id	sid	tid	token	lemma	pos
ejpam-4498	1	1	european	european	PROPN
ejpam-4498	1	2	journal	journal	PROPN
ejpam-4498	1	3	of	of	ADP
ejpam-4498	1	4	pure	pure	ADJ
ejpam-4498	1	5	and	and	CCONJ
ejpam-4498	1	6	applied	apply	VERB
ejpam-4498	1	7	mathematics	mathematic	NOUN
ejpam-4498	1	8	vol	vol	NOUN
ejpam-4498	1	9	.	.	PROPN
ejpam-4498	2	1	15	15	NUM
ejpam-4498	2	2	,	,	PUNCT
ejpam-4498	2	3	no	no	INTJ
ejpam-4498	2	4	.	.	NOUN
ejpam-4498	2	5	3	3	NUM
ejpam-4498	2	6	,	,	PUNCT
ejpam-4498	2	7	2022	2022	NUM
ejpam-4498	2	8	,	,	PUNCT
ejpam-4498	2	9	1237	1237	NUM
ejpam-4498	2	10	-	-	SYM
ejpam-4498	2	11	1253	1253	NUM
ejpam-4498	2	12	issn	issn	PROPN
ejpam-4498	2	13	1307	1307	NUM
ejpam-4498	2	14	-	-	SYM
ejpam-4498	2	15	5543	5543	NUM
ejpam-4498	2	16	–	–	PUNCT
ejpam-4498	2	17	ejpam.com	ejpam.com	X
ejpam-4498	2	18	published	publish	VERB
ejpam-4498	2	19	by	by	ADP
ejpam-4498	2	20	new	new	PROPN
ejpam-4498	2	21	york	york	PROPN
ejpam-4498	2	22	business	business	PROPN
ejpam-4498	2	23	global	global	ADJ
ejpam-4498	2	24	using	use	VERB
ejpam-4498	2	25	the	the	DET
ejpam-4498	2	26	sum	sum	NOUN
ejpam-4498	2	27	of	of	ADP
ejpam-4498	2	28	triangular	triangular	NOUN
ejpam-4498	2	29	numbers	number	NOUN
ejpam-4498	2	30	as	as	ADP
ejpam-4498	2	31	the	the	DET
ejpam-4498	2	32	most	most	ADV
ejpam-4498	2	33	fundamental	fundamental	ADJ
ejpam-4498	2	34	number	number	NOUN
ejpam-4498	2	35	of	of	ADP
ejpam-4498	2	36	combinations	combination	NOUN
ejpam-4498	2	37	found	find	VERB
ejpam-4498	2	38	using	use	VERB
ejpam-4498	2	39	heuristic	heuristic	ADJ
ejpam-4498	2	40	methods	method	NOUN
ejpam-4498	2	41	yukio	yukio	PROPN
ejpam-4498	2	42	kobayashi	kobayashi	PROPN
ejpam-4498	2	43	department	department	PROPN
ejpam-4498	2	44	of	of	ADP
ejpam-4498	2	45	information	information	NOUN
ejpam-4498	2	46	systems	system	NOUN
ejpam-4498	2	47	science	science	NOUN
ejpam-4498	2	48	,	,	PUNCT
ejpam-4498	2	49	faculty	faculty	NOUN
ejpam-4498	2	50	of	of	ADP
ejpam-4498	2	51	science	science	NOUN
ejpam-4498	2	52	and	and	CCONJ
ejpam-4498	2	53	engineering	engineering	NOUN
ejpam-4498	2	54	,	,	PUNCT
ejpam-4498	2	55	soka	soka	PROPN
ejpam-4498	2	56	university	university	NOUN
ejpam-4498	2	57	1	1	NUM
ejpam-4498	2	58	-	-	SYM
ejpam-4498	2	59	236	236	NUM
ejpam-4498	2	60	tangi	tangi	NOUN
ejpam-4498	2	61	-	-	PUNCT
ejpam-4498	2	62	machi	machi	NOUN
ejpam-4498	2	63	,	,	PUNCT
ejpam-4498	2	64	hachioji	hachioji	PROPN
ejpam-4498	2	65	-	-	PUNCT
ejpam-4498	2	66	shi	shi	PROPN
ejpam-4498	2	67	,	,	PUNCT
ejpam-4498	2	68	tokyo	tokyo	PROPN
ejpam-4498	2	69	192	192	NUM
ejpam-4498	2	70	-	-	SYM
ejpam-4498	2	71	8577	8577	NUM
ejpam-4498	2	72	,	,	PUNCT
ejpam-4498	2	73	japan	japan	PROPN
ejpam-4498	2	74	abstract	abstract	PROPN
ejpam-4498	2	75	.	.	PUNCT
ejpam-4498	3	1	the	the	DET
ejpam-4498	3	2	same	same	ADJ
ejpam-4498	3	3	mathematical	mathematical	ADJ
ejpam-4498	3	4	meaning	meaning	NOUN
ejpam-4498	3	5	can	can	AUX
ejpam-4498	3	6	be	be	AUX
ejpam-4498	3	7	expressed	express	VERB
ejpam-4498	3	8	in	in	ADP
ejpam-4498	3	9	many	many	ADJ
ejpam-4498	3	10	different	different	ADJ
ejpam-4498	3	11	ways	way	NOUN
ejpam-4498	3	12	.	.	PUNCT
ejpam-4498	4	1	students	student	NOUN
ejpam-4498	4	2	should	should	AUX
ejpam-4498	4	3	deduce	deduce	VERB
ejpam-4498	4	4	this	this	PRON
ejpam-4498	4	5	and	and	CCONJ
ejpam-4498	4	6	thus	thus	ADV
ejpam-4498	4	7	perceive	perceive	ADJ
ejpam-4498	4	8	that	that	SCONJ
ejpam-4498	4	9	the	the	DET
ejpam-4498	4	10	different	different	ADJ
ejpam-4498	4	11	expressions	expression	NOUN
ejpam-4498	4	12	have	have	VERB
ejpam-4498	4	13	the	the	DET
ejpam-4498	4	14	same	same	ADJ
ejpam-4498	4	15	meaning	meaning	NOUN
ejpam-4498	4	16	.	.	PUNCT
ejpam-4498	5	1	in	in	ADP
ejpam-4498	5	2	this	this	DET
ejpam-4498	5	3	work	work	NOUN
ejpam-4498	5	4	,	,	PUNCT
ejpam-4498	5	5	a	a	DET
ejpam-4498	5	6	pedagogical	pedagogical	ADJ
ejpam-4498	5	7	approach	approach	NOUN
ejpam-4498	5	8	to	to	ADP
ejpam-4498	5	9	the	the	DET
ejpam-4498	5	10	determination	determination	NOUN
ejpam-4498	5	11	of	of	ADP
ejpam-4498	5	12	various	various	ADJ
ejpam-4498	5	13	equations	equation	NOUN
ejpam-4498	5	14	related	relate	VERB
ejpam-4498	5	15	to	to	ADP
ejpam-4498	5	16	the	the	DET
ejpam-4498	5	17	sum	sum	NOUN
ejpam-4498	5	18	of	of	ADP
ejpam-4498	5	19	triangular	triangular	NOUN
ejpam-4498	5	20	numbers	number	NOUN
ejpam-4498	5	21	is	be	AUX
ejpam-4498	5	22	presented	present	VERB
ejpam-4498	5	23	as	as	ADP
ejpam-4498	5	24	an	an	DET
ejpam-4498	5	25	example	example	NOUN
ejpam-4498	5	26	to	to	PART
ejpam-4498	5	27	show	show	VERB
ejpam-4498	5	28	how	how	SCONJ
ejpam-4498	5	29	students	student	NOUN
ejpam-4498	5	30	can	can	AUX
ejpam-4498	5	31	deduce	deduce	VERB
ejpam-4498	5	32	these	these	DET
ejpam-4498	5	33	different	different	ADJ
ejpam-4498	5	34	expressions	expression	NOUN
ejpam-4498	5	35	and	and	CCONJ
ejpam-4498	5	36	their	their	PRON
ejpam-4498	5	37	meanings	meaning	NOUN
ejpam-4498	5	38	.	.	PUNCT
ejpam-4498	6	1	the	the	DET
ejpam-4498	6	2	sum	sum	NOUN
ejpam-4498	6	3	of	of	ADP
ejpam-4498	6	4	triangular	triangular	NOUN
ejpam-4498	6	5	numbers	number	NOUN
ejpam-4498	6	6	is	be	AUX
ejpam-4498	6	7	closely	closely	ADV
ejpam-4498	6	8	related	relate	VERB
ejpam-4498	6	9	to	to	ADP
ejpam-4498	6	10	the	the	DET
ejpam-4498	6	11	sums	sum	NOUN
ejpam-4498	6	12	of	of	ADP
ejpam-4498	6	13	natural	natural	ADJ
ejpam-4498	6	14	numbers	number	NOUN
ejpam-4498	6	15	,	,	PUNCT
ejpam-4498	6	16	square	square	ADJ
ejpam-4498	6	17	numbers	number	NOUN
ejpam-4498	6	18	,	,	PUNCT
ejpam-4498	6	19	and	and	CCONJ
ejpam-4498	6	20	two	two	NUM
ejpam-4498	6	21	consecutive	consecutive	ADJ
ejpam-4498	6	22	numbers	number	NOUN
ejpam-4498	6	23	.	.	PUNCT
ejpam-4498	7	1	these	these	DET
ejpam-4498	7	2	relationships	relationship	NOUN
ejpam-4498	7	3	can	can	AUX
ejpam-4498	7	4	be	be	AUX
ejpam-4498	7	5	found	find	VERB
ejpam-4498	7	6	using	use	VERB
ejpam-4498	7	7	a	a	DET
ejpam-4498	7	8	heuristic	heuristic	ADJ
ejpam-4498	7	9	method	method	NOUN
ejpam-4498	7	10	in	in	ADP
ejpam-4498	7	11	which	which	PRON
ejpam-4498	7	12	the	the	DET
ejpam-4498	7	13	equations	equation	NOUN
ejpam-4498	7	14	representing	represent	VERB
ejpam-4498	7	15	the	the	DET
ejpam-4498	7	16	sum	sum	NOUN
ejpam-4498	7	17	of	of	ADP
ejpam-4498	7	18	natural	natural	ADJ
ejpam-4498	7	19	numbers	number	NOUN
ejpam-4498	7	20	are	be	AUX
ejpam-4498	7	21	simply	simply	ADV
ejpam-4498	7	22	listed	list	VERB
ejpam-4498	7	23	.	.	PUNCT
ejpam-4498	8	1	guiding	guide	VERB
ejpam-4498	8	2	students	student	NOUN
ejpam-4498	8	3	to	to	PART
ejpam-4498	8	4	discover	discover	VERB
ejpam-4498	8	5	the	the	DET
ejpam-4498	8	6	formula	formula	NOUN
ejpam-4498	8	7	by	by	ADP
ejpam-4498	8	8	themselves	themselves	PRON
ejpam-4498	8	9	is	be	AUX
ejpam-4498	8	10	important	important	ADJ
ejpam-4498	8	11	,	,	PUNCT
ejpam-4498	8	12	rather	rather	ADV
ejpam-4498	8	13	than	than	ADP
ejpam-4498	8	14	giving	give	VERB
ejpam-4498	8	15	them	they	PRON
ejpam-4498	8	16	the	the	DET
ejpam-4498	8	17	formula	formula	NOUN
ejpam-4498	8	18	from	from	ADP
ejpam-4498	8	19	the	the	DET
ejpam-4498	8	20	beginning	beginning	NOUN
ejpam-4498	8	21	and	and	CCONJ
ejpam-4498	8	22	asking	ask	VERB
ejpam-4498	8	23	them	they	PRON
ejpam-4498	8	24	to	to	PART
ejpam-4498	8	25	prove	prove	VERB
ejpam-4498	8	26	it	it	PRON
ejpam-4498	8	27	.	.	PUNCT
ejpam-4498	9	1	triangular	triangular	NOUN
ejpam-4498	9	2	numbers	number	NOUN
ejpam-4498	9	3	are	be	AUX
ejpam-4498	9	4	the	the	DET
ejpam-4498	9	5	fundamental	fundamental	ADJ
ejpam-4498	9	6	numbers	number	NOUN
ejpam-4498	9	7	of	of	ADP
ejpam-4498	9	8	combinations	combination	NOUN
ejpam-4498	9	9	.	.	PUNCT
ejpam-4498	10	1	the	the	DET
ejpam-4498	10	2	sum	sum	NOUN
ejpam-4498	10	3	of	of	ADP
ejpam-4498	10	4	n	n	CCONJ
ejpam-4498	10	5	consecutive	consecutive	ADJ
ejpam-4498	10	6	triangular	triangular	NOUN
ejpam-4498	10	7	numbers	number	NOUN
ejpam-4498	10	8	is	be	AUX
ejpam-4498	10	9	equal	equal	ADJ
ejpam-4498	10	10	to	to	ADP
ejpam-4498	10	11	n+2c3	n+2c3	PROPN
ejpam-4498	10	12	,	,	PUNCT
ejpam-4498	10	13	as	as	SCONJ
ejpam-4498	10	14	shown	show	VERB
ejpam-4498	10	15	by	by	ADP
ejpam-4498	10	16	combinatoric	combinatoric	ADJ
ejpam-4498	10	17	tree	tree	NOUN
ejpam-4498	10	18	diagrams	diagram	NOUN
ejpam-4498	10	19	.	.	PUNCT
ejpam-4498	11	1	the	the	DET
ejpam-4498	11	2	sum	sum	NOUN
ejpam-4498	11	3	of	of	ADP
ejpam-4498	11	4	the	the	DET
ejpam-4498	11	5	triangular	triangular	NOUN
ejpam-4498	11	6	numbers	number	NOUN
ejpam-4498	11	7	can	can	AUX
ejpam-4498	11	8	be	be	AUX
ejpam-4498	11	9	regarded	regard	VERB
ejpam-4498	11	10	as	as	ADP
ejpam-4498	11	11	an	an	DET
ejpam-4498	11	12	extended	extended	ADJ
ejpam-4498	11	13	version	version	NOUN
ejpam-4498	11	14	of	of	ADP
ejpam-4498	11	15	the	the	DET
ejpam-4498	11	16	sum	sum	NOUN
ejpam-4498	11	17	of	of	ADP
ejpam-4498	11	18	the	the	DET
ejpam-4498	11	19	natural	natural	ADJ
ejpam-4498	11	20	numbers	number	NOUN
ejpam-4498	11	21	.	.	PUNCT
ejpam-4498	12	1	the	the	DET
ejpam-4498	12	2	sum	sum	NOUN
ejpam-4498	12	3	of	of	ADP
ejpam-4498	12	4	square	square	ADJ
ejpam-4498	12	5	numbers	number	NOUN
ejpam-4498	12	6	is	be	AUX
ejpam-4498	12	7	also	also	ADV
ejpam-4498	12	8	related	relate	VERB
ejpam-4498	12	9	to	to	ADP
ejpam-4498	12	10	the	the	DET
ejpam-4498	12	11	sum	sum	NOUN
ejpam-4498	12	12	of	of	ADP
ejpam-4498	12	13	triangular	triangular	NOUN
ejpam-4498	12	14	numbers	number	NOUN
ejpam-4498	12	15	,	,	PUNCT
ejpam-4498	12	16	which	which	PRON
ejpam-4498	12	17	is	be	AUX
ejpam-4498	12	18	also	also	ADV
ejpam-4498	12	19	easy	easy	ADJ
ejpam-4498	12	20	to	to	PART
ejpam-4498	12	21	understand	understand	VERB
ejpam-4498	12	22	in	in	ADP
ejpam-4498	12	23	terms	term	NOUN
ejpam-4498	12	24	of	of	ADP
ejpam-4498	12	25	combinatorics	combinatoric	NOUN
ejpam-4498	12	26	.	.	PUNCT
ejpam-4498	13	1	the	the	DET
ejpam-4498	13	2	combinatoric	combinatoric	ADJ
ejpam-4498	13	3	expression	expression	NOUN
ejpam-4498	13	4	for	for	ADP
ejpam-4498	13	5	the	the	DET
ejpam-4498	13	6	sum	sum	NOUN
ejpam-4498	13	7	of	of	ADP
ejpam-4498	13	8	triangular	triangular	NOUN
ejpam-4498	13	9	numbers	number	NOUN
ejpam-4498	13	10	can	can	AUX
ejpam-4498	13	11	be	be	AUX
ejpam-4498	13	12	extended	extend	VERB
ejpam-4498	13	13	to	to	PART
ejpam-4498	13	14	k−1∑	k−1∑	VERB
ejpam-4498	13	15	ℓ=1	ℓ=1	PROPN
ejpam-4498	13	16	n−ℓck−1	n−ℓck−1	PROPN
ejpam-4498	13	17	=	=	SYM
ejpam-4498	13	18	nck	nck	PROPN
ejpam-4498	13	19	,	,	PUNCT
ejpam-4498	13	20	which	which	PRON
ejpam-4498	13	21	can	can	AUX
ejpam-4498	13	22	be	be	AUX
ejpam-4498	13	23	expressed	express	VERB
ejpam-4498	13	24	as	as	ADP
ejpam-4498	13	25	the	the	DET
ejpam-4498	13	26	sum	sum	NOUN
ejpam-4498	13	27	of	of	ADP
ejpam-4498	13	28	triangular	triangular	NOUN
ejpam-4498	13	29	numbers	number	NOUN
ejpam-4498	13	30	.	.	PUNCT
ejpam-4498	14	1	this	this	DET
ejpam-4498	14	2	study	study	NOUN
ejpam-4498	14	3	provides	provide	VERB
ejpam-4498	14	4	an	an	DET
ejpam-4498	14	5	example	example	NOUN
ejpam-4498	14	6	of	of	ADP
ejpam-4498	14	7	teaching	teaching	NOUN
ejpam-4498	14	8	material	material	NOUN
ejpam-4498	14	9	that	that	PRON
ejpam-4498	14	10	broadens	broaden	VERB
ejpam-4498	14	11	students	student	NOUN
ejpam-4498	14	12	’	'	PUNCT
ejpam-4498	14	13	view	view	NOUN
ejpam-4498	14	14	of	of	ADP
ejpam-4498	14	15	a	a	DET
ejpam-4498	14	16	given	give	VERB
ejpam-4498	14	17	equation	equation	NOUN
ejpam-4498	14	18	by	by	ADP
ejpam-4498	14	19	crossing	cross	VERB
ejpam-4498	14	20	different	different	ADJ
ejpam-4498	14	21	learning	learning	NOUN
ejpam-4498	14	22	units	unit	NOUN
ejpam-4498	14	23	such	such	ADJ
ejpam-4498	14	24	as	as	ADP
ejpam-4498	14	25	combinatorics	combinatoric	NOUN
ejpam-4498	14	26	and	and	CCONJ
ejpam-4498	14	27	algebra	algebra	NOUN
ejpam-4498	14	28	.	.	PUNCT
ejpam-4498	15	1	the	the	DET
ejpam-4498	15	2	sum	sum	NOUN
ejpam-4498	15	3	of	of	ADP
ejpam-4498	15	4	triangular	triangular	NOUN
ejpam-4498	15	5	numbers	number	NOUN
ejpam-4498	15	6	discussed	discuss	VERB
ejpam-4498	15	7	in	in	ADP
ejpam-4498	15	8	this	this	DET
ejpam-4498	15	9	study	study	NOUN
ejpam-4498	15	10	is	be	AUX
ejpam-4498	15	11	simply	simply	ADV
ejpam-4498	15	12	an	an	DET
ejpam-4498	15	13	example	example	NOUN
ejpam-4498	15	14	of	of	ADP
ejpam-4498	15	15	developing	develop	VERB
ejpam-4498	15	16	the	the	DET
ejpam-4498	15	17	habit	habit	NOUN
ejpam-4498	15	18	of	of	ADP
ejpam-4498	15	19	considering	consider	VERB
ejpam-4498	15	20	various	various	ADJ
ejpam-4498	15	21	interpretations	interpretation	NOUN
ejpam-4498	15	22	of	of	ADP
ejpam-4498	15	23	the	the	DET
ejpam-4498	15	24	same	same	ADJ
ejpam-4498	15	25	formula	formula	NOUN
ejpam-4498	15	26	,	,	PUNCT
ejpam-4498	15	27	but	but	CCONJ
ejpam-4498	15	28	even	even	ADV
ejpam-4498	15	29	well	well	ADV
ejpam-4498	15	30	-	-	PUNCT
ejpam-4498	15	31	known	know	VERB
ejpam-4498	15	32	formulas	formula	NOUN
ejpam-4498	15	33	can	can	AUX
ejpam-4498	15	34	be	be	AUX
ejpam-4498	15	35	utilized	utilize	VERB
ejpam-4498	15	36	for	for	ADP
ejpam-4498	15	37	this	this	DET
ejpam-4498	15	38	type	type	NOUN
ejpam-4498	15	39	of	of	ADP
ejpam-4498	15	40	subject	subject	ADJ
ejpam-4498	15	41	matter	matter	NOUN
ejpam-4498	15	42	.	.	PUNCT
ejpam-4498	16	1	2020	2020	NUM
ejpam-4498	16	2	mathematics	mathematic	NOUN
ejpam-4498	16	3	subject	subject	NOUN
ejpam-4498	16	4	classifications	classification	NOUN
ejpam-4498	16	5	:	:	PUNCT
ejpam-4498	16	6	97h20	97h20	NUM
ejpam-4498	16	7	key	key	ADJ
ejpam-4498	16	8	words	word	NOUN
ejpam-4498	16	9	and	and	CCONJ
ejpam-4498	16	10	phrases	phrase	NOUN
ejpam-4498	16	11	:	:	PUNCT
ejpam-4498	16	12	sum	sum	NOUN
ejpam-4498	16	13	of	of	ADP
ejpam-4498	16	14	natural	natural	ADJ
ejpam-4498	16	15	numbers	number	NOUN
ejpam-4498	16	16	,	,	PUNCT
ejpam-4498	16	17	sum	sum	NOUN
ejpam-4498	16	18	of	of	ADP
ejpam-4498	16	19	triangular	triangular	NOUN
ejpam-4498	16	20	numbers	number	NOUN
ejpam-4498	16	21	,	,	PUNCT
ejpam-4498	16	22	sum	sum	NOUN
ejpam-4498	16	23	of	of	ADP
ejpam-4498	16	24	square	square	ADJ
ejpam-4498	16	25	numbers	number	NOUN
ejpam-4498	16	26	,	,	PUNCT
ejpam-4498	16	27	tree	tree	NOUN
ejpam-4498	16	28	diagram	diagram	NOUN
ejpam-4498	16	29	,	,	PUNCT
ejpam-4498	16	30	combinatorics	combinatoric	NOUN
ejpam-4498	16	31	,	,	PUNCT
ejpam-4498	16	32	heuristic	heuristic	ADJ
ejpam-4498	16	33	method	method	NOUN
ejpam-4498	16	34	doi	doi	NOUN
ejpam-4498	16	35	:	:	PUNCT
ejpam-4498	16	36	https://doi.org/10.29020/nybg.ejpam.v15i3.4498	https://doi.org/10.29020/nybg.ejpam.v15i3.4498	PROPN
ejpam-4498	16	37	email	email	NOUN
ejpam-4498	16	38	address	address	NOUN
ejpam-4498	16	39	:	:	PUNCT
ejpam-4498	16	40	koba@t.soka.ac.jp	koba@t.soka.ac.jp	PROPN
ejpam-4498	16	41	(	(	PUNCT
ejpam-4498	16	42	y.	y.	PROPN
ejpam-4498	16	43	kobayashi	kobayashi	PROPN
ejpam-4498	16	44	)	)	PUNCT
ejpam-4498	16	45	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4498	16	46	1237	1237	NUM
ejpam-4498	17	1	©	©	PROPN
ejpam-4498	17	2	2022	2022	NUM
ejpam-4498	17	3	ejpam	ejpam	VERB
ejpam-4498	17	4	all	all	DET
ejpam-4498	17	5	rights	right	NOUN
ejpam-4498	17	6	reserved	reserve	VERB
ejpam-4498	17	7	.	.	PUNCT
ejpam-4498	18	1	yukio	yukio	PROPN
ejpam-4498	18	2	kobayashi	kobayashi	PROPN
ejpam-4498	18	3	/	/	SYM
ejpam-4498	18	4	eur	eur	PROPN
ejpam-4498	18	5	.	.	PUNCT
ejpam-4498	19	1	j.	j.	PROPN
ejpam-4498	19	2	pure	pure	PROPN
ejpam-4498	19	3	appl	appl	PROPN
ejpam-4498	19	4	.	.	PROPN
ejpam-4498	19	5	math	math	PROPN
ejpam-4498	19	6	,	,	PUNCT
ejpam-4498	19	7	15	15	NUM
ejpam-4498	19	8	(	(	PUNCT
ejpam-4498	19	9	3	3	NUM
ejpam-4498	19	10	)	)	PUNCT
ejpam-4498	19	11	(	(	PUNCT
ejpam-4498	19	12	2022	2022	NUM
ejpam-4498	19	13	)	)	PUNCT
ejpam-4498	19	14	,	,	PUNCT
ejpam-4498	19	15	1237	1237	NUM
ejpam-4498	19	16	-	-	SYM
ejpam-4498	19	17	1253	1253	NUM
ejpam-4498	19	18	1238	1238	NUM
ejpam-4498	19	19	1	1	NUM
ejpam-4498	19	20	.	.	PUNCT
ejpam-4498	19	21	introduction	introduction	NOUN
ejpam-4498	19	22	in	in	ADP
ejpam-4498	19	23	previous	previous	ADJ
ejpam-4498	19	24	studies	study	NOUN
ejpam-4498	19	25	[	[	X
ejpam-4498	19	26	3–5	3–5	NOUN
ejpam-4498	19	27	]	]	PUNCT
ejpam-4498	19	28	,	,	PUNCT
ejpam-4498	19	29	we	we	PRON
ejpam-4498	19	30	considered	consider	VERB
ejpam-4498	19	31	the	the	DET
ejpam-4498	19	32	geometrical	geometrical	ADJ
ejpam-4498	19	33	meaning	meaning	NOUN
ejpam-4498	19	34	of	of	ADP
ejpam-4498	19	35	the	the	DET
ejpam-4498	19	36	sums	sum	NOUN
ejpam-4498	19	37	of	of	ADP
ejpam-4498	19	38	natural	natural	ADJ
ejpam-4498	19	39	numbers	number	NOUN
ejpam-4498	19	40	,	,	PUNCT
ejpam-4498	19	41	square	square	ADJ
ejpam-4498	19	42	numbers	number	NOUN
ejpam-4498	19	43	,	,	PUNCT
ejpam-4498	19	44	and	and	CCONJ
ejpam-4498	19	45	cubic	cubic	ADJ
ejpam-4498	19	46	numbers	number	NOUN
ejpam-4498	19	47	in	in	ADP
ejpam-4498	19	48	terms	term	NOUN
ejpam-4498	19	49	of	of	ADP
ejpam-4498	19	50	elementary	elementary	ADJ
ejpam-4498	19	51	combinatorics	combinatoric	NOUN
ejpam-4498	19	52	or	or	CCONJ
ejpam-4498	19	53	tree	tree	NOUN
ejpam-4498	19	54	diagrams	diagram	NOUN
ejpam-4498	19	55	.	.	PUNCT
ejpam-4498	20	1	this	this	DET
ejpam-4498	20	2	basic	basic	ADJ
ejpam-4498	20	3	concept	concept	NOUN
ejpam-4498	20	4	involves	involve	VERB
ejpam-4498	20	5	counting	count	VERB
ejpam-4498	20	6	the	the	DET
ejpam-4498	20	7	number	number	NOUN
ejpam-4498	20	8	of	of	ADP
ejpam-4498	20	9	points	point	NOUN
ejpam-4498	20	10	or	or	CCONJ
ejpam-4498	20	11	edges	edge	NOUN
ejpam-4498	20	12	.	.	PUNCT
ejpam-4498	21	1	adopting	adopt	VERB
ejpam-4498	21	2	a	a	DET
ejpam-4498	21	3	different	different	ADJ
ejpam-4498	21	4	perspective	perspective	NOUN
ejpam-4498	21	5	and	and	CCONJ
ejpam-4498	21	6	focusing	focus	VERB
ejpam-4498	21	7	on	on	ADP
ejpam-4498	21	8	triangular	triangular	NOUN
ejpam-4498	21	9	numbers	number	NOUN
ejpam-4498	21	10	,	,	PUNCT
ejpam-4498	21	11	it	it	PRON
ejpam-4498	21	12	may	may	AUX
ejpam-4498	21	13	be	be	AUX
ejpam-4498	21	14	observed	observe	VERB
ejpam-4498	21	15	that	that	SCONJ
ejpam-4498	21	16	the	the	DET
ejpam-4498	21	17	same	same	ADJ
ejpam-4498	21	18	number	number	NOUN
ejpam-4498	21	19	of	of	ADP
ejpam-4498	21	20	points	point	NOUN
ejpam-4498	21	21	or	or	CCONJ
ejpam-4498	21	22	edges	edge	NOUN
ejpam-4498	21	23	can	can	AUX
ejpam-4498	21	24	be	be	AUX
ejpam-4498	21	25	expressed	express	VERB
ejpam-4498	21	26	by	by	ADP
ejpam-4498	21	27	different	different	ADJ
ejpam-4498	21	28	formulas	formula	NOUN
ejpam-4498	21	29	,	,	PUNCT
ejpam-4498	21	30	depending	depend	VERB
ejpam-4498	21	31	on	on	ADP
ejpam-4498	21	32	how	how	SCONJ
ejpam-4498	21	33	they	they	PRON
ejpam-4498	21	34	are	be	AUX
ejpam-4498	21	35	counted	count	VERB
ejpam-4498	21	36	.	.	PUNCT
ejpam-4498	22	1	counting	counting	NOUN
ejpam-4498	22	2	is	be	AUX
ejpam-4498	22	3	an	an	DET
ejpam-4498	22	4	appropriate	appropriate	ADJ
ejpam-4498	22	5	subject	subject	NOUN
ejpam-4498	22	6	to	to	PART
ejpam-4498	22	7	help	help	VERB
ejpam-4498	22	8	students	student	NOUN
ejpam-4498	22	9	realise	realise	VERB
ejpam-4498	22	10	that	that	SCONJ
ejpam-4498	22	11	an	an	DET
ejpam-4498	22	12	equation	equation	NOUN
ejpam-4498	22	13	is	be	AUX
ejpam-4498	22	14	both	both	PRON
ejpam-4498	22	15	a	a	DET
ejpam-4498	22	16	means	means	NOUN
ejpam-4498	22	17	of	of	ADP
ejpam-4498	22	18	writing	write	VERB
ejpam-4498	22	19	the	the	DET
ejpam-4498	22	20	process	process	NOUN
ejpam-4498	22	21	of	of	ADP
ejpam-4498	22	22	a	a	DET
ejpam-4498	22	23	calculation	calculation	NOUN
ejpam-4498	22	24	and	and	CCONJ
ejpam-4498	22	25	a	a	DET
ejpam-4498	22	26	composition	composition	NOUN
ejpam-4498	22	27	for	for	ADP
ejpam-4498	22	28	expressing	express	VERB
ejpam-4498	22	29	numbers	number	NOUN
ejpam-4498	22	30	.	.	PUNCT
ejpam-4498	23	1	as	as	SCONJ
ejpam-4498	23	2	students	student	NOUN
ejpam-4498	23	3	become	become	VERB
ejpam-4498	23	4	familiar	familiar	ADJ
ejpam-4498	23	5	with	with	ADP
ejpam-4498	23	6	various	various	ADJ
ejpam-4498	23	7	expressions	expression	NOUN
ejpam-4498	23	8	of	of	ADP
ejpam-4498	23	9	the	the	DET
ejpam-4498	23	10	sum	sum	NOUN
ejpam-4498	23	11	of	of	ADP
ejpam-4498	23	12	triangular	triangular	NOUN
ejpam-4498	23	13	numbers	number	NOUN
ejpam-4498	23	14	,	,	PUNCT
ejpam-4498	23	15	they	they	PRON
ejpam-4498	23	16	can	can	AUX
ejpam-4498	23	17	discover	discover	VERB
ejpam-4498	23	18	the	the	DET
ejpam-4498	23	19	close	close	ADJ
ejpam-4498	23	20	relationships	relationship	NOUN
ejpam-4498	23	21	between	between	ADP
ejpam-4498	23	22	the	the	DET
ejpam-4498	23	23	sums	sum	NOUN
ejpam-4498	23	24	of	of	ADP
ejpam-4498	23	25	natural	natural	ADJ
ejpam-4498	23	26	numbers	number	NOUN
ejpam-4498	23	27	,	,	PUNCT
ejpam-4498	23	28	square	square	ADJ
ejpam-4498	23	29	numbers	number	NOUN
ejpam-4498	23	30	,	,	PUNCT
ejpam-4498	23	31	and	and	CCONJ
ejpam-4498	23	32	triangular	triangular	NOUN
ejpam-4498	23	33	numbers	number	NOUN
ejpam-4498	23	34	.	.	PUNCT
ejpam-4498	24	1	in	in	ADP
ejpam-4498	24	2	contrast	contrast	NOUN
ejpam-4498	24	3	to	to	ADP
ejpam-4498	24	4	passive	passive	ADJ
ejpam-4498	24	5	learning	learning	NOUN
ejpam-4498	24	6	,	,	PUNCT
ejpam-4498	24	7	guiding	guide	VERB
ejpam-4498	24	8	students	student	NOUN
ejpam-4498	24	9	to	to	PART
ejpam-4498	24	10	independently	independently	ADV
ejpam-4498	24	11	discover	discover	VERB
ejpam-4498	24	12	the	the	DET
ejpam-4498	24	13	characteristics	characteristic	NOUN
ejpam-4498	24	14	of	of	ADP
ejpam-4498	24	15	numbers	number	NOUN
ejpam-4498	24	16	or	or	CCONJ
ejpam-4498	24	17	equations	equation	NOUN
ejpam-4498	24	18	using	use	VERB
ejpam-4498	24	19	heuristic	heuristic	ADJ
ejpam-4498	24	20	methods	method	NOUN
ejpam-4498	24	21	is	be	AUX
ejpam-4498	24	22	essential	essential	ADJ
ejpam-4498	24	23	[	[	X
ejpam-4498	24	24	6	6	NUM
ejpam-4498	24	25	]	]	PUNCT
ejpam-4498	24	26	.	.	PUNCT
ejpam-4498	25	1	through	through	ADP
ejpam-4498	25	2	these	these	DET
ejpam-4498	25	3	learning	learning	NOUN
ejpam-4498	25	4	experiences	experience	NOUN
ejpam-4498	25	5	,	,	PUNCT
ejpam-4498	25	6	students	student	NOUN
ejpam-4498	25	7	should	should	AUX
ejpam-4498	25	8	deduce	deduce	VERB
ejpam-4498	25	9	and	and	CCONJ
ejpam-4498	25	10	thus	thus	ADV
ejpam-4498	25	11	perceive	perceive	ADJ
ejpam-4498	25	12	that	that	SCONJ
ejpam-4498	25	13	the	the	DET
ejpam-4498	25	14	different	different	ADJ
ejpam-4498	25	15	expressions	expression	NOUN
ejpam-4498	25	16	have	have	VERB
ejpam-4498	25	17	the	the	DET
ejpam-4498	25	18	same	same	ADJ
ejpam-4498	25	19	meaning	meaning	NOUN
ejpam-4498	25	20	.	.	PUNCT
ejpam-4498	26	1	proofs	proof	NOUN
ejpam-4498	26	2	without	without	ADP
ejpam-4498	26	3	words	word	NOUN
ejpam-4498	26	4	can	can	AUX
ejpam-4498	26	5	provide	provide	VERB
ejpam-4498	26	6	a	a	DET
ejpam-4498	26	7	very	very	ADV
ejpam-4498	26	8	useful	useful	ADJ
ejpam-4498	26	9	support	support	NOUN
ejpam-4498	26	10	in	in	ADP
ejpam-4498	26	11	this	this	DET
ejpam-4498	26	12	process	process	NOUN
ejpam-4498	26	13	to	to	PART
ejpam-4498	26	14	improve	improve	VERB
ejpam-4498	26	15	students	student	NOUN
ejpam-4498	26	16	’	'	PUNCT
ejpam-4498	26	17	learning	learn	VERB
ejpam-4498	26	18	performance	performance	NOUN
ejpam-4498	26	19	[	[	X
ejpam-4498	26	20	1	1	NUM
ejpam-4498	26	21	]	]	PUNCT
ejpam-4498	26	22	.	.	PUNCT
ejpam-4498	27	1	in	in	ADP
ejpam-4498	27	2	this	this	DET
ejpam-4498	27	3	study	study	NOUN
ejpam-4498	27	4	,	,	PUNCT
ejpam-4498	27	5	we	we	PRON
ejpam-4498	27	6	consider	consider	VERB
ejpam-4498	27	7	the	the	DET
ejpam-4498	27	8	relationship	relationship	NOUN
ejpam-4498	27	9	between	between	ADP
ejpam-4498	27	10	the	the	DET
ejpam-4498	27	11	sums	sum	NOUN
ejpam-4498	27	12	of	of	ADP
ejpam-4498	27	13	natural	natural	ADJ
ejpam-4498	27	14	numbers	number	NOUN
ejpam-4498	27	15	,	,	PUNCT
ejpam-4498	27	16	square	square	ADJ
ejpam-4498	27	17	numbers	number	NOUN
ejpam-4498	27	18	,	,	PUNCT
ejpam-4498	27	19	and	and	CCONJ
ejpam-4498	27	20	triangular	triangular	NOUN
ejpam-4498	27	21	numbers	number	NOUN
ejpam-4498	27	22	using	use	VERB
ejpam-4498	27	23	various	various	ADJ
ejpam-4498	27	24	equations	equation	NOUN
ejpam-4498	27	25	to	to	PART
ejpam-4498	27	26	represent	represent	VERB
ejpam-4498	27	27	the	the	DET
ejpam-4498	27	28	sum	sum	NOUN
ejpam-4498	27	29	of	of	ADP
ejpam-4498	27	30	the	the	DET
ejpam-4498	27	31	triangular	triangular	NOUN
ejpam-4498	27	32	numbers	number	NOUN
ejpam-4498	27	33	.	.	PUNCT
ejpam-4498	28	1	in	in	ADP
ejpam-4498	28	2	standard	standard	ADJ
ejpam-4498	28	3	teaching	teaching	NOUN
ejpam-4498	28	4	methods	method	NOUN
ejpam-4498	28	5	,	,	PUNCT
ejpam-4498	28	6	combinatorics	combinatoric	NOUN
ejpam-4498	28	7	and	and	CCONJ
ejpam-4498	28	8	algebra	algebra	NOUN
ejpam-4498	28	9	are	be	AUX
ejpam-4498	28	10	treated	treat	VERB
ejpam-4498	28	11	as	as	ADP
ejpam-4498	28	12	separate	separate	ADJ
ejpam-4498	28	13	subjects	subject	NOUN
ejpam-4498	28	14	.	.	PUNCT
ejpam-4498	29	1	however	however	ADV
ejpam-4498	29	2	,	,	PUNCT
ejpam-4498	29	3	in	in	ADP
ejpam-4498	29	4	this	this	DET
ejpam-4498	29	5	study	study	NOUN
ejpam-4498	29	6	,	,	PUNCT
ejpam-4498	29	7	the	the	DET
ejpam-4498	29	8	sum	sum	NOUN
ejpam-4498	29	9	of	of	ADP
ejpam-4498	29	10	triangular	triangular	NOUN
ejpam-4498	29	11	numbers	number	NOUN
ejpam-4498	29	12	is	be	AUX
ejpam-4498	29	13	regarded	regard	VERB
ejpam-4498	29	14	as	as	ADP
ejpam-4498	29	15	the	the	DET
ejpam-4498	29	16	basis	basis	NOUN
ejpam-4498	29	17	of	of	ADP
ejpam-4498	29	18	combinatorics	combinatoric	NOUN
ejpam-4498	29	19	and	and	CCONJ
ejpam-4498	29	20	the	the	DET
ejpam-4498	29	21	relationships	relationship	NOUN
ejpam-4498	29	22	between	between	ADP
ejpam-4498	29	23	series	series	NOUN
ejpam-4498	29	24	and	and	CCONJ
ejpam-4498	29	25	combinatorics	combinatoric	NOUN
ejpam-4498	29	26	are	be	AUX
ejpam-4498	29	27	shown	show	VERB
ejpam-4498	29	28	by	by	ADP
ejpam-4498	29	29	a	a	DET
ejpam-4498	29	30	simple	simple	ADJ
ejpam-4498	29	31	approach	approach	NOUN
ejpam-4498	29	32	.	.	PUNCT
ejpam-4498	30	1	this	this	DET
ejpam-4498	30	2	paper	paper	NOUN
ejpam-4498	30	3	presents	present	VERB
ejpam-4498	30	4	an	an	DET
ejpam-4498	30	5	example	example	NOUN
ejpam-4498	30	6	of	of	ADP
ejpam-4498	30	7	how	how	SCONJ
ejpam-4498	30	8	students	student	NOUN
ejpam-4498	30	9	can	can	AUX
ejpam-4498	30	10	deduce	deduce	VERB
ejpam-4498	30	11	mathematical	mathematical	ADJ
ejpam-4498	30	12	expressions	expression	NOUN
ejpam-4498	30	13	;	;	PUNCT
ejpam-4498	30	14	thus	thus	ADV
ejpam-4498	30	15	,	,	PUNCT
ejpam-4498	30	16	the	the	DET
ejpam-4498	30	17	pedagogical	pedagogical	ADJ
ejpam-4498	30	18	approach	approach	NOUN
ejpam-4498	30	19	can	can	AUX
ejpam-4498	30	20	be	be	AUX
ejpam-4498	30	21	used	use	VERB
ejpam-4498	30	22	to	to	PART
ejpam-4498	30	23	determine	determine	VERB
ejpam-4498	30	24	the	the	DET
ejpam-4498	30	25	mathematical	mathematical	ADJ
ejpam-4498	30	26	meaning	meaning	NOUN
ejpam-4498	30	27	of	of	ADP
ejpam-4498	30	28	various	various	ADJ
ejpam-4498	30	29	equations	equation	NOUN
ejpam-4498	30	30	related	relate	VERB
ejpam-4498	30	31	to	to	ADP
ejpam-4498	30	32	the	the	DET
ejpam-4498	30	33	sum	sum	NOUN
ejpam-4498	30	34	of	of	ADP
ejpam-4498	30	35	triangular	triangular	NOUN
ejpam-4498	30	36	numbers	number	NOUN
ejpam-4498	30	37	.	.	PUNCT
ejpam-4498	31	1	although	although	SCONJ
ejpam-4498	31	2	they	they	PRON
ejpam-4498	31	3	are	be	AUX
ejpam-4498	31	4	beyond	beyond	ADP
ejpam-4498	31	5	the	the	DET
ejpam-4498	31	6	scope	scope	NOUN
ejpam-4498	31	7	of	of	ADP
ejpam-4498	31	8	the	the	DET
ejpam-4498	31	9	present	present	ADJ
ejpam-4498	31	10	work	work	NOUN
ejpam-4498	31	11	,	,	PUNCT
ejpam-4498	31	12	some	some	DET
ejpam-4498	31	13	advanced	advanced	ADJ
ejpam-4498	31	14	studies	study	NOUN
ejpam-4498	31	15	that	that	PRON
ejpam-4498	31	16	extend	extend	VERB
ejpam-4498	31	17	this	this	DET
ejpam-4498	31	18	idea	idea	NOUN
ejpam-4498	31	19	beyond	beyond	ADP
ejpam-4498	31	20	mathematics	mathematics	NOUN
ejpam-4498	31	21	education	education	NOUN
ejpam-4498	31	22	are	be	AUX
ejpam-4498	31	23	also	also	ADV
ejpam-4498	31	24	in	in	ADP
ejpam-4498	31	25	progress	progress	NOUN
ejpam-4498	31	26	,	,	PUNCT
ejpam-4498	31	27	which	which	PRON
ejpam-4498	31	28	consider	consider	VERB
ejpam-4498	31	29	approaches	approach	NOUN
ejpam-4498	31	30	such	such	ADJ
ejpam-4498	31	31	as	as	ADP
ejpam-4498	31	32	representation	representation	NOUN
ejpam-4498	31	33	of	of	ADP
ejpam-4498	31	34	integers	integer	NOUN
ejpam-4498	31	35	as	as	ADP
ejpam-4498	31	36	sums	sum	NOUN
ejpam-4498	31	37	of	of	ADP
ejpam-4498	31	38	triangular	triangular	NOUN
ejpam-4498	31	39	numbers	number	NOUN
ejpam-4498	31	40	[	[	X
ejpam-4498	31	41	10	10	NUM
ejpam-4498	31	42	]	]	PUNCT
ejpam-4498	31	43	.	.	PUNCT
ejpam-4498	32	1	2	2	X
ejpam-4498	32	2	.	.	X
ejpam-4498	32	3	equations	equation	NOUN
ejpam-4498	32	4	for	for	ADP
ejpam-4498	32	5	the	the	DET
ejpam-4498	32	6	sum	sum	NOUN
ejpam-4498	32	7	of	of	ADP
ejpam-4498	32	8	triangular	triangular	NOUN
ejpam-4498	32	9	numbers	number	NOUN
ejpam-4498	32	10	to	to	PART
ejpam-4498	32	11	begin	begin	VERB
ejpam-4498	32	12	,	,	PUNCT
ejpam-4498	32	13	we	we	PRON
ejpam-4498	32	14	consider	consider	VERB
ejpam-4498	32	15	the	the	DET
ejpam-4498	32	16	various	various	ADJ
ejpam-4498	32	17	ways	way	NOUN
ejpam-4498	32	18	of	of	ADP
ejpam-4498	32	19	representing	represent	VERB
ejpam-4498	32	20	triangular	triangular	NOUN
ejpam-4498	32	21	numbers	number	NOUN
ejpam-4498	32	22	.	.	PUNCT
ejpam-4498	33	1	the	the	DET
ejpam-4498	33	2	sum	sum	NOUN
ejpam-4498	33	3	of	of	ADP
ejpam-4498	33	4	triangular	triangular	NOUN
ejpam-4498	33	5	numbers	number	NOUN
ejpam-4498	33	6	can	can	AUX
ejpam-4498	33	7	be	be	AUX
ejpam-4498	33	8	expressed	express	VERB
ejpam-4498	33	9	in	in	ADP
ejpam-4498	33	10	four	four	NUM
ejpam-4498	33	11	ways	way	NOUN
ejpam-4498	33	12	[	[	X
ejpam-4498	33	13	7	7	NUM
ejpam-4498	33	14	,	,	PUNCT
ejpam-4498	33	15	9	9	NUM
ejpam-4498	33	16	]	]	PUNCT
ejpam-4498	33	17	,	,	PUNCT
ejpam-4498	33	18	as	as	SCONJ
ejpam-4498	33	19	given	give	VERB
ejpam-4498	33	20	below	below	ADV
ejpam-4498	33	21	.	.	PUNCT
ejpam-4498	34	1	1	1	NUM
ejpam-4498	34	2	2	2	NUM
ejpam-4498	34	3	[	[	SYM
ejpam-4498	34	4	1	1	NUM
ejpam-4498	34	5	·	·	SYM
ejpam-4498	34	6	2	2	NUM
ejpam-4498	34	7	+	+	CCONJ
ejpam-4498	34	8	2	2	NUM
ejpam-4498	34	9	·	·	SYM
ejpam-4498	34	10	3	3	NUM
ejpam-4498	34	11	+	+	CCONJ
ejpam-4498	34	12	3	3	NUM
ejpam-4498	34	13	·	·	SYM
ejpam-4498	34	14	4	4	NUM
ejpam-4498	34	15	+	+	CCONJ
ejpam-4498	34	16	·	·	PUNCT
ejpam-4498	34	17	·	·	PUNCT
ejpam-4498	34	18	·	·	PUNCT
ejpam-4498	35	1	+	+	PUNCT
ejpam-4498	35	2	(	(	PUNCT
ejpam-4498	35	3	k	k	NOUN
ejpam-4498	35	4	−	−	PROPN
ejpam-4498	35	5	1	1	NUM
ejpam-4498	35	6	)	)	PUNCT
ejpam-4498	35	7	·	·	PUNCT
ejpam-4498	36	1	k	k	X
ejpam-4498	36	2	+	+	PUNCT
ejpam-4498	36	3	·	·	PUNCT
ejpam-4498	36	4	·	·	PUNCT
ejpam-4498	36	5	·	·	PUNCT
ejpam-4498	36	6	+	+	CCONJ
ejpam-4498	36	7	(	(	PUNCT
ejpam-4498	36	8	n−	n−	NOUN
ejpam-4498	36	9	1	1	NUM
ejpam-4498	36	10	)	)	PUNCT
ejpam-4498	36	11	·	·	PUNCT
ejpam-4498	36	12	n+	n+	X
ejpam-4498	36	13	n	n	X
ejpam-4498	36	14	·	·	PUNCT
ejpam-4498	36	15	(	(	PUNCT
ejpam-4498	36	16	n+	n+	NOUN
ejpam-4498	36	17	1	1	NUM
ejpam-4498	36	18	)	)	PUNCT
ejpam-4498	36	19	]	]	PUNCT
ejpam-4498	37	1	(	(	PUNCT
ejpam-4498	37	2	1	1	X
ejpam-4498	37	3	)	)	PUNCT
ejpam-4498	37	4	=	=	NOUN
ejpam-4498	37	5	2c2	2c2	NUM
ejpam-4498	37	6	+	+	CCONJ
ejpam-4498	37	7	3c2	3c2	NUM
ejpam-4498	37	8	+	+	CCONJ
ejpam-4498	37	9	4c2	4c2	NUM
ejpam-4498	37	10	+	+	X
ejpam-4498	37	11	·	·	PUNCT
ejpam-4498	37	12	·	·	PUNCT
ejpam-4498	37	13	·	·	PUNCT
ejpam-4498	38	1	+	+	NUM
ejpam-4498	38	2	kc2	kc2	X
ejpam-4498	38	3	+	+	X
ejpam-4498	38	4	·	·	PUNCT
ejpam-4498	38	5	·	·	PUNCT
ejpam-4498	38	6	·	·	PUNCT
ejpam-4498	38	7	+	+	NUM
ejpam-4498	38	8	nc2	nc2	PROPN
ejpam-4498	38	9	+	+	CCONJ
ejpam-4498	38	10	n+1c2	n+1c2	PROPN
ejpam-4498	38	11	(	(	PUNCT
ejpam-4498	38	12	2	2	NUM
ejpam-4498	38	13	)	)	PUNCT
ejpam-4498	38	14	=	=	SYM
ejpam-4498	38	15	1	1	NUM
ejpam-4498	38	16	+	+	CCONJ
ejpam-4498	38	17	(	(	PUNCT
ejpam-4498	38	18	1	1	NUM
ejpam-4498	38	19	+	+	NUM
ejpam-4498	38	20	2	2	NUM
ejpam-4498	38	21	)	)	PUNCT
ejpam-4498	38	22	+	+	CCONJ
ejpam-4498	38	23	(	(	PUNCT
ejpam-4498	38	24	1	1	NUM
ejpam-4498	38	25	+	+	NUM
ejpam-4498	38	26	2	2	NUM
ejpam-4498	38	27	+	+	NUM
ejpam-4498	38	28	3	3	NUM
ejpam-4498	38	29	)	)	PUNCT
ejpam-4498	38	30	+	+	NUM
ejpam-4498	38	31	·	·	PUNCT
ejpam-4498	38	32	·	·	PUNCT
ejpam-4498	38	33	·	·	PUNCT
ejpam-4498	38	34	+	+	CCONJ
ejpam-4498	38	35	(	(	PUNCT
ejpam-4498	38	36	1	1	NUM
ejpam-4498	38	37	+	+	NUM
ejpam-4498	38	38	2	2	NUM
ejpam-4498	38	39	+	+	NUM
ejpam-4498	38	40	3	3	NUM
ejpam-4498	38	41	+	+	NUM
ejpam-4498	38	42	·	·	PUNCT
ejpam-4498	38	43	·	·	PUNCT
ejpam-4498	38	44	·	·	PUNCT
ejpam-4498	38	45	+	+	NUM
ejpam-4498	38	46	n	n	CCONJ
ejpam-4498	38	47	)	)	PUNCT
ejpam-4498	38	48	(	(	PUNCT
ejpam-4498	38	49	3	3	X
ejpam-4498	38	50	)	)	PUNCT
ejpam-4498	38	51	=	=	SYM
ejpam-4498	38	52	1	1	X
ejpam-4498	38	53	·	·	PUNCT
ejpam-4498	38	54	n+	n+	NUM
ejpam-4498	38	55	2	2	NUM
ejpam-4498	38	56	·	·	PUNCT
ejpam-4498	38	57	(	(	PUNCT
ejpam-4498	38	58	n−	n−	NOUN
ejpam-4498	38	59	1	1	NUM
ejpam-4498	38	60	)	)	PUNCT
ejpam-4498	38	61	+	+	CCONJ
ejpam-4498	38	62	·	·	PUNCT
ejpam-4498	38	63	·	·	PUNCT
ejpam-4498	38	64	·	·	PUNCT
ejpam-4498	38	65	+	+	NUM
ejpam-4498	38	66	n	n	CCONJ
ejpam-4498	38	67	·	·	SYM
ejpam-4498	38	68	1	1	X
ejpam-4498	38	69	.	.	PUNCT
ejpam-4498	38	70	(	(	PUNCT
ejpam-4498	38	71	4	4	X
ejpam-4498	38	72	)	)	PUNCT
ejpam-4498	38	73	this	this	PRON
ejpam-4498	38	74	is	be	AUX
ejpam-4498	38	75	a	a	DET
ejpam-4498	38	76	useful	useful	ADJ
ejpam-4498	38	77	pedagogical	pedagogical	ADJ
ejpam-4498	38	78	exercise	exercise	NOUN
ejpam-4498	38	79	for	for	ADP
ejpam-4498	38	80	students	student	NOUN
ejpam-4498	38	81	,	,	PUNCT
ejpam-4498	38	82	in	in	ADP
ejpam-4498	38	83	which	which	PRON
ejpam-4498	38	84	they	they	PRON
ejpam-4498	38	85	consider	consider	VERB
ejpam-4498	38	86	why	why	SCONJ
ejpam-4498	38	87	the	the	DET
ejpam-4498	38	88	values	value	NOUN
ejpam-4498	38	89	of	of	ADP
ejpam-4498	38	90	these	these	DET
ejpam-4498	38	91	four	four	NUM
ejpam-4498	38	92	equations	equation	NOUN
ejpam-4498	38	93	are	be	AUX
ejpam-4498	38	94	equal	equal	ADJ
ejpam-4498	38	95	despite	despite	SCONJ
ejpam-4498	38	96	their	their	PRON
ejpam-4498	38	97	different	different	ADJ
ejpam-4498	38	98	forms	form	NOUN
ejpam-4498	38	99	.	.	PUNCT
ejpam-4498	39	1	computational	computational	ADJ
ejpam-4498	39	2	skills	skill	NOUN
ejpam-4498	39	3	are	be	AUX
ejpam-4498	39	4	important	important	ADJ
ejpam-4498	39	5	to	to	PART
ejpam-4498	39	6	help	help	VERB
ejpam-4498	39	7	students	student	NOUN
ejpam-4498	39	8	cultivate	cultivate	VERB
ejpam-4498	39	9	the	the	DET
ejpam-4498	39	10	ability	ability	NOUN
ejpam-4498	39	11	to	to	PART
ejpam-4498	39	12	observe	observe	VERB
ejpam-4498	39	13	expressions	expression	NOUN
ejpam-4498	39	14	of	of	ADP
ejpam-4498	39	15	equations	equation	NOUN
ejpam-4498	39	16	.	.	PUNCT
ejpam-4498	40	1	to	to	ADP
ejpam-4498	40	2	yukio	yukio	PROPN
ejpam-4498	40	3	kobayashi	kobayashi	PROPN
ejpam-4498	40	4	/	/	SYM
ejpam-4498	40	5	eur	eur	PROPN
ejpam-4498	40	6	.	.	PUNCT
ejpam-4498	41	1	j.	j.	PROPN
ejpam-4498	41	2	pure	pure	PROPN
ejpam-4498	41	3	appl	appl	PROPN
ejpam-4498	41	4	.	.	PROPN
ejpam-4498	41	5	math	math	PROPN
ejpam-4498	41	6	,	,	PUNCT
ejpam-4498	41	7	15	15	NUM
ejpam-4498	41	8	(	(	PUNCT
ejpam-4498	41	9	3	3	NUM
ejpam-4498	41	10	)	)	PUNCT
ejpam-4498	41	11	(	(	PUNCT
ejpam-4498	41	12	2022	2022	NUM
ejpam-4498	41	13	)	)	PUNCT
ejpam-4498	41	14	,	,	PUNCT
ejpam-4498	41	15	1237	1237	NUM
ejpam-4498	41	16	-	-	SYM
ejpam-4498	41	17	1253	1253	NUM
ejpam-4498	41	18	1239	1239	NUM
ejpam-4498	41	19	this	this	DET
ejpam-4498	41	20	end	end	NOUN
ejpam-4498	41	21	,	,	PUNCT
ejpam-4498	41	22	becoming	become	VERB
ejpam-4498	41	23	sufficiently	sufficiently	ADV
ejpam-4498	41	24	skilled	skilled	ADJ
ejpam-4498	41	25	such	such	ADJ
ejpam-4498	41	26	that	that	SCONJ
ejpam-4498	41	27	one	one	NOUN
ejpam-4498	41	28	can	can	AUX
ejpam-4498	41	29	clearly	clearly	ADV
ejpam-4498	41	30	perceive	perceive	VERB
ejpam-4498	41	31	the	the	DET
ejpam-4498	41	32	values	value	NOUN
ejpam-4498	41	33	of	of	ADP
ejpam-4498	41	34	these	these	DET
ejpam-4498	41	35	equations	equation	NOUN
ejpam-4498	41	36	as	as	ADP
ejpam-4498	41	37	equal	equal	ADJ
ejpam-4498	41	38	is	be	AUX
ejpam-4498	41	39	an	an	DET
ejpam-4498	41	40	appropriate	appropriate	ADJ
ejpam-4498	41	41	learning	learning	NOUN
ejpam-4498	41	42	goal	goal	NOUN
ejpam-4498	41	43	.	.	PUNCT
ejpam-4498	42	1	first	first	ADV
ejpam-4498	42	2	,	,	PUNCT
ejpam-4498	42	3	instructors	instructor	NOUN
ejpam-4498	42	4	should	should	AUX
ejpam-4498	42	5	begin	begin	VERB
ejpam-4498	42	6	by	by	ADP
ejpam-4498	42	7	having	have	VERB
ejpam-4498	42	8	students	student	NOUN
ejpam-4498	42	9	explain	explain	VERB
ejpam-4498	42	10	how	how	SCONJ
ejpam-4498	42	11	(	(	PUNCT
ejpam-4498	42	12	1	1	X
ejpam-4498	42	13	)	)	PUNCT
ejpam-4498	42	14	and	and	CCONJ
ejpam-4498	42	15	(	(	PUNCT
ejpam-4498	42	16	2	2	X
ejpam-4498	42	17	)	)	PUNCT
ejpam-4498	42	18	are	be	AUX
ejpam-4498	42	19	equivalent	equivalent	ADJ
ejpam-4498	42	20	.	.	PUNCT
ejpam-4498	43	1	if	if	SCONJ
ejpam-4498	43	2	students	student	NOUN
ejpam-4498	43	3	notice	notice	VERB
ejpam-4498	43	4	that	that	SCONJ
ejpam-4498	43	5	the	the	DET
ejpam-4498	43	6	denominator	denominator	NOUN
ejpam-4498	43	7	2	2	NUM
ejpam-4498	43	8	is	be	AUX
ejpam-4498	43	9	equal	equal	ADJ
ejpam-4498	43	10	to	to	ADP
ejpam-4498	43	11	2	2	NUM
ejpam-4498	43	12	!	!	PUNCT
ejpam-4498	43	13	,	,	PUNCT
ejpam-4498	43	14	then	then	ADV
ejpam-4498	43	15	it	it	PRON
ejpam-4498	43	16	is	be	AUX
ejpam-4498	43	17	clear	clear	ADJ
ejpam-4498	43	18	that	that	SCONJ
ejpam-4498	43	19	(	(	PUNCT
ejpam-4498	43	20	1	1	X
ejpam-4498	43	21	)	)	PUNCT
ejpam-4498	43	22	expresses	express	VERB
ejpam-4498	43	23	the	the	DET
ejpam-4498	43	24	sum	sum	NOUN
ejpam-4498	43	25	of	of	ADP
ejpam-4498	43	26	combinations	combination	NOUN
ejpam-4498	44	1	n∑	n∑	NOUN
ejpam-4498	44	2	k=1	k=1	PROPN
ejpam-4498	45	1	k+1c2	k+1c2	PROPN
ejpam-4498	46	1	[	[	X
ejpam-4498	46	2	7	7	NUM
ejpam-4498	46	3	,	,	PUNCT
ejpam-4498	46	4	9	9	NUM
ejpam-4498	46	5	,	,	PUNCT
ejpam-4498	46	6	11	11	NUM
ejpam-4498	46	7	]	]	PUNCT
ejpam-4498	46	8	.	.	PUNCT
ejpam-4498	47	1	second	second	ADJ
ejpam-4498	47	2	,	,	PUNCT
ejpam-4498	47	3	students	student	NOUN
ejpam-4498	47	4	should	should	AUX
ejpam-4498	47	5	proceed	proceed	VERB
ejpam-4498	47	6	to	to	PART
ejpam-4498	47	7	explain	explain	VERB
ejpam-4498	47	8	how	how	SCONJ
ejpam-4498	47	9	(	(	PUNCT
ejpam-4498	47	10	2	2	NUM
ejpam-4498	47	11	)	)	PUNCT
ejpam-4498	47	12	and	and	CCONJ
ejpam-4498	47	13	(	(	PUNCT
ejpam-4498	47	14	3	3	X
ejpam-4498	47	15	)	)	PUNCT
ejpam-4498	47	16	are	be	AUX
ejpam-4498	47	17	equivalent	equivalent	ADJ
ejpam-4498	47	18	.	.	PUNCT
ejpam-4498	48	1	if	if	SCONJ
ejpam-4498	48	2	students	student	NOUN
ejpam-4498	48	3	draw	draw	VERB
ejpam-4498	48	4	combinations	combination	NOUN
ejpam-4498	48	5	k+1c2	k+1c2	PROPN
ejpam-4498	48	6	in	in	ADP
ejpam-4498	48	7	terms	term	NOUN
ejpam-4498	48	8	of	of	ADP
ejpam-4498	48	9	tree	tree	NOUN
ejpam-4498	48	10	diagrams	diagram	NOUN
ejpam-4498	48	11	,	,	PUNCT
ejpam-4498	48	12	they	they	PRON
ejpam-4498	48	13	can	can	AUX
ejpam-4498	48	14	determine	determine	VERB
ejpam-4498	48	15	that	that	SCONJ
ejpam-4498	48	16	the	the	DET
ejpam-4498	48	17	total	total	ADJ
ejpam-4498	48	18	number	number	NOUN
ejpam-4498	48	19	of	of	ADP
ejpam-4498	48	20	edges	edge	NOUN
ejpam-4498	48	21	is	be	AUX
ejpam-4498	48	22	1	1	NUM
ejpam-4498	48	23	+	+	CCONJ
ejpam-4498	48	24	2	2	NUM
ejpam-4498	48	25	+	+	NUM
ejpam-4498	48	26	·	·	PUNCT
ejpam-4498	48	27	·	·	PUNCT
ejpam-4498	48	28	·	·	PUNCT
ejpam-4498	49	1	+	+	CCONJ
ejpam-4498	49	2	k.	k.	NOUN
ejpam-4498	49	3	the	the	DET
ejpam-4498	49	4	equivalence	equivalence	NOUN
ejpam-4498	49	5	of	of	ADP
ejpam-4498	49	6	(	(	PUNCT
ejpam-4498	49	7	1	1	NUM
ejpam-4498	49	8	)	)	PUNCT
ejpam-4498	49	9	and	and	CCONJ
ejpam-4498	49	10	(	(	PUNCT
ejpam-4498	49	11	3	3	X
ejpam-4498	49	12	)	)	PUNCT
ejpam-4498	49	13	can	can	AUX
ejpam-4498	49	14	also	also	ADV
ejpam-4498	49	15	be	be	AUX
ejpam-4498	49	16	explained	explain	VERB
ejpam-4498	49	17	by	by	ADP
ejpam-4498	49	18	rewriting	rewrite	VERB
ejpam-4498	49	19	(	(	PUNCT
ejpam-4498	49	20	1	1	NUM
ejpam-4498	49	21	)	)	PUNCT
ejpam-4498	49	22	as	as	ADP
ejpam-4498	49	23	the	the	DET
ejpam-4498	49	24	sum	sum	NOUN
ejpam-4498	49	25	of	of	ADP
ejpam-4498	49	26	the	the	DET
ejpam-4498	49	27	arithmetical	arithmetical	ADJ
ejpam-4498	49	28	series	series	NOUN
ejpam-4498	49	29	1(1	1(1	NUM
ejpam-4498	49	30	+	+	CCONJ
ejpam-4498	49	31	1	1	NUM
ejpam-4498	49	32	)	)	PUNCT
ejpam-4498	49	33	2	2	NUM
ejpam-4498	49	34	+	+	NUM
ejpam-4498	49	35	2(2	2(2	NUM
ejpam-4498	50	1	+	+	CCONJ
ejpam-4498	50	2	1	1	NUM
ejpam-4498	50	3	)	)	PUNCT
ejpam-4498	50	4	2	2	NUM
ejpam-4498	51	1	+	+	CCONJ
ejpam-4498	51	2	·	·	PUNCT
ejpam-4498	51	3	·	·	PUNCT
ejpam-4498	51	4	·	·	PUNCT
ejpam-4498	51	5	+	+	NUM
ejpam-4498	51	6	n(n+	n(n+	NUM
ejpam-4498	51	7	1	1	NUM
ejpam-4498	51	8	)	)	PUNCT
ejpam-4498	51	9	2	2	NUM
ejpam-4498	51	10	.	.	PUNCT
ejpam-4498	52	1	(	(	PUNCT
ejpam-4498	52	2	5	5	NUM
ejpam-4498	52	3	)	)	PUNCT
ejpam-4498	52	4	third	third	NOUN
ejpam-4498	52	5	,	,	PUNCT
ejpam-4498	52	6	(	(	PUNCT
ejpam-4498	52	7	4	4	X
ejpam-4498	52	8	)	)	PUNCT
ejpam-4498	52	9	is	be	AUX
ejpam-4498	52	10	derived	derive	VERB
ejpam-4498	52	11	by	by	ADP
ejpam-4498	52	12	rewriting	rewrite	VERB
ejpam-4498	52	13	(	(	PUNCT
ejpam-4498	52	14	3	3	NUM
ejpam-4498	52	15	)	)	PUNCT
ejpam-4498	52	16	as	as	ADP
ejpam-4498	52	17	1	1	NUM
ejpam-4498	52	18	+	+	CCONJ
ejpam-4498	52	19	(	(	PUNCT
ejpam-4498	52	20	1	1	NUM
ejpam-4498	52	21	+	+	NUM
ejpam-4498	52	22	2	2	NUM
ejpam-4498	52	23	)	)	PUNCT
ejpam-4498	52	24	+	+	CCONJ
ejpam-4498	52	25	(	(	PUNCT
ejpam-4498	52	26	1	1	NUM
ejpam-4498	52	27	+	+	NUM
ejpam-4498	52	28	2	2	NUM
ejpam-4498	52	29	+	+	NUM
ejpam-4498	52	30	3	3	NUM
ejpam-4498	52	31	)	)	PUNCT
ejpam-4498	52	32	+	+	NUM
ejpam-4498	52	33	·	·	PUNCT
ejpam-4498	52	34	·	·	PUNCT
ejpam-4498	52	35	·	·	PUNCT
ejpam-4498	53	1	+	+	PUNCT
ejpam-4498	54	1	[	[	X
ejpam-4498	54	2	1	1	NUM
ejpam-4498	54	3	+	+	NUM
ejpam-4498	54	4	2	2	NUM
ejpam-4498	54	5	+	+	NUM
ejpam-4498	54	6	3	3	NUM
ejpam-4498	54	7	+	+	NUM
ejpam-4498	54	8	·	·	PUNCT
ejpam-4498	54	9	·	·	PUNCT
ejpam-4498	54	10	·	·	PUNCT
ejpam-4498	54	11	+	+	CCONJ
ejpam-4498	54	12	(	(	PUNCT
ejpam-4498	54	13	n−	n−	NOUN
ejpam-4498	54	14	1	1	NUM
ejpam-4498	54	15	)	)	PUNCT
ejpam-4498	54	16	]	]	PUNCT
ejpam-4498	55	1	+	+	CCONJ
ejpam-4498	55	2	[	[	X
ejpam-4498	55	3	1	1	NUM
ejpam-4498	55	4	+	+	NUM
ejpam-4498	55	5	2	2	NUM
ejpam-4498	55	6	+	+	NUM
ejpam-4498	55	7	3	3	NUM
ejpam-4498	55	8	+	+	NUM
ejpam-4498	55	9	·	·	PUNCT
ejpam-4498	55	10	·	·	PUNCT
ejpam-4498	55	11	·	·	PUNCT
ejpam-4498	55	12	+	+	CCONJ
ejpam-4498	55	13	(	(	PUNCT
ejpam-4498	55	14	n−	n−	NOUN
ejpam-4498	55	15	1	1	NUM
ejpam-4498	55	16	)	)	PUNCT
ejpam-4498	55	17	+	+	NUM
ejpam-4498	55	18	n	n	CCONJ
ejpam-4498	55	19	]	]	X
ejpam-4498	55	20	=	=	PUNCT
ejpam-4498	55	21	(	(	PUNCT
ejpam-4498	55	22	1	1	NUM
ejpam-4498	55	23	+	+	NUM
ejpam-4498	55	24	1	1	NUM
ejpam-4498	55	25	+	+	CCONJ
ejpam-4498	55	26	·	·	PUNCT
ejpam-4498	55	27	·	·	PUNCT
ejpam-4498	55	28	·	·	PUNCT
ejpam-4498	55	29	+	+	NUM
ejpam-4498	55	30	1)︸	1)︸	NUM
ejpam-4498	55	31	︷︷	︷︷	PROPN
ejpam-4498	55	32	︸	︸	ADP
ejpam-4498	55	33	n	n	PROPN
ejpam-4498	55	34	terms	term	NOUN
ejpam-4498	55	35	+	+	PROPN
ejpam-4498	55	36	(	(	PUNCT
ejpam-4498	55	37	2	2	NUM
ejpam-4498	55	38	+	+	SYM
ejpam-4498	55	39	2	2	NUM
ejpam-4498	55	40	+	+	CCONJ
ejpam-4498	55	41	·	·	PUNCT
ejpam-4498	55	42	·	·	PUNCT
ejpam-4498	55	43	·	·	PUNCT
ejpam-4498	55	44	+	+	NUM
ejpam-4498	55	45	2)︸	2)︸	NUM
ejpam-4498	55	46	︷︷	︷︷	PROPN
ejpam-4498	55	47	︸	︸	X
ejpam-4498	55	48	(	(	PUNCT
ejpam-4498	55	49	n−1	n−1	PROPN
ejpam-4498	55	50	)	)	PUNCT
ejpam-4498	55	51	terms	term	NOUN
ejpam-4498	55	52	+	+	X
ejpam-4498	55	53	·	·	PUNCT
ejpam-4498	55	54	·	·	PUNCT
ejpam-4498	55	55	·	·	PUNCT
ejpam-4498	56	1	+	+	PUNCT
ejpam-4498	56	2	[	[	X
ejpam-4498	56	3	(	(	PUNCT
ejpam-4498	56	4	n−	n−	NOUN
ejpam-4498	56	5	1	1	NUM
ejpam-4498	56	6	)	)	PUNCT
ejpam-4498	56	7	+	+	CCONJ
ejpam-4498	56	8	(	(	PUNCT
ejpam-4498	56	9	n−	n−	NOUN
ejpam-4498	56	10	1)]︸	1)]︸	NUM
ejpam-4498	56	11	︷︷	︷︷	PROPN
ejpam-4498	56	12	︸	︸	ADP
ejpam-4498	56	13	two	two	NUM
ejpam-4498	56	14	terms	term	NOUN
ejpam-4498	56	15	+	+	CCONJ
ejpam-4498	56	16	n︸︷︷︸	n︸︷︷︸	ADP
ejpam-4498	56	17	one	one	NUM
ejpam-4498	56	18	term	term	NOUN
ejpam-4498	56	19	.	.	PUNCT
ejpam-4498	57	1	(	(	PUNCT
ejpam-4498	57	2	6	6	NUM
ejpam-4498	57	3	)	)	PUNCT
ejpam-4498	57	4	equations	equation	NOUN
ejpam-4498	57	5	(	(	PUNCT
ejpam-4498	57	6	1	1	NUM
ejpam-4498	57	7	)	)	PUNCT
ejpam-4498	57	8	,	,	PUNCT
ejpam-4498	57	9	(	(	PUNCT
ejpam-4498	57	10	2	2	NUM
ejpam-4498	57	11	)	)	PUNCT
ejpam-4498	57	12	,	,	PUNCT
ejpam-4498	57	13	(	(	PUNCT
ejpam-4498	57	14	3	3	NUM
ejpam-4498	57	15	)	)	PUNCT
ejpam-4498	57	16	,	,	PUNCT
ejpam-4498	57	17	and	and	CCONJ
ejpam-4498	57	18	(	(	PUNCT
ejpam-4498	57	19	4	4	X
ejpam-4498	57	20	)	)	PUNCT
ejpam-4498	57	21	can	can	AUX
ejpam-4498	57	22	be	be	AUX
ejpam-4498	57	23	also	also	ADV
ejpam-4498	57	24	represented	represent	VERB
ejpam-4498	57	25	by	by	ADP
ejpam-4498	57	26	a	a	DET
ejpam-4498	57	27	pictorial	pictorial	ADJ
ejpam-4498	57	28	proof	proof	NOUN
ejpam-4498	57	29	because	because	SCONJ
ejpam-4498	57	30	the	the	DET
ejpam-4498	57	31	value	value	NOUN
ejpam-4498	57	32	of	of	ADP
ejpam-4498	57	33	these	these	DET
ejpam-4498	57	34	equations	equation	NOUN
ejpam-4498	57	35	is	be	AUX
ejpam-4498	57	36	equal	equal	ADJ
ejpam-4498	57	37	to	to	ADP
ejpam-4498	57	38	the	the	DET
ejpam-4498	57	39	total	total	ADJ
ejpam-4498	57	40	number	number	NOUN
ejpam-4498	57	41	of	of	ADP
ejpam-4498	57	42	tree	tree	NOUN
ejpam-4498	57	43	diagrams	diagram	NOUN
ejpam-4498	57	44	in	in	ADP
ejpam-4498	57	45	figure	figure	NOUN
ejpam-4498	57	46	1	1	NUM
ejpam-4498	57	47	.	.	PUNCT
ejpam-4498	58	1	if	if	SCONJ
ejpam-4498	58	2	the	the	DET
ejpam-4498	58	3	edges	edge	NOUN
ejpam-4498	58	4	are	be	AUX
ejpam-4498	58	5	replaced	replace	VERB
ejpam-4498	58	6	with	with	ADP
ejpam-4498	58	7	filled	fill	VERB
ejpam-4498	58	8	circles	circle	NOUN
ejpam-4498	58	9	and	and	CCONJ
ejpam-4498	58	10	the	the	DET
ejpam-4498	58	11	points	point	NOUN
ejpam-4498	58	12	are	be	AUX
ejpam-4498	58	13	rearranged	rearrange	VERB
ejpam-4498	58	14	into	into	ADP
ejpam-4498	58	15	equilateral	equilateral	ADJ
ejpam-4498	58	16	triangles	triangle	NOUN
ejpam-4498	58	17	,	,	PUNCT
ejpam-4498	58	18	students	student	NOUN
ejpam-4498	58	19	should	should	AUX
ejpam-4498	58	20	be	be	AUX
ejpam-4498	58	21	convinced	convince	VERB
ejpam-4498	58	22	that	that	SCONJ
ejpam-4498	58	23	the	the	DET
ejpam-4498	58	24	number	number	NOUN
ejpam-4498	58	25	of	of	ADP
ejpam-4498	58	26	filled	fill	VERB
ejpam-4498	58	27	circles	circle	NOUN
ejpam-4498	58	28	is	be	AUX
ejpam-4498	58	29	triangular	triangular	ADJ
ejpam-4498	58	30	.	.	PUNCT
ejpam-4498	59	1	3	3	X
ejpam-4498	59	2	.	.	X
ejpam-4498	59	3	relationship	relationship	NOUN
ejpam-4498	59	4	between	between	ADP
ejpam-4498	59	5	the	the	DET
ejpam-4498	59	6	sums	sum	NOUN
ejpam-4498	59	7	of	of	ADP
ejpam-4498	59	8	natural	natural	ADJ
ejpam-4498	59	9	numbers	number	NOUN
ejpam-4498	59	10	,	,	PUNCT
ejpam-4498	59	11	triangular	triangular	NOUN
ejpam-4498	59	12	numbers	number	NOUN
ejpam-4498	59	13	,	,	PUNCT
ejpam-4498	59	14	and	and	CCONJ
ejpam-4498	59	15	the	the	DET
ejpam-4498	59	16	product	product	NOUN
ejpam-4498	59	17	of	of	ADP
ejpam-4498	59	18	consecutive	consecutive	ADJ
ejpam-4498	59	19	numbers	number	NOUN
ejpam-4498	59	20	having	have	VERB
ejpam-4498	59	21	students	student	NOUN
ejpam-4498	59	22	first	first	ADV
ejpam-4498	59	23	determine	determine	VERB
ejpam-4498	59	24	the	the	DET
ejpam-4498	59	25	characteristics	characteristic	NOUN
ejpam-4498	59	26	of	of	ADP
ejpam-4498	59	27	concrete	concrete	ADJ
ejpam-4498	59	28	numbers	number	NOUN
ejpam-4498	59	29	or	or	CCONJ
ejpam-4498	59	30	equations	equation	NOUN
ejpam-4498	59	31	by	by	ADP
ejpam-4498	59	32	arranging	arrange	VERB
ejpam-4498	59	33	them	they	PRON
ejpam-4498	59	34	is	be	AUX
ejpam-4498	59	35	essential	essential	ADJ
ejpam-4498	59	36	,	,	PUNCT
ejpam-4498	59	37	rather	rather	ADV
ejpam-4498	59	38	than	than	ADP
ejpam-4498	59	39	providing	provide	VERB
ejpam-4498	59	40	formulas	formula	NOUN
ejpam-4498	59	41	and	and	CCONJ
ejpam-4498	59	42	having	have	VERB
ejpam-4498	59	43	students	student	NOUN
ejpam-4498	59	44	prove	prove	VERB
ejpam-4498	59	45	them	they	PRON
ejpam-4498	59	46	.	.	PUNCT
ejpam-4498	60	1	instructors	instructor	NOUN
ejpam-4498	60	2	can	can	AUX
ejpam-4498	60	3	adopt	adopt	VERB
ejpam-4498	60	4	the	the	DET
ejpam-4498	60	5	following	following	ADJ
ejpam-4498	60	6	heuristic	heuristic	ADJ
ejpam-4498	60	7	method	method	NOUN
ejpam-4498	60	8	to	to	PART
ejpam-4498	60	9	implement	implement	VERB
ejpam-4498	60	10	this	this	DET
ejpam-4498	60	11	policy	policy	NOUN
ejpam-4498	60	12	.	.	PUNCT
ejpam-4498	61	1	instructors	instructor	NOUN
ejpam-4498	61	2	should	should	AUX
ejpam-4498	61	3	display	display	VERB
ejpam-4498	61	4	figure	figure	NOUN
ejpam-4498	61	5	2	2	NUM
ejpam-4498	61	6	and	and	CCONJ
ejpam-4498	61	7	ask	ask	VERB
ejpam-4498	61	8	students	student	NOUN
ejpam-4498	61	9	to	to	PART
ejpam-4498	61	10	consider	consider	VERB
ejpam-4498	61	11	the	the	DET
ejpam-4498	61	12	problem	problem	NOUN
ejpam-4498	61	13	of	of	ADP
ejpam-4498	61	14	finding	find	VERB
ejpam-4498	61	15	the	the	DET
ejpam-4498	61	16	relationship	relationship	NOUN
ejpam-4498	61	17	between	between	ADP
ejpam-4498	61	18	the	the	DET
ejpam-4498	61	19	sums	sum	NOUN
ejpam-4498	61	20	of	of	ADP
ejpam-4498	61	21	natural	natural	ADJ
ejpam-4498	61	22	numbers	number	NOUN
ejpam-4498	61	23	,	,	PUNCT
ejpam-4498	61	24	triangular	triangular	NOUN
ejpam-4498	61	25	numbers	number	NOUN
ejpam-4498	61	26	,	,	PUNCT
ejpam-4498	61	27	and	and	CCONJ
ejpam-4498	61	28	the	the	DET
ejpam-4498	61	29	product	product	NOUN
ejpam-4498	61	30	of	of	ADP
ejpam-4498	61	31	consecutive	consecutive	ADJ
ejpam-4498	61	32	numbers	number	NOUN
ejpam-4498	61	33	,	,	PUNCT
ejpam-4498	61	34	referring	refer	VERB
ejpam-4498	61	35	to	to	ADP
ejpam-4498	61	36	the	the	DET
ejpam-4498	61	37	equations	equation	NOUN
ejpam-4498	61	38	presented	present	VERB
ejpam-4498	61	39	in	in	ADP
ejpam-4498	61	40	section	section	NOUN
ejpam-4498	61	41	2	2	NUM
ejpam-4498	61	42	.	.	PUNCT
ejpam-4498	61	43	considering	consider	VERB
ejpam-4498	61	44	the	the	DET
ejpam-4498	61	45	sum	sum	NOUN
ejpam-4498	61	46	of	of	ADP
ejpam-4498	61	47	the	the	DET
ejpam-4498	61	48	natural	natural	ADJ
ejpam-4498	61	49	numbers	number	NOUN
ejpam-4498	61	50	from	from	ADP
ejpam-4498	61	51	1	1	NUM
ejpam-4498	61	52	to	to	ADP
ejpam-4498	61	53	n	n	CCONJ
ejpam-4498	61	54	,	,	PUNCT
ejpam-4498	61	55	the	the	DET
ejpam-4498	61	56	following	follow	VERB
ejpam-4498	61	57	sequence	sequence	NOUN
ejpam-4498	61	58	of	of	ADP
ejpam-4498	61	59	n	n	PRON
ejpam-4498	61	60	equations	equation	NOUN
ejpam-4498	61	61	can	can	AUX
ejpam-4498	61	62	be	be	AUX
ejpam-4498	61	63	provided	provide	VERB
ejpam-4498	61	64	(	(	PUNCT
ejpam-4498	61	65	instead	instead	ADV
ejpam-4498	61	66	of	of	ADP
ejpam-4498	61	67	the	the	DET
ejpam-4498	61	68	five	five	NUM
ejpam-4498	61	69	equations	equation	NOUN
ejpam-4498	61	70	in	in	ADP
ejpam-4498	61	71	figure	figure	NOUN
ejpam-4498	61	72	2	2	NUM
ejpam-4498	61	73	)	)	PUNCT
ejpam-4498	61	74	.	.	PUNCT
ejpam-4498	62	1	depending	depend	VERB
ejpam-4498	62	2	on	on	ADP
ejpam-4498	62	3	how	how	SCONJ
ejpam-4498	62	4	we	we	PRON
ejpam-4498	62	5	interpret	interpret	VERB
ejpam-4498	62	6	the	the	DET
ejpam-4498	62	7	sequence	sequence	NOUN
ejpam-4498	62	8	of	of	ADP
ejpam-4498	62	9	equations	equation	NOUN
ejpam-4498	62	10	in	in	ADP
ejpam-4498	62	11	figure	figure	NOUN
ejpam-4498	62	12	2	2	NUM
ejpam-4498	62	13	,	,	PUNCT
ejpam-4498	62	14	we	we	PRON
ejpam-4498	62	15	can	can	AUX
ejpam-4498	62	16	determine	determine	VERB
ejpam-4498	62	17	the	the	DET
ejpam-4498	62	18	features	feature	NOUN
ejpam-4498	62	19	shown	show	VERB
ejpam-4498	62	20	in	in	ADP
ejpam-4498	62	21	figures	figure	NOUN
ejpam-4498	62	22	3	3	NUM
ejpam-4498	62	23	and	and	CCONJ
ejpam-4498	62	24	4	4	NUM
ejpam