-1174	113	20	these	these	DET
cana-1174	113	21	candies	candy	NOUN
cana-1174	113	22	to	to	ADP
cana-1174	113	23	𝑑	𝑑	PROPN
cana-1174	113	24	+	+	NOUN
cana-1174	113	25	1	1	NUM
cana-1174	113	26	people	people	NOUN
cana-1174	113	27	.	.	PUNCT
cana-1174	114	1	then	then	ADV
cana-1174	114	2	,	,	PUNCT
cana-1174	114	3	instead	instead	ADV
cana-1174	114	4	of	of	ADP
cana-1174	114	5	considering	consider	VERB
cana-1174	114	6	how	how	SCONJ
cana-1174	114	7	to	to	PART
cana-1174	114	8	distribute	distribute	VERB
cana-1174	114	9	these	these	DET
cana-1174	114	10	candies	candy	NOUN
cana-1174	114	11	,	,	PUNCT
cana-1174	114	12	we	we	PRON
cana-1174	114	13	consider	consider	VERB
cana-1174	114	14	𝑑	𝑑	PRON
cana-1174	114	15	partitions	partition	NOUN
cana-1174	114	16	to	to	PART
cana-1174	114	17	divide	divide	VERB
cana-1174	114	18	these	these	DET
cana-1174	114	19	𝑛	𝑛	DET
cana-1174	114	20	−	−	NUM
cana-1174	114	21	1	1	NUM
cana-1174	114	22	candies	candy	NOUN
cana-1174	114	23	.	.	PUNCT
cana-1174	115	1	if	if	SCONJ
cana-1174	115	2	we	we	PRON
cana-1174	115	3	insert	insert	VERB
cana-1174	115	4	these	these	DET
cana-1174	115	5	𝑑	𝑑	PROPN
cana-1174	115	6	partitions	partition	NOUN
cana-1174	115	7	between	between	ADP
cana-1174	115	8	communications	communication	NOUN
cana-1174	115	9	on	on	ADP
cana-1174	115	10	applied	apply	VERB
cana-1174	115	11	nonlinear	nonlinear	ADJ
cana-1174	115	12	analysis	analysis	NOUN
cana-1174	115	13	issn	issn	NOUN
cana-1174	115	14	:	:	PUNCT
cana-1174	115	15	1074	1074	NUM
cana-1174	115	16	-	-	PUNCT
cana-1174	115	17	133x	133x	NUM
cana-1174	115	18	vol	vol	NOUN
cana-1174	115	19	31	31	NUM
cana-1174	115	20	no	no	NOUN
cana-1174	115	21	.	.	PUNCT
cana-1174	116	1	6s	6s	NUM
cana-1174	116	2	(	(	PUNCT
cana-1174	116	3	2024	2024	NUM
cana-1174	116	4	)	)	PUNCT
cana-1174	116	5	157	157	NUM
cana-1174	116	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1174	116	7	𝑛	𝑛	DET
cana-1174	116	8	−	−	NUM
cana-1174	116	9	1	1	NUM
cana-1174	116	10	candies	candy	NOUN
cana-1174	116	11	then	then	ADV
cana-1174	116	12	the	the	DET
cana-1174	116	13	number	number	NOUN
cana-1174	116	14	of	of	ADP
cana-1174	116	15	candies	candy	NOUN
cana-1174	116	16	for	for	ADP
cana-1174	116	17	each	each	DET
cana-1174	116	18	person	person	NOUN
cana-1174	116	19	will	will	AUX
cana-1174	116	20	be	be	AUX
cana-1174	116	21	determined	determine	VERB
cana-1174	116	22	.	.	PUNCT
cana-1174	117	1	instead	instead	ADV
cana-1174	117	2	of	of	ADP
cana-1174	117	3	inserting	insert	VERB
cana-1174	117	4	𝑑	𝑑	PRON
cana-1174	117	5	partitions	partition	NOUN
cana-1174	117	6	between	between	ADP
cana-1174	117	7	candies	candy	NOUN
cana-1174	117	8	,	,	PUNCT
cana-1174	117	9	we	we	PRON
cana-1174	117	10	add	add	VERB
cana-1174	117	11	𝑑	𝑑	ADP
cana-1174	117	12	candies	candy	NOUN
cana-1174	117	13	so	so	ADV
cana-1174	117	14	considering	consider	VERB
cana-1174	117	15	𝑛	𝑛	PROPN
cana-1174	117	16	+	+	CCONJ
cana-1174	117	17	𝑑	𝑑	PROPN
cana-1174	117	18	−	−	NUM
cana-1174	117	19	1	1	NUM
cana-1174	117	20	candies	candy	NOUN
cana-1174	117	21	first	first	ADV
cana-1174	117	22	.	.	PUNCT
cana-1174	118	1	after	after	ADP
cana-1174	118	2	that	that	PRON
cana-1174	118	3	,	,	PUNCT
cana-1174	118	4	we	we	PRON
cana-1174	118	5	choose	choose	VERB
cana-1174	118	6	𝑑	𝑑	NOUN
cana-1174	118	7	candies	candy	NOUN
cana-1174	118	8	and	and	CCONJ
cana-1174	118	9	replace	replace	VERB
cana-1174	118	10	them	they	PRON
cana-1174	118	11	with	with	ADP
cana-1174	118	12	partitions	partition	NOUN
cana-1174	118	13	.	.	PUNCT
cana-1174	119	1	so	so	ADV
cana-1174	119	2	,	,	PUNCT
cana-1174	119	3	the	the	DET
cana-1174	119	4	number	number	NOUN
cana-1174	119	5	of	of	ADP
cana-1174	119	6	ways	way	NOUN
cana-1174	119	7	to	to	PART
cana-1174	119	8	insert	insert	VERB
cana-1174	119	9	partitions	partition	NOUN
cana-1174	119	10	is	be	AUX
cana-1174	119	11	(	(	PUNCT
cana-1174	119	12	𝑛+𝑑−1	𝑛+𝑑−1	ADJ
cana-1174	119	13	)	)	PUNCT
cana-1174	119	14	!	!	PUNCT
cana-1174	120	1	𝐶	𝐶	PROPN
cana-1174	120	2	=	=	PRON
cana-1174	120	3	.	.	PUNCT
cana-1174	121	1	finally	finally	ADV
cana-1174	121	2	,	,	PUNCT
cana-1174	121	3	we	we	PRON
cana-1174	121	4	get	get	VERB
cana-1174	121	5	the	the	DET
cana-1174	121	6	explicit	explicit	ADJ
cana-1174	121	7	form	form	NOUN
cana-1174	121	8	of	of	ADP
cana-1174	121	9	𝑇	𝑇	PROPN
cana-1174	121	10	(	(	PUNCT
cana-1174	121	11	𝑛	𝑛	NOUN
cana-1174	121	12	)	)	PUNCT
cana-1174	121	13	as	as	SCONJ
cana-1174	121	14	follows	follow	VERB
cana-1174	121	15	:	:	PUNCT
cana-1174	121	16	𝑛+𝑑−2	𝑛+𝑑−2	PROPN
cana-1174	121	17	𝑑	𝑑	X
cana-1174	121	18	(	(	PUNCT
cana-1174	121	19	𝑛−1)!𝑑	𝑛−1)!𝑑	NOUN
cana-1174	121	20	!	!	PUNCT
cana-1174	122	1	𝑇	𝑇	PROPN
cana-1174	122	2	(	(	PUNCT
cana-1174	122	3	𝑛	𝑛	NOUN
cana-1174	122	4	)	)	PUNCT
cana-1174	122	5	=	=	SYM
cana-1174	122	6	𝑑	𝑑	PROPN
cana-1174	122	7	𝑑	𝑑	PROPN
cana-1174	122	8	(	(	PUNCT
cana-1174	122	9	𝑛+𝑑−1	𝑛+𝑑−1	PROPN
cana-1174	122	10	)	)	PUNCT
cana-1174	122	11	!	!	PUNCT
cana-1174	123	1	(	(	PUNCT
cana-1174	123	2	𝑛−1)!𝑑	𝑛−1)!𝑑	PROPN
cana-1174	123	3	!	!	PUNCT
cana-1174	124	1	when	when	SCONJ
cana-1174	124	2	𝑑	𝑑	PROPN
cana-1174	124	3	=	=	SYM
cana-1174	124	4	1	1	NUM
cana-1174	124	5	,	,	PUNCT
cana-1174	124	6	we	we	PRON
cana-1174	124	7	get	get	VERB
cana-1174	124	8	𝑇	𝑇	PROPN
cana-1174	124	9	(	(	PUNCT
cana-1174	124	10	𝑛	𝑛	NOUN
cana-1174	124	11	)	)	PUNCT
cana-1174	124	12	=	=	SYM
cana-1174	124	13	1	1	NUM
cana-1174	124	14	𝑛	𝑛	NOUN
cana-1174	124	15	!	!	PUNCT
cana-1174	124	16	(	(	PUNCT
cana-1174	124	17	𝑛−1)!1	𝑛−1)!1	NOUN
cana-1174	124	18	!	!	PUNCT
cana-1174	125	1	=	=	SYM
cana-1174	125	2	𝑛	𝑛	PROPN
cana-1174	126	1	and	and	CCONJ
cana-1174	126	2	it	it	PRON
cana-1174	126	3	implies	imply	VERB
cana-1174	126	4	one	one	NUM
cana-1174	126	5	-	-	PUNCT
cana-1174	126	6	dimensional	dimensional	ADJ
cana-1174	126	7	triangular	triangular	NOUN
cana-1174	126	8	numbers	number	NOUN
cana-1174	126	9	are	be	AUX
cana-1174	126	10	just	just	ADV
cana-1174	126	11	natural	natural	ADJ
cana-1174	126	12	numbers	number	NOUN
cana-1174	126	13	.	.	PUNCT
cana-1174	127	1	so	so	ADV
cana-1174	127	2	,	,	PUNCT
cana-1174	127	3	we	we	PRON
cana-1174	127	4	can	can	AUX
cana-1174	127	5	conclude	conclude	VERB
cana-1174	127	6	that	that	PRON
cana-1174	127	7	𝑆	𝑆	PROPN
cana-1174	127	8	(	(	PUNCT
cana-1174	127	9	𝑛	𝑛	PROPN
cana-1174	127	10	)	)	PUNCT
cana-1174	127	11	=	=	SYM
cana-1174	127	12	𝑇	𝑇	PROPN
cana-1174	127	13	(	(	PUNCT
cana-1174	127	14	𝑛	𝑛	NOUN
cana-1174	127	15	)	)	PUNCT
cana-1174	127	16	=	=	SYM
cana-1174	127	17	𝑛	𝑛	PRON
cana-1174	127	18	1	1	NUM
cana-1174	127	19	1	1	NUM
cana-1174	127	20	where	where	SCONJ
cana-1174	127	21	𝑆	𝑆	PROPN
cana-1174	127	22	(	(	PUNCT
cana-1174	127	23	𝑛	𝑛	NOUN
cana-1174	127	24	)	)	PUNCT
cana-1174	127	25	is	be	AUX
cana-1174	127	26	a	a	DET
cana-1174	127	27	one	one	NUM
cana-1174	127	28	-	-	PUNCT
cana-1174	127	29	dimensional	dimensional	ADJ
cana-1174	127	30	square	square	ADJ
cana-1174	127	31	number	number	NOUN
cana-1174	127	32	.	.	PUNCT
cana-1174	127	33	1	1	NUM
cana-1174	127	34	when	when	SCONJ
cana-1174	127	35	𝑑	𝑑	PROPN
cana-1174	127	36	=	=	SYM
cana-1174	127	37	2	2	NUM
cana-1174	127	38	,	,	PUNCT
cana-1174	127	39	we	we	PRON
cana-1174	127	40	get	get	VERB
cana-1174	127	41	𝑇	𝑇	PROPN
cana-1174	127	42	(	(	PUNCT
cana-1174	127	43	𝑛	𝑛	NOUN
cana-1174	127	44	)	)	PUNCT
cana-1174	127	45	=	=	SYM
cana-1174	127	46	2	2	NUM
cana-1174	127	47	(	(	PUNCT
cana-1174	127	48	𝑛+1	𝑛+1	NOUN
cana-1174	127	49	)	)	PUNCT
cana-1174	127	50	!	!	PUNCT
cana-1174	128	1	(	(	PUNCT
cana-1174	128	2	𝑛−1)!2	𝑛−1)!2	NOUN
cana-1174	128	3	!	!	PUNCT
cana-1174	128	4	=	=	PUNCT
cana-1174	129	1	𝑛(𝑛+1	𝑛(𝑛+1	NUM
cana-1174	129	2	)	)	PUNCT
cana-1174	129	3	2	2	NUM
cana-1174	129	4	and	and	CCONJ
cana-1174	129	5	we	we	PRON
cana-1174	129	6	can	can	AUX
cana-1174	129	7	recover	recover	VERB
cana-1174	129	8	the	the	DET
cana-1174	129	9	result	result	NOUN
cana-1174	129	10	of	of	ADP
cana-1174	129	11	the	the	DET
cana-1174	129	12	original	original	ADJ
cana-1174	129	13	triangular	triangular	NOUN
cana-1174	129	14	numbers	number	NOUN
cana-1174	129	15	.	.	PUNCT
cana-1174	130	1	when	when	SCONJ
cana-1174	130	2	𝑑	𝑑	PROPN
cana-1174	130	3	=	=	SYM
cana-1174	130	4	3	3	NUM
cana-1174	130	5	,	,	PUNCT
cana-1174	130	6	we	we	PRON
cana-1174	130	7	can	can	AUX
cana-1174	130	8	obtain	obtain	VERB
cana-1174	130	9	𝑇	𝑇	PROPN
cana-1174	130	10	(	(	PUNCT
cana-1174	130	11	𝑛	𝑛	NOUN
cana-1174	130	12	)	)	PUNCT
cana-1174	130	13	=	=	SYM
cana-1174	130	14	3	3	NUM
cana-1174	130	15	(	(	PUNCT
cana-1174	130	16	𝑛+2	𝑛+2	NUM
cana-1174	130	17	)	)	PUNCT
cana-1174	130	18	!	!	PUNCT
cana-1174	131	1	(	(	PUNCT
cana-1174	131	2	𝑛−1)!3	𝑛−1)!3	NOUN
cana-1174	131	3	!	!	PUNCT
cana-1174	132	1	=	=	SYM
cana-1174	132	2	𝑛(𝑛+1)(𝑛+2	𝑛(𝑛+1)(𝑛+2	PROPN
cana-1174	132	3	)	)	PUNCT
cana-1174	132	4	6	6	NUM
cana-1174	132	5	recall	recall	NOUN
cana-1174	132	6	that	that	PRON
cana-1174	132	7	we	we	PRON
cana-1174	132	8	proved	prove	VERB
cana-1174	132	9	𝑇	𝑇	PROPN
cana-1174	132	10	(	(	PUNCT
cana-1174	132	11	𝑛	𝑛	PROPN
cana-1174	132	12	+	+	NOUN
cana-1174	132	13	1	1	NUM
cana-1174	132	14	)	)	PUNCT
cana-1174	132	15	=	=	SYM
cana-1174	132	16	𝑇	𝑇	PROPN
cana-1174	132	17	(	(	PUNCT
cana-1174	132	18	𝑛	𝑛	NOUN
cana-1174	132	19	)	)	PUNCT
cana-1174	132	20	+	+	CCONJ
cana-1174	132	21	𝑇	𝑇	PROPN
cana-1174	132	22	(	(	PUNCT
cana-1174	132	23	𝑛	𝑛	PROPN
cana-1174	132	24	+	+	NOUN
cana-1174	132	25	1	1	NUM
cana-1174	132	26	)	)	PUNCT
cana-1174	132	27	.	.	PUNCT
cana-1174	133	1	if	if	SCONJ
cana-1174	133	2	we	we	PRON
cana-1174	133	3	use	use	VERB
cana-1174	133	4	𝑑	𝑑	PRON
cana-1174	133	5	𝑑	𝑑	PROPN
cana-1174	133	6	𝑑−1	𝑑−1	PROPN
cana-1174	133	7	(	(	PUNCT
cana-1174	133	8	𝑛+𝑑−1	𝑛+𝑑−1	PROPN
cana-1174	133	9	)	)	PUNCT
cana-1174	133	10	!	!	PUNCT
cana-1174	134	1	𝑇	𝑇	PROPN
cana-1174	134	2	(	(	PUNCT
cana-1174	134	3	𝑛	𝑛	NOUN
cana-1174	134	4	)	)	PUNCT
cana-1174	134	5	=	=	NOUN
cana-1174	134	6	=	=	SYM
cana-1174	134	7	𝐶	𝐶	PROPN
cana-1174	134	8	,	,	PUNCT
cana-1174	134	9	𝑑	𝑑	PROPN
cana-1174	134	10	(	(	PUNCT
cana-1174	134	11	𝑛−1)!𝑑	𝑛−1)!𝑑	NOUN
cana-1174	134	12	!	!	PUNCT
cana-1174	135	1	𝑛+𝑑−2	𝑛+𝑑−2	VERB
cana-1174	136	1	𝑑	𝑑	NOUN
cana-1174	136	2	then	then	ADV
cana-1174	136	3	we	we	PRON
cana-1174	136	4	can	can	AUX
cana-1174	136	5	prove	prove	VERB
cana-1174	136	6	the	the	DET
cana-1174	136	7	following	follow	VERB
cana-1174	136	8	identity	identity	NOUN
cana-1174	136	9	using	use	VERB
cana-1174	136	10	the	the	DET
cana-1174	136	11	generalization	generalization	NOUN
cana-1174	136	12	of	of	ADP
cana-1174	136	13	triangular	triangular	NOUN
cana-1174	136	14	numbers	number	NOUN
cana-1174	136	15	:	:	PUNCT
cana-1174	136	16	𝐶	𝐶	PROPN
cana-1174	136	17	=	=	PUNCT
cana-1174	136	18	𝐶	𝐶	PROPN
cana-1174	136	19	+	+	CCONJ
cana-1174	136	20	𝐶	𝐶	PROPN
cana-1174	136	21	.	.	PUNCT
cana-1174	137	1	𝑛+𝑑−1	𝑛+𝑑−1	PROPN
cana-1174	137	2	𝑑	𝑑	PROPN
cana-1174	137	3	𝑛+𝑑−2	𝑛+𝑑−2	PROPN
cana-1174	137	4	𝑑	𝑑	NOUN
cana-1174	137	5	𝑛+𝑑−2	𝑛+𝑑−2	VERB
cana-1174	137	6	𝑑−1	𝑑−1	PROPN
cana-1174	137	7	5	5	NUM
cana-1174	137	8	.	.	PUNCT
cana-1174	137	9	comparison	comparison	NOUN
cana-1174	137	10	between	between	ADP
cana-1174	137	11	generalization	generalization	NOUN
cana-1174	137	12	of	of	ADP
cana-1174	137	13	square	square	ADJ
cana-1174	137	14	and	and	CCONJ
cana-1174	137	15	triangular	triangular	NOUN
cana-1174	137	16	numbers	number	NOUN
cana-1174	137	17	in	in	ADP
cana-1174	137	18	this	this	DET
cana-1174	137	19	section	section	NOUN
cana-1174	137	20	,	,	PUNCT
cana-1174	137	21	we	we	PRON
cana-1174	137	22	compare	compare	VERB
cana-1174	137	23	two	two	NUM
cana-1174	137	24	generalized	generalized	ADJ
cana-1174	137	25	numbers	number	NOUN
cana-1174	137	26	.	.	PUNCT
cana-1174	138	1	after	after	SCONJ
cana-1174	138	2	we	we	PRON
cana-1174	138	3	expand	expand	VERB
cana-1174	138	4	all	all	DET
cana-1174	138	5	factorial	factorial	ADJ
cana-1174	138	6	terms	term	NOUN
cana-1174	138	7	in	in	ADP
cana-1174	138	8	𝑇	𝑇	PROPN
cana-1174	138	9	𝑑	𝑑	PROPN
cana-1174	138	10	(	(	PUNCT
cana-1174	138	11	𝑛	𝑛	NOUN
cana-1174	138	12	)	)	PUNCT
cana-1174	138	13	,	,	PUNCT
cana-1174	138	14	we	we	PRON
cana-1174	138	15	can	can	AUX
cana-1174	138	16	rewrite	rewrite	VERB
cana-1174	138	17	𝑇	𝑇	PROPN
cana-1174	138	18	𝑑	𝑑	PROPN
cana-1174	138	19	(	(	PUNCT
cana-1174	138	20	𝑛	𝑛	NOUN
cana-1174	138	21	)	)	PUNCT
cana-1174	138	22	as	as	SCONJ
cana-1174	138	23	follows	follow	VERB
cana-1174	138	24	:	:	PUNCT
cana-1174	138	25	𝑇	𝑇	PROPN
cana-1174	138	26	(	(	PUNCT
cana-1174	138	27	𝑛	𝑛	NOUN
cana-1174	138	28	)	)	PUNCT
cana-1174	138	29	=	=	SYM
cana-1174	138	30	𝑑	𝑑	PROPN
cana-1174	138	31	𝑛(𝑛+1	𝑛(𝑛+1	PROPN
cana-1174	138	32	)	)	PUNCT
cana-1174	138	33	...	...	PUNCT
cana-1174	139	1	(	(	PUNCT
cana-1174	139	2	𝑛+𝑑−1	𝑛+𝑑−1	ADJ
cana-1174	139	3	)	)	PUNCT
cana-1174	139	4	𝑑	𝑑	NOUN
cana-1174	139	5	!	!	PROPN
cana-1174	139	6	.	.	PUNCT
cana-1174	140	1	communications	communication	NOUN
cana-1174	140	2	on	on	ADP
cana-1174	140	3	applied	apply	VERB
cana-1174	140	4	nonlinear	nonlinear	ADJ
cana-1174	140	5	analysis	analysis	NOUN
cana-1174	140	6	issn	issn	NOUN
cana-1174	140	7	:	:	PUNCT
cana-1174	140	8	1074	1074	NUM
cana-1174	140	9	-	-	PUNCT
cana-1174	140	10	133x	133x	NUM
cana-1174	140	11	vol	vol	NOUN
cana-1174	140	12	31	31	NUM
cana-1174	140	13	no	no	NOUN
cana-1174	140	14	.	.	PUNCT
cana-1174	141	1	6s	6s	NUM
cana-1174	141	2	(	(	PUNCT
cana-1174	141	3	2024	2024	NUM
cana-1174	141	4	)	)	PUNCT
cana-1174	141	5	158	158	NUM
cana-1174	141	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1174	141	7	,	,	PUNCT
cana-1174	141	8	the	the	DET
cana-1174	141	9	ratio	ratio	NOUN
cana-1174	141	10	between	between	ADP
cana-1174	141	11	𝑇	𝑇	PROPN
cana-1174	141	12	(	(	PUNCT
cana-1174	141	13	𝑛	𝑛	NOUN
cana-1174	141	14	)	)	PUNCT
cana-1174	141	15	and	and	CCONJ
cana-1174	141	16	𝑆	𝑆	PROPN
cana-1174	141	17	(	(	PUNCT
cana-1174	141	18	𝑛	𝑛	NOUN
cana-1174	141	19	)	)	PUNCT
cana-1174	141	20	can	can	AUX
cana-1174	141	21	be	be	AUX
cana-1174	141	22	considered	consider	VERB
cana-1174	141	23	as	as	SCONJ
cana-1174	141	24	follows	follow	VERB
cana-1174	141	25	:	:	PUNCT
cana-1174	141	26	𝑑	𝑑	PROPN
cana-1174	141	27	𝑑	𝑑	PROPN
cana-1174	141	28	lim	lim	NOUN
cana-1174	141	29	𝑛	𝑛	PROPN
cana-1174	141	30	→	→	SYM
cana-1174	141	31	∞	∞	NUM
cana-1174	141	32	𝑇	𝑇	PROPN
cana-1174	141	33	(	(	PUNCT
cana-1174	141	34	𝑛	𝑛	NOUN
cana-1174	141	35	)	)	PUNCT
cana-1174	141	36	𝑑	𝑑	PROPN
cana-1174	141	37	𝑆	𝑆	PROPN
cana-1174	141	38	(	(	PUNCT
cana-1174	141	39	𝑛	𝑛	NOUN
cana-1174	141	40	)	)	PUNCT
cana-1174	141	41	=	=	SYM
cana-1174	141	42	𝑑	𝑑	PROPN
cana-1174	141	43	1	1	NUM
cana-1174	141	44	𝑑	𝑑	NOUN
cana-1174	141	45	!	!	NOUN
cana-1174	142	1	when	when	SCONJ
cana-1174	142	2	𝑛	𝑛	PRON
cana-1174	142	3	approaches	approach	VERB
cana-1174	142	4	infinity	infinity	VERB
cana-1174	142	5	1	1	NUM
cana-1174	142	6	2	2	NUM
cana-1174	142	7	,	,	PUNCT
cana-1174	142	8	...	...	PUNCT
cana-1174	142	9	,	,	PUNCT
cana-1174	142	10	𝑑−1	𝑑−1	PROPN
cana-1174	142	11	approaches	approach	VERB
cana-1174	142	12	zero	zero	NUM
cana-1174	142	13	.	.	PUNCT
cana-1174	143	1	the	the	DET
cana-1174	143	2	meaning	meaning	NOUN
cana-1174	143	3	of	of	ADP
cana-1174	143	4	the	the	DET
cana-1174	143	5	above	above	ADJ
cana-1174	143	6	limit	limit	NOUN
cana-1174	143	7	𝑛	𝑛	PRON
cana-1174	143	8	𝑛	𝑛	NOUN
cana-1174	143	9	𝑛	𝑛	NOUN
cana-1174	143	10	can	can	AUX
cana-1174	143	11	be	be	AUX
cana-1174	143	12	interpreted	interpret	VERB
cana-1174	143	13	geometrically	geometrically	ADV
cana-1174	143	14	.	.	PUNCT
cana-1174	144	1	we	we	PRON
cana-1174	144	2	consider	consider	VERB
cana-1174	144	3	a	a	DET
cana-1174	144	4	square	square	NOUN
cana-1174	144	5	.	.	PUNCT
cana-1174	145	1	if	if	SCONJ
cana-1174	145	2	we	we	PRON
cana-1174	145	3	divide	divide	VERB
cana-1174	145	4	the	the	DET
cana-1174	145	5	square	square	NOUN
cana-1174	145	6	into	into	ADP
cana-1174	145	7	two	two	NUM
cana-1174	145	8	parts	part	NOUN
cana-1174	145	9	,	,	PUNCT
cana-1174	145	10	then	then	ADV
cana-1174	145	11	we	we	PRON
cana-1174	145	12	get	get	VERB
cana-1174	145	13	the	the	DET
cana-1174	145	14	triangle	triangle	NOUN
cana-1174	145	15	.	.	PUNCT
cana-1174	146	1	the	the	DET
cana-1174	146	2	ratio	ratio	NOUN
cana-1174	146	3	between	between	ADP
cana-1174	146	4	two	two	NUM
cana-1174	146	5	areas	area	NOUN
cana-1174	146	6	of	of	ADP
cana-1174	146	7	a	a	DET
cana-1174	146	8	triangle	triangle	NOUN
cana-1174	146	9	and	and	CCONJ
cana-1174	146	10	the	the	DET
cana-1174	146	11	square	square	NOUN
cana-1174	146	12	is	be	AUX
cana-1174	146	13	1	1	NUM
cana-1174	146	14	.	.	SYM
cana-1174	146	15	2	2	NUM
cana-1174	146	16	similarly	similarly	ADV
cana-1174	146	17	,	,	PUNCT
cana-1174	146	18	we	we	PRON
cana-1174	146	19	consider	consider	VERB
cana-1174	146	20	the	the	DET
cana-1174	146	21	cube	cube	NOUN
cana-1174	146	22	.	.	PUNCT
cana-1174	147	1	we	we	PRON
cana-1174	147	2	can	can	AUX
cana-1174	147	3	divide	divide	VERB
cana-1174	147	4	this	this	DET
cana-1174	147	5	cube	cube	NOUN
cana-1174	147	6	into	into	ADP
cana-1174	147	7	six	six	NUM
cana-1174	147	8	congruent	congruent	ADJ
cana-1174	147	9	tetrahedrons	tetrahedron	NOUN
cana-1174	147	10	.	.	PUNCT
cana-1174	148	1	then	then	ADV
cana-1174	148	2	,	,	PUNCT
cana-1174	148	3	the	the	DET
cana-1174	148	4	ratio	ratio	NOUN
cana-1174	148	5	between	between	ADP
cana-1174	148	6	two	two	NUM
cana-1174	148	7	volumes	volume	NOUN
cana-1174	148	8	of	of	ADP
cana-1174	148	9	a	a	DET
cana-1174	148	10	tetrahedron	tetrahedron	NOUN
cana-1174	148	11	and	and	CCONJ
cana-1174	148	12	the	the	DET
cana-1174	148	13	cube	cube	NOUN
cana-1174	148	14	is	be	AUX
cana-1174	148	15	1	1	NUM
cana-1174	148	16	=	=	SYM
cana-1174	148	17	1	1	NUM
cana-1174	148	18	.	.	PUNCT
cana-1174	149	1	in	in	ADP
cana-1174	149	2	general	general	ADJ
cana-1174	149	3	,	,	PUNCT
cana-1174	149	4	if	if	SCONJ
cana-1174	149	5	we	we	PRON
cana-1174	149	6	3	3	NUM
cana-1174	149	7	!	!	NOUN
cana-1174	149	8	6	6	NUM
cana-1174	149	9	consider	consider	VERB
cana-1174	149	10	𝑑-dimensional	𝑑-dimensional	ADJ
cana-1174	149	11	cube	cube	NOUN
cana-1174	149	12	,	,	PUNCT
cana-1174	149	13	then	then	ADV
cana-1174	149	14	we	we	PRON
cana-1174	149	15	can	can	AUX
cana-1174	149	16	divide	divide	VERB
cana-1174	149	17	this	this	DET
cana-1174	149	18	cube	cube	NOUN
cana-1174	149	19	into	into	ADP
cana-1174	149	20	𝑑	𝑑	PROPN
cana-1174	149	21	!	!	X
cana-1174	149	22	congruent	congruent	ADJ
cana-1174	149	23	generalized	generalize	VERB
cana-1174	149	24	tetrahedrons	tetrahedron	NOUN
cana-1174	149	25	such	such	ADJ
cana-1174	149	26	that	that	SCONJ
cana-1174	149	27	the	the	DET
cana-1174	149	28	volume	volume	NOUN
cana-1174	149	29	ratio	ratio	NOUN
cana-1174	149	30	between	between	ADP
cana-1174	149	31	the	the	DET
cana-1174	149	32	tetrahedron	tetrahedron	NOUN
cana-1174	149	33	and	and	CCONJ
cana-1174	149	34	the	the	DET
cana-1174	149	35	cube	cube	NOUN
cana-1174	149	36	is	be	AUX
cana-1174	149	37	6	6	NUM
cana-1174	149	38	.	.	PUNCT
cana-1174	149	39	conclusion	conclusion	NOUN
cana-1174	149	40	and	and	CCONJ
cana-1174	149	41	discussion	discussion	NOUN
cana-1174	149	42	1	1	NUM
cana-1174	149	43	𝑑	𝑑	NOUN
cana-1174	149	44	!	!	PROPN
cana-1174	149	45	.	.	PUNCT
cana-1174	150	1	this	this	DET
cana-1174	150	2	paper	paper	NOUN
cana-1174	150	3	explored	explore	VERB
cana-1174	150	4	the	the	DET
cana-1174	150	5	generalization	generalization	NOUN
cana-1174	150	6	of	of	ADP
cana-1174	150	7	triangular	triangular	NOUN
cana-1174	150	8	and	and	CCONJ
cana-1174	150	9	square	square	ADJ
cana-1174	150	10	numbers	number	NOUN
cana-1174	150	11	to	to	ADP
cana-1174	150	12	higher	higher	ADV
cana-1174	150	13	-	-	PUNCT
cana-1174	150	14	dimensional	dimensional	ADJ
cana-1174	150	15	spaces	space	NOUN
cana-1174	150	16	through	through	ADP
cana-1174	150	17	coordinate	coordinate	NOUN
cana-1174	150	18	systems	system	NOUN
cana-1174	150	19	,	,	PUNCT
cana-1174	150	20	building	build	VERB
cana-1174	150	21	upon	upon	SCONJ
cana-1174	150	22	their	their	PRON
cana-1174	150	23	initial	initial	ADJ
cana-1174	150	24	geometric	geometric	ADJ
cana-1174	150	25	definitions	definition	NOUN
cana-1174	150	26	in	in	ADP
cana-1174	150	27	two	two	NUM
cana-1174	150	28	dimensions	dimension	NOUN
cana-1174	150	29	.	.	PUNCT
cana-1174	151	1	square	square	ADJ
cana-1174	151	2	numbers	number	NOUN
cana-1174	151	3	and	and	CCONJ
cana-1174	151	4	triangular	triangular	NOUN
cana-1174	151	5	numbers	number	NOUN
cana-1174	151	6	in	in	ADP
cana-1174	151	7	two	two	NUM
cana-1174	151	8	-	-	PUNCT
cana-1174	151	9	dimensional	dimensional	ADJ
cana-1174	151	10	case	case	NOUN
cana-1174	151	11	are	be	AUX
cana-1174	151	12	respectively	respectively	ADV
cana-1174	151	13	defined	define	VERB
cana-1174	151	14	as	as	ADP
cana-1174	151	15	|{(𝑎	|{(𝑎	NOUN
cana-1174	151	16	,	,	PUNCT
cana-1174	151	17	𝑏	𝑏	NOUN
cana-1174	151	18	)	)	PUNCT
cana-1174	151	19	:	:	PUNCT
cana-1174	152	1	0	0	NUM
cana-1174	152	2	≤	≤	NUM
cana-1174	152	3	𝑎	𝑎	ADJ
cana-1174	152	4	,	,	PUNCT
cana-1174	152	5	𝑏	𝑏	PROPN
cana-1174	152	6	≤	≤	NUM
cana-1174	152	7	𝑛	𝑛	PRON
cana-1174	152	8	−	−	PROPN
cana-1174	152	9	1}|	1}|	NUM
cana-1174	152	10	and	and	CCONJ
cana-1174	152	11	|{(𝑎	|{(𝑎	NOUN
cana-1174	152	12	,	,	PUNCT
cana-1174	152	13	𝑏	𝑏	NOUN
cana-1174	152	14	)	)	PUNCT
cana-1174	152	15	:	:	PUNCT
cana-1174	152	16	0	0	NUM
cana-1174	152	17	≤	≤	NUM
cana-1174	152	18	𝑎	𝑎	ADJ
cana-1174	152	19	,	,	PUNCT
cana-1174	152	20	𝑏	𝑏	NOUN
cana-1174	152	21	,	,	PUNCT
cana-1174	152	22	0	0	NUM
cana-1174	152	23	≤	≤	NOUN
cana-1174	152	24	𝑎	𝑎	X
cana-1174	152	25	+	+	NOUN
cana-1174	152	26	𝑏	𝑏	NOUN
cana-1174	152	27	≤	≤	NOUN
cana-1174	152	28	𝑛	𝑛	PRON
cana-1174	152	29	−	−	PROPN
cana-1174	152	30	1}|	1}|	NUM
cana-1174	152	31	.	.	PUNCT
cana-1174	152	32	for	for	ADP
cana-1174	152	33	square	square	ADJ
cana-1174	152	34	numbers	number	NOUN
cana-1174	152	35	,	,	PUNCT
cana-1174	152	36	this	this	DET
cana-1174	152	37	concept	concept	NOUN
cana-1174	152	38	is	be	AUX
cana-1174	152	39	then	then	ADV
cana-1174	152	40	generalized	generalize	VERB
cana-1174	152	41	to	to	ADP
cana-1174	152	42	𝑑-dimension	𝑑-dimension	NOUN
cana-1174	152	43	,	,	PUNCT
cana-1174	152	44	resulting	result	VERB
cana-1174	152	45	in	in	ADP
cana-1174	152	46	𝑑	𝑑	PROPN
cana-1174	152	47	𝑆	𝑆	PROPN
cana-1174	152	48	(	(	PUNCT
cana-1174	152	49	𝑛	𝑛	NOUN
cana-1174	152	50	)	)	PUNCT
cana-1174	152	51	=	=	SYM
cana-1174	153	1	|{(𝑥	|{(𝑥	NOUN
cana-1174	153	2	,	,	PUNCT
cana-1174	153	3	𝑥	𝑥	X
cana-1174	153	4	,	,	PUNCT
cana-1174	153	5	...	...	PUNCT
cana-1174	153	6	,	,	PUNCT
cana-1174	153	7	𝑥	𝑥	PROPN
cana-1174	153	8	)	)	PUNCT
cana-1174	153	9	:	:	PUNCT
cana-1174	153	10	0	0	NUM
cana-1174	153	11	≤	≤	NUM
cana-1174	154	1	𝑥	𝑥	PRON
cana-1174	154	2	,	,	PUNCT
cana-1174	154	3	𝑥	𝑥	X
cana-1174	154	4	,	,	PUNCT
cana-1174	154	5	...	...	PUNCT
cana-1174	154	6	,	,	PUNCT
cana-1174	154	7	𝑥	𝑥	PROPN
cana-1174	154	8	≤	≤	NUM
cana-1174	154	9	𝑛	𝑛	PRON
cana-1174	154	10	−	−	PROPN
cana-1174	154	11	1}|	1}|	NUM
cana-1174	154	12	=	=	SYM
cana-1174	154	13	𝑛	𝑛	PROPN
cana-1174	154	14	.	.	PUNCT
cana-1174	155	1	𝑑	𝑑	PROPN
cana-1174	155	2	1	1	NUM
cana-1174	155	3	2	2	NUM
cana-1174	155	4	𝑑	𝑑	PROPN
cana-1174	155	5	1	1	NUM
cana-1174	155	6	2	2	NUM
cana-1174	155	7	𝑑	𝑑	NOUN
cana-1174	155	8	triangular	triangular	NOUN
cana-1174	155	9	numbers	number	NOUN
cana-1174	155	10	follow	follow	VERB
cana-1174	155	11	a	a	DET
cana-1174	155	12	similar	similar	ADJ
cana-1174	155	13	pattern	pattern	NOUN
cana-1174	155	14	in	in	ADP
cana-1174	155	15	𝑑-dimension	𝑑-dimension	NOUN
cana-1174	155	16	,	,	PUNCT
cana-1174	155	17	resulting	result	VERB
cana-1174	155	18	in	in	ADP
cana-1174	155	19	(	(	PUNCT
cana-1174	155	20	𝑛+𝑑−1	𝑛+𝑑−1	ADJ
cana-1174	155	21	)	)	PUNCT
cana-1174	155	22	!	!	PUNCT
cana-1174	156	1	𝑇	𝑇	PROPN
cana-1174	156	2	(	(	PUNCT
cana-1174	156	3	𝑛	𝑛	NOUN
cana-1174	156	4	)	)	PUNCT
cana-1174	156	5	=	=	SYM
cana-1174	156	6	|{(𝑥	|{(𝑥	NOUN
cana-1174	156	7	,	,	PUNCT
cana-1174	156	8	𝑥	𝑥	X
cana-1174	156	9	,	,	PUNCT
cana-1174	156	10	...	...	PUNCT
cana-1174	156	11	,	,	PUNCT
cana-1174	156	12	𝑥	𝑥	PROPN
cana-1174	156	13	)	)	PUNCT
cana-1174	156	14	:	:	PUNCT
cana-1174	156	15	0	0	NUM
cana-1174	156	16	≤	≤	NUM
cana-1174	156	17	𝑥	𝑥	PRON
cana-1174	156	18	,	,	PUNCT
cana-1174	156	19	𝑥	𝑥	X
cana-1174	156	20	,	,	PUNCT
cana-1174	156	21	...	...	PUNCT
cana-1174	156	22	,	,	PUNCT
cana-1174	156	23	𝑥	𝑥	X
cana-1174	156	24	,	,	PUNCT
cana-1174	156	25	0	0	NUM
cana-1174	156	26	≤	≤	NUM
cana-1174	156	27	𝑥	𝑥	PRON
cana-1174	156	28	+	+	NOUN
cana-1174	156	29	...	...	PUNCT
cana-1174	156	30	+	+	CCONJ
cana-1174	156	31	𝑥	𝑥	PROPN
cana-1174	156	32	≤	≤	NUM
cana-1174	156	33	𝑛	𝑛	PRON
cana-1174	156	34	−	−	PROPN
cana-1174	156	35	1}|	1}|	NUM
cana-1174	156	36	=	=	PUNCT
cana-1174	156	37	.	.	PUNCT
cana-1174	157	1	𝑑	𝑑	PROPN
cana-1174	157	2	1	1	NUM
cana-1174	157	3	2	2	NUM
cana-1174	157	4	𝑑	𝑑	PROPN
cana-1174	157	5	1	1	NUM
cana-1174	157	6	2	2	NUM
cana-1174	157	7	𝑑	𝑑	PROPN
cana-1174	157	8	1	1	NUM
cana-1174	157	9	𝑑	𝑑	NOUN
cana-1174	157	10	(	(	PUNCT
cana-1174	157	11	𝑛−1)!𝑑	𝑛−1)!𝑑	NOUN
cana-1174	157	12	!	!	PUNCT
cana-1174	158	1	the	the	DET
cana-1174	158	2	paper	paper	NOUN
cana-1174	158	3	establishes	establish	VERB
cana-1174	158	4	the	the	DET
cana-1174	158	5	identity	identity	NOUN
cana-1174	158	6	𝑇	𝑇	NOUN
cana-1174	158	7	(	(	PUNCT
cana-1174	158	8	𝑛	𝑛	PROPN
cana-1174	158	9	+	+	NOUN
cana-1174	158	10	1	1	NUM
cana-1174	158	11	)	)	PUNCT
cana-1174	158	12	=	=	SYM
cana-1174	158	13	𝑇	𝑇	PROPN
cana-1174	158	14	(	(	PUNCT
cana-1174	158	15	𝑛	𝑛	NOUN
cana-1174	158	16	)	)	PUNCT
cana-1174	159	1	+	+	CCONJ
cana-1174	159	2	𝑇	𝑇	PROPN
cana-1174	159	3	(	(	PUNCT
cana-1174	159	4	𝑛	𝑛	PROPN
cana-1174	159	5	+	+	NOUN
cana-1174	159	6	1	1	NUM
cana-1174	159	7	)	)	PUNCT
cana-1174	159	8	,	,	PUNCT
cana-1174	159	9	proving	prove	VERB
cana-1174	159	10	the	the	DET
cana-1174	159	11	recursive	recursive	NOUN
cana-1174	159	12	𝑑	𝑑	NOUN
cana-1174	159	13	𝑑	𝑑	NOUN
cana-1174	159	14	𝑑−1	𝑑−1	PROPN
cana-1174	159	15	relationship	relationship	NOUN
cana-1174	159	16	between	between	ADP
cana-1174	159	17	triangular	triangular	NOUN
cana-1174	159	18	numbers	number	NOUN
cana-1174	159	19	of	of	ADP
cana-1174	159	20	different	different	ADJ
cana-1174	159	21	dimensions	dimension	NOUN
cana-1174	159	22	.	.	PUNCT
cana-1174	160	1	communications	communication	NOUN
cana-1174	160	2	on	on	ADP
cana-1174	160	3	applied	apply	VERB
cana-1174	160	4	nonlinear	nonlinear	ADJ
cana-1174	160	5	analysis	analysis	NOUN
cana-1174	160	6	issn	issn	NOUN
cana-1174	160	7	:	:	PUNCT
cana-1174	160	8	1074	1074	NUM
cana-1174	160	9	-	-	PUNCT
cana-1174	160	10	133x	133x	NUM
cana-1174	160	11	vol	vol	NOUN
cana-1174	160	12	31	31	NUM
cana-1174	160	13	no	no	NOUN
cana-1174	160	14	.	.	PUNCT
cana-1174	161	1	6s	6s	NUM
cana-1174	161	2	(	(	PUNCT
cana-1174	161	3	2024	2024	NUM
cana-1174	161	4	)	)	PUNCT
cana-1174	161	5	159	159	NUM
cana-1174	161	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1174	161	7	another	another	DET
cana-1174	161	8	significant	significant	ADJ
cana-1174	161	9	finding	finding	NOUN
cana-1174	161	10	is	be	AUX
cana-1174	161	11	the	the	DET
cana-1174	161	12	comparison	comparison	NOUN
cana-1174	161	13	between	between	ADP
cana-1174	161	14	generalized	generalized	ADJ
cana-1174	161	15	triangular	triangular	NOUN
cana-1174	161	16	and	and	CCONJ
cana-1174	161	17	square	square	ADJ
cana-1174	161	18	numbers	number	NOUN
cana-1174	161	19	.	.	PUNCT
cana-1174	162	1	the	the	DET
cana-1174	162	2	ratio	ratio	NOUN
cana-1174	162	3	between	between	ADP
cana-1174	162	4	these	these	DET
cana-1174	162	5	numbers	number	NOUN
cana-1174	162	6	approaches	approach	VERB
cana-1174	162	7	1	1	NUM
cana-1174	162	8	as	as	ADP
cana-1174	162	9	n	n	NUM
cana-1174	162	10	approaches	approach	NOUN
cana-1174	162	11	infinity	infinity	NOUN
cana-1174	162	12	.	.	PUNCT
cana-1174	163	1	this	this	DET
cana-1174	163	2	ratio	ratio	NOUN
cana-1174	163	3	𝑑	𝑑	VERB
cana-1174	163	4	!	!	PROPN
cana-1174	163	5	has	have	VERB
cana-1174	163	6	a	a	DET
cana-1174	163	7	geometric	geometric	ADJ
cana-1174	163	8	interpretation	interpretation	NOUN
cana-1174	163	9	:	:	PUNCT
cana-1174	163	10	dividing	divide	VERB
cana-1174	163	11	a	a	DET
cana-1174	163	12	d	d	ADJ
cana-1174	163	13	-	-	ADJ
cana-1174	163	14	dimensional	dimensional	ADJ
cana-1174	163	15	cube	cube	NOUN
cana-1174	163	16	into	into	ADP
cana-1174	163	17	congruent	congruent	ADJ
cana-1174	163	18	generalized	generalize	VERB
cana-1174	163	19	tetrahedrons	tetrahedron	NOUN
cana-1174	163	20	reveals	reveal	VERB
cana-1174	163	21	that	that	SCONJ
cana-1174	163	22	the	the	DET
cana-1174	163	23	volume	volume	NOUN
cana-1174	163	24	ratio	ratio	NOUN
cana-1174	163	25	between	between	ADP
cana-1174	163	26	a	a	DET
cana-1174	163	27	tetrahedron	tetrahedron	NOUN
cana-1174	163	28	and	and	CCONJ
cana-1174	163	29	the	the	DET
cana-1174	163	30	cube	cube	NOUN
cana-1174	163	31	is	be	AUX
cana-1174	163	32	1	1	NUM
cana-1174	163	33	.	.	PUNCT
cana-1174	164	1	𝑑	𝑑	X
cana-1174	164	2	!	!	NOUN
cana-1174	164	3	triangular	triangular	NOUN
cana-1174	164	4	numbers	number	NOUN
cana-1174	164	5	also	also	ADV
cana-1174	164	6	have	have	VERB
cana-1174	164	7	practical	practical	ADJ
cana-1174	164	8	applications	application	NOUN
cana-1174	164	9	in	in	ADP
cana-1174	164	10	addition	addition	NOUN
cana-1174	164	11	to	to	ADP
cana-1174	164	12	combinatorial	combinatorial	ADJ
cana-1174	164	13	identities	identity	NOUN
cana-1174	164	14	and	and	CCONJ
cana-1174	164	15	can	can	AUX
cana-1174	164	16	be	be	AUX
cana-1174	164	17	particularly	particularly	ADV
cana-1174	164	18	insightful	insightful	ADJ
cana-1174	164	19	when	when	SCONJ
cana-1174	164	20	generalized	generalized	ADJ
cana-1174	164	21	to	to	ADP
cana-1174	164	22	higher	high	ADJ
cana-1174	164	23	dimensions	dimension	NOUN
cana-1174	164	24	.	.	PUNCT
cana-1174	165	1	in	in	ADP
cana-1174	165	2	a	a	DET
cana-1174	165	3	fully	fully	ADV
cana-1174	165	4	connected	connected	ADJ
cana-1174	165	5	network	network	NOUN
cana-1174	165	6	of	of	ADP
cana-1174	165	7	n	n	NOUN
cana-1174	165	8	computing	computing	NOUN
cana-1174	165	9	devices	device	NOUN
cana-1174	165	10	,	,	PUNCT
cana-1174	165	11	the	the	DET
cana-1174	165	12	number	number	NOUN
cana-1174	165	13	of	of	ADP
cana-1174	165	14	necessary	necessary	ADJ
cana-1174	165	15	connections	connection	NOUN
cana-1174	165	16	corresponds	correspond	VERB
cana-1174	165	17	to	to	ADP
cana-1174	165	18	the	the	DET
cana-1174	165	19	triangular	triangular	NOUN
cana-1174	165	20	number	number	NOUN
cana-1174	165	21	𝑇	𝑇	PROPN
cana-1174	165	22	,	,	PUNCT
cana-1174	165	23	akin	akin	ADJ
cana-1174	165	24	to	to	ADP
cana-1174	165	25	the	the	DET
cana-1174	165	26	handshake	handshake	ADJ
cana-1174	165	27	problem	problem	NOUN
cana-1174	165	28	.	.	PUNCT
cana-1174	166	1	in	in	ADP
cana-1174	166	2	a	a	DET
cana-1174	166	3	three	three	NUM
cana-1174	166	4	-	-	PUNCT
cana-1174	166	5	dimensional	dimensional	ADJ
cana-1174	166	6	lattice	lattice	NOUN
cana-1174	166	7	of	of	ADP
cana-1174	166	8	𝑛−1	𝑛−1	PROPN
cana-1174	166	9	computing	computing	NOUN
cana-1174	166	10	nodes	node	NOUN
cana-1174	166	11	,	,	PUNCT
cana-1174	166	12	each	each	DET
cana-1174	166	13	node	node	NOUN
cana-1174	166	14	's	's	PART
cana-1174	166	15	connections	connection	NOUN
cana-1174	166	16	can	can	AUX
cana-1174	166	17	be	be	AUX
cana-1174	166	18	shown	show	VERB
cana-1174	166	19	as	as	ADP
cana-1174	166	20	a	a	DET
cana-1174	166	21	higher	higher	ADV
cana-1174	166	22	-	-	PUNCT
cana-1174	166	23	dimensional	dimensional	ADJ
cana-1174	166	24	analog	analog	NOUN
cana-1174	166	25	of	of	ADP
cana-1174	166	26	triangular	triangular	NOUN
cana-1174	166	27	numbers	number	NOUN
cana-1174	166	28	,	,	PUNCT
cana-1174	166	29	such	such	ADJ
cana-1174	166	30	as	as	ADP
cana-1174	166	31	tetrahedral	tetrahedral	ADJ
cana-1174	166	32	numbers	number	NOUN
cana-1174	166	33	,	,	PUNCT
cana-1174	166	34	𝑇	𝑇	PROPN
cana-1174	166	35	,	,	PUNCT
cana-1174	166	36	representing	represent	VERB
cana-1174	166	37	the	the	DET
cana-1174	166	38	sum	sum	NOUN
cana-1174	166	39	of	of	ADP
cana-1174	166	40	combinations	combination	NOUN
cana-1174	166	41	𝑛−1	𝑛−1	PROPN
cana-1174	166	42	,	,	PUNCT
cana-1174	166	43	3	3	NUM
cana-1174	166	44	𝐶	𝐶	PROPN
cana-1174	166	45	.	.	PUNCT
cana-1174	167	1	𝑛−1	𝑛−1	NUM
cana-1174	167	2	3	3	NUM
cana-1174	167	3	this	this	DET
cana-1174	167	4	approach	approach	NOUN
cana-1174	167	5	minimizes	minimize	VERB
cana-1174	167	6	the	the	DET
cana-1174	167	7	total	total	ADJ
cana-1174	167	8	number	number	NOUN
cana-1174	167	9	of	of	ADP
cana-1174	167	10	connections	connection	NOUN
cana-1174	167	11	and	and	CCONJ
cana-1174	167	12	ensures	ensure	VERB
cana-1174	167	13	efficient	efficient	ADJ
cana-1174	167	14	communication	communication	NOUN
cana-1174	167	15	pathways	pathway	NOUN
cana-1174	167	16	.	.	PUNCT
cana-1174	168	1	another	another	DET
cana-1174	168	2	application	application	NOUN
cana-1174	168	3	can	can	AUX
cana-1174	168	4	be	be	AUX
cana-1174	168	5	that	that	SCONJ
cana-1174	168	6	the	the	DET
cana-1174	168	7	maximum	maximum	ADJ
cana-1174	168	8	number	number	NOUN
cana-1174	168	9	of	of	ADP
cana-1174	168	10	pieces	piece	NOUN
cana-1174	168	11	,	,	PUNCT
cana-1174	168	12	𝑝	𝑝	NOUN
cana-1174	168	13	,	,	PUNCT
cana-1174	168	14	obtainable	obtainable	ADJ
cana-1174	168	15	with	with	ADP
cana-1174	168	16	𝑛	𝑛	DET
cana-1174	168	17	straight	straight	ADJ
cana-1174	168	18	cuts	cut	NOUN
cana-1174	168	19	in	in	ADP
cana-1174	168	20	higher	high	ADJ
cana-1174	168	21	-	-	PUNCT
cana-1174	168	22	dimensional	dimensional	ADJ
cana-1174	168	23	spaces	space	NOUN
cana-1174	168	24	aligns	align	VERB
cana-1174	168	25	with	with	ADP
cana-1174	168	26	the	the	DET
cana-1174	168	27	generalized	generalize	VERB
cana-1174	168	28	triangular	triangular	NOUN
cana-1174	168	29	number	number	NOUN
cana-1174	168	30	𝑇	𝑇	NOUN
cana-1174	168	31	+	+	CCONJ
cana-1174	168	32	1	1	NUM
cana-1174	168	33	.	.	X
cana-1174	168	34	𝑛	𝑛	PROPN
cana-1174	168	35	,	,	PUNCT
cana-1174	168	36	𝑑	𝑑	PROPN
cana-1174	168	37	this	this	DET
cana-1174	168	38	principle	principle	NOUN
cana-1174	168	39	,	,	PUNCT
cana-1174	168	40	known	know	VERB
cana-1174	168	41	in	in	ADP
cana-1174	168	42	two	two	NUM
cana-1174	168	43	dimensions	dimension	NOUN
cana-1174	168	44	as	as	ADP
cana-1174	168	45	the	the	DET
cana-1174	168	46	"	"	PUNCT
cana-1174	168	47	lazy	lazy	ADJ
cana-1174	168	48	caterer	caterer	NOUN
cana-1174	168	49	's	's	PART
cana-1174	168	50	sequence	sequence	NOUN
cana-1174	168	51	,	,	PUNCT
cana-1174	168	52	"	"	PUNCT
cana-1174	168	53	becomes	become	VERB
cana-1174	168	54	a	a	DET
cana-1174	168	55	powerful	powerful	ADJ
cana-1174	168	56	tool	tool	NOUN
cana-1174	168	57	for	for	ADP
cana-1174	168	58	solving	solve	VERB
cana-1174	168	59	problems	problem	NOUN
cana-1174	168	60	in	in	ADP
cana-1174	168	61	higher	higher	ADV
cana-1174	168	62	-	-	PUNCT
cana-1174	168	63	dimensional	dimensional	ADJ
cana-1174	168	64	cutting	cutting	NOUN
cana-1174	168	65	and	and	CCONJ
cana-1174	168	66	partitioning	partitioning	NOUN
cana-1174	168	67	tasks	task	NOUN
cana-1174	168	68	.	.	PUNCT
cana-1174	169	1	for	for	ADP
cana-1174	169	2	instance	instance	NOUN
cana-1174	169	3	,	,	PUNCT
cana-1174	169	4	consider	consider	VERB
cana-1174	169	5	making	make	VERB
cana-1174	169	6	cuts	cut	NOUN
cana-1174	169	7	in	in	ADP
cana-1174	169	8	a	a	DET
cana-1174	169	9	three	three	NUM
cana-1174	169	10	-	-	PUNCT
cana-1174	169	11	dimensional	dimensional	ADJ
cana-1174	169	12	space	space	NOUN
cana-1174	169	13	(	(	PUNCT
cana-1174	169	14	a	a	DET
cana-1174	169	15	cube	cube	NOUN
cana-1174	169	16	)	)	PUNCT
cana-1174	169	17	.	.	PUNCT
cana-1174	170	1	with	with	ADP
cana-1174	170	2	𝑛	𝑛	PROPN
cana-1174	170	3	=	=	SYM
cana-1174	170	4	3	3	NUM
cana-1174	170	5	straight	straight	ADJ
cana-1174	170	6	cuts	cut	NOUN
cana-1174	170	7	,	,	PUNCT
cana-1174	170	8	the	the	DET
cana-1174	170	9	number	number	NOUN
cana-1174	170	10	of	of	ADP
cana-1174	170	11	pieces	piece	NOUN
cana-1174	170	12	,	,	PUNCT
cana-1174	170	13	𝑝	𝑝	NOUN
cana-1174	170	14	,	,	PUNCT
cana-1174	170	15	can	can	AUX
cana-1174	170	16	be	be	AUX
cana-1174	170	17	calculated	calculate	VERB
cana-1174	170	18	using	use	VERB
cana-1174	170	19	the	the	DET
cana-1174	170	20	tetrahedral	tetrahedral	ADJ
cana-1174	170	21	number	number	NOUN
cana-1174	170	22	plus	plus	CCONJ
cana-1174	170	23	one	one	NUM
cana-1174	170	24	.	.	PUNCT
cana-1174	171	1	𝑝	𝑝	NOUN
cana-1174	171	2	=	=	SYM
cana-1174	171	3	𝑇	𝑇	PROPN
cana-1174	171	4	+	+	CCONJ
cana-1174	171	5	1	1	NUM
cana-1174	171	6	=	=	SYM
cana-1174	171	7	3	3	NUM
cana-1174	171	8	*	*	PUNCT
cana-1174	171	9	(	(	PUNCT
cana-1174	171	10	3	3	NUM
cana-1174	171	11	+	+	NUM
cana-1174	171	12	1	1	NUM
cana-1174	171	13	)	)	PUNCT
cana-1174	171	14	*	*	PUNCT
cana-1174	172	1	(	(	PUNCT
cana-1174	172	2	3	3	NUM
cana-1174	172	3	+	+	NUM
cana-1174	172	4	2)/6	2)/6	NUM
cana-1174	173	1	+	+	CCONJ
cana-1174	173	2	1	1	NUM
cana-1174	173	3	=	=	SYM
cana-1174	173	4	11	11	NUM
cana-1174	173	5	.	.	X
cana-1174	174	1	3	3	NUM
cana-1174	174	2	,	,	PUNCT
cana-1174	174	3	3	3	NUM
cana-1174	174	4	this	this	PRON
cana-1174	174	5	can	can	AUX
cana-1174	174	6	be	be	AUX
cana-1174	174	7	extended	extend	VERB
cana-1174	174	8	to	to	ADP
cana-1174	174	9	higher	high	ADJ
cana-1174	174	10	dimensions	dimension	NOUN
cana-1174	174	11	to	to	PART
cana-1174	174	12	determine	determine	VERB
cana-1174	174	13	the	the	DET
cana-1174	174	14	maximum	maximum	ADJ
cana-1174	174	15	number	number	NOUN
cana-1174	174	16	of	of	ADP
cana-1174	174	17	regions	region	NOUN
cana-1174	174	18	formed	form	VERB
cana-1174	174	19	by	by	ADP
cana-1174	174	20	𝑛	𝑛	PROPN
cana-1174	174	21	cuts	cut	NOUN
cana-1174	174	22	in	in	ADP
cana-1174	174	23	𝑑-dimensional	𝑑-dimensional	ADJ
cana-1174	174	24	spaces	space	NOUN
cana-1174	174	25	as	as	ADV
cana-1174	174	26	well	well	ADV
cana-1174	174	27	.	.	PUNCT
cana-1174	175	1	the	the	DET
cana-1174	175	2	importance	importance	NOUN
cana-1174	175	3	and	and	CCONJ
cana-1174	175	4	meaning	meaning	NOUN
cana-1174	175	5	of	of	ADP
cana-1174	175	6	triangular	triangular	NOUN
cana-1174	175	7	numbers	number	NOUN
cana-1174	175	8	lie	lie	VERB
cana-1174	175	9	in	in	ADP
cana-1174	175	10	their	their	PRON
cana-1174	175	11	fundamental	fundamental	ADJ
cana-1174	175	12	role	role	NOUN
cana-1174	175	13	in	in	ADP
cana-1174	175	14	understanding	understand	VERB
cana-1174	175	15	geometric	geometric	ADJ
cana-1174	175	16	and	and	CCONJ
cana-1174	175	17	arithmetic	arithmetic	ADJ
cana-1174	175	18	relationships	relationship	NOUN
cana-1174	175	19	,	,	PUNCT
cana-1174	175	20	their	their	PRON
cana-1174	175	21	applications	application	NOUN
cana-1174	175	22	in	in	ADP
cana-1174	175	23	various	various	ADJ
cana-1174	175	24	fields	field	NOUN
cana-1174	175	25	,	,	PUNCT
cana-1174	175	26	and	and	CCONJ
cana-1174	175	27	their	their	PRON
cana-1174	175	28	elegant	elegant	ADJ
cana-1174	175	29	recursive	recursive	ADJ
cana-1174	175	30	properties	property	NOUN
cana-1174	175	31	that	that	PRON
cana-1174	175	32	extend	extend	VERB
cana-1174	175	33	into	into	ADP
cana-1174	175	34	higher	high	ADJ
cana-1174	175	35	dimensions	dimension	NOUN
cana-1174	175	36	.	.	PUNCT
cana-1174	176	1	references	reference	NOUN
cana-1174	176	2	[	[	X
cana-1174	176	3	1	1	NUM
cana-1174	176	4	]	]	PUNCT
cana-1174	176	5	andreescu	andreescu	NOUN
cana-1174	176	6	,	,	PUNCT
cana-1174	176	7	t.	t.	PROPN
cana-1174	176	8	,	,	PUNCT
cana-1174	176	9	d.	d.	PROPN
cana-1174	176	10	andrica	andrica	PROPN
cana-1174	176	11	,	,	PUNCT
cana-1174	176	12	and	and	CCONJ
cana-1174	176	13	i.	i.	NOUN
cana-1174	176	14	cucurezeanu	cucurezeanu	PROPN
cana-1174	176	15	,	,	PUNCT
cana-1174	176	16	an	an	DET
cana-1174	176	17	introduction	introduction	NOUN
cana-1174	176	18	to	to	PART
cana-1174	176	19	diophantine	diophantine	VERB
cana-1174	176	20	equations	equation	NOUN
cana-1174	176	21	:	:	PUNCT
cana-1174	176	22	a	a	DET
cana-1174	176	23	problem	problem	NOUN
cana-1174	176	24	-	-	PUNCT
cana-1174	176	25	based	base	VERB
cana-1174	176	26	approach	approach	NOUN
cana-1174	176	27	,	,	PUNCT
cana-1174	176	28	birkhäuser	birkhäuser	ADJ
cana-1174	176	29	verlag	verlag	PROPN
cana-1174	176	30	,	,	PUNCT
cana-1174	176	31	new	new	PROPN
cana-1174	176	32	york	york	PROPN
cana-1174	176	33	,	,	PUNCT
cana-1174	176	34	2010	2010	NUM
cana-1174	176	35	.	.	PUNCT
cana-1174	177	1	[	[	X
cana-1174	177	2	2	2	NUM
cana-1174	177	3	]	]	X
cana-1174	177	4	andrews	andrews	PROPN
cana-1174	177	5	,	,	PUNCT
cana-1174	177	6	g.	g.	PROPN
cana-1174	177	7	e.	e.	PROPN
cana-1174	177	8	1971	1971	NUM
cana-1174	177	9	,	,	PUNCT
cana-1174	177	10	number	number	NOUN
cana-1174	177	11	theory	theory	NOUN
cana-1174	177	12	,	,	PUNCT
cana-1174	177	13	w.	w.	PROPN
cana-1174	177	14	b.	b.	PROPN
cana-1174	177	15	saunders	saunders	PROPN
cana-1174	177	16	co.	co.	PROPN
cana-1174	177	17	,	,	PUNCT
cana-1174	177	18	philadelphia	philadelphia	PROPN
cana-1174	177	19	,	,	PUNCT
cana-1174	177	20	pa.-london	pa.-london	PROPN
cana-1174	177	21	-	-	PUNCT
cana-1174	177	22	toronto	toronto	PROPN
cana-1174	177	23	,	,	PUNCT
cana-1174	177	24	ont	ont	PROPN
cana-1174	177	25	.	.	PUNCT
cana-1174	178	1	[	[	X
cana-1174	178	2	3	3	X
cana-1174	178	3	]	]	X
cana-1174	178	4	flexa	flexa	NOUN
cana-1174	178	5	et	et	PROPN
cana-1174	178	6	al	al	PROPN
cana-1174	178	7	.	.	PROPN
cana-1174	178	8	,	,	PUNCT
cana-1174	178	9	caio	caio	PROPN
cana-1174	178	10	.	.	PUNCT
cana-1174	179	1	“	"	PUNCT
cana-1174	179	2	polygonal	polygonal	ADJ
cana-1174	179	3	coordinate	coordinate	NOUN
cana-1174	179	4	system	system	NOUN
cana-1174	179	5	:	:	PUNCT
cana-1174	179	6	visualizing	visualize	VERB
cana-1174	179	7	high	high	ADJ
cana-1174	179	8	-	-	PUNCT
cana-1174	179	9	dimensional	dimensional	ADJ
cana-1174	179	10	data	datum	NOUN
cana-1174	179	11	using	use	VERB
cana-1174	179	12	geometric	geometric	ADJ
cana-1174	179	13	dr	dr	PROPN
cana-1174	179	14	,	,	PUNCT
cana-1174	179	15	and	and	CCONJ
cana-1174	179	16	a	a	DET
cana-1174	179	17	deterministic	deterministic	ADJ
cana-1174	179	18	version	version	NOUN
cana-1174	179	19	of	of	ADP
cana-1174	179	20	t	t	PROPN
cana-1174	179	21	-	-	PUNCT
cana-1174	179	22	sne	sne	NOUN
cana-1174	179	23	.	.	PUNCT
cana-1174	179	24	”	"	PUNCT
cana-1174	180	1	expert	expert	NOUN
cana-1174	180	2	systems	system	NOUN
cana-1174	180	3	with	with	ADP
cana-1174	180	4	applications	application	NOUN
cana-1174	180	5	,	,	PUNCT
cana-1174	180	6	vol	vol	NOUN
cana-1174	180	7	.	.	PROPN
cana-1174	180	8	175	175	NUM
cana-1174	180	9	,	,	PUNCT
cana-1174	180	10	2021	2021	NUM
cana-1174	180	11	.	.	PUNCT
cana-1174	181	1	[	[	X
cana-1174	181	2	4	4	X
cana-1174	181	3	]	]	PUNCT
cana-1174	181	4	he	he	PRON
cana-1174	181	5	,	,	PUNCT
cana-1174	181	6	yang	yang	PROPN
cana-1174	181	7	-	-	PUNCT
cana-1174	181	8	hui	hui	PROPN
cana-1174	181	9	.	.	PUNCT
cana-1174	182	1	“	"	PUNCT
cana-1174	182	2	machine	machine	NOUN
cana-1174	182	3	-	-	PUNCT
cana-1174	182	4	learning	learn	VERB
cana-1174	182	5	mathematical	mathematical	ADJ
cana-1174	182	6	structures	structure	NOUN
cana-1174	182	7	.	.	PUNCT
cana-1174	182	8	”	"	PUNCT
cana-1174	183	1	arxiv	arxiv	PROPN
cana-1174	183	2	,	,	PUNCT
cana-1174	183	3	8	8	NUM
cana-1174	183	4	april	april	PROPN
cana-1174	183	5	2021	2021	NUM
cana-1174	183	6	,	,	PUNCT
cana-1174	183	7	https://arxiv.org/pdf/2101.06317	https://arxiv.org/pdf/2101.06317	NOUN
cana-1174	183	8	.	.	PUNCT
cana-1174	184	1	[	[	X
cana-1174	184	2	5	5	NUM
cana-1174	184	3	]	]	X
cana-1174	184	4	r.	r.	PROPN
cana-1174	184	5	sivaraman	sivaraman	PROPN
cana-1174	184	6	,	,	PUNCT
cana-1174	184	7	triangular	triangular	NOUN
cana-1174	184	8	–	–	PUNCT
cana-1174	184	9	pentagonal	pentagonal	ADJ
cana-1174	184	10	numbers	number	NOUN
cana-1174	184	11	and	and	CCONJ
cana-1174	184	12	interesting	interesting	ADJ
cana-1174	184	13	puzzle	puzzle	NOUN
cana-1174	184	14	,	,	PUNCT
cana-1174	184	15	international	international	ADJ
cana-1174	184	16	journal	journal	NOUN
cana-1174	184	17	of	of	ADP
cana-1174	184	18	engineering	engineering	NOUN
cana-1174	184	19	research	research	NOUN
cana-1174	184	20	and	and	CCONJ
cana-1174	184	21	modern	modern	ADJ
cana-1174	184	22	education	education	NOUN
cana-1174	184	23	,	,	PUNCT
cana-1174	184	24	volume	volume	NOUN
cana-1174	184	25	7	7	NUM
cana-1174	184	26	,	,	PUNCT
cana-1174	184	27	issue	issue	NOUN
cana-1174	184	28	1	1	NUM
cana-1174	184	29	,	,	PUNCT
cana-1174	184	30	1	1	NUM
cana-1174	184	31	–	–	SYM
cana-1174	184	32	3	3	NUM
cana-1174	184	33	,	,	PUNCT
cana-1174	184	34	2022	2022	NUM
cana-1174	184	35	.	.	PUNCT
cana-1174	185	1	[	[	X
cana-1174	185	2	6	6	NUM
cana-1174	185	3	]	]	PUNCT
cana-1174	185	4	sarp	sarp	NOUN
cana-1174	185	5	,	,	PUNCT
cana-1174	185	6	umit	umit	VERB
cana-1174	185	7	.	.	PUNCT
cana-1174	186	1	“	"	PUNCT
cana-1174	186	2	visualising	visualise	VERB
cana-1174	186	3	connections	connection	NOUN
cana-1174	186	4	between	between	ADP
cana-1174	186	5	types	type	NOUN
cana-1174	186	6	of	of	ADP
cana-1174	186	7	polygonal	polygonal	ADJ
cana-1174	186	8	number	number	NOUN
cana-1174	186	9	.	.	PUNCT
cana-1174	186	10	”	"	PUNCT
cana-1174	187	1	cambridge	cambridge	PROPN
cana-1174	187	2	core	core	NOUN
cana-1174	187	3	,	,	PUNCT
cana-1174	187	4	2023	2023	NUM
cana-1174	187	5	,	,	PUNCT
cana-1174	187	6	https://doi.org/10.1017/mag.2023.7	https://doi.org/10.1017/mag.2023.7	NOUN
cana-1174	187	7	.	.	PUNCT
cana-1174	188	1	[	[	X
cana-1174	188	2	7	7	X
cana-1174	188	3	]	]	X
cana-1174	188	4	zoppis	zoppis	X
cana-1174	188	5	et	et	PROPN
cana-1174	188	6	al	al	PROPN
cana-1174	188	7	.	.	PROPN
cana-1174	188	8	,	,	PUNCT
cana-1174	188	9	italo	italo	PROPN
cana-1174	188	10	.	.	PUNCT
cana-1174	189	1	“	"	PUNCT
cana-1174	189	2	kernel	kernel	PROPN
cana-1174	189	3	methods	method	NOUN
cana-1174	189	4	:	:	PUNCT
cana-1174	189	5	support	support	VERB
cana-1174	189	6	vector	vector	NOUN
cana-1174	189	7	machines	machine	NOUN
cana-1174	189	8	.	.	PUNCT
cana-1174	189	9	”	"	PUNCT
cana-1174	190	1	reference	reference	NOUN
cana-1174	190	2	module	module	NOUN
cana-1174	190	3	in	in	ADP
cana-1174	190	4	life	life	NOUN
cana-1174	190	5	sciences	science	NOUN
cana-1174	190	6	,	,	PUNCT
cana-1174	190	7	vol	vol	NOUN
cana-1174	190	8	.	.	PROPN
cana-1174	190	9	1	1	NUM
cana-1174	190	10	,	,	PUNCT
cana-1174	190	11	2019	2019	NUM
cana-1174	190	12	,	,	PUNCT
cana-1174	190	13	pp	pp	ADJ
cana-1174	190	14	.	.	PUNCT
cana-1174	191	1	503	503	NUM
cana-1174	191	2	-	-	SYM
cana-1174	191	3	510	510	NUM
cana-1174	191	4	.	.	PUNCT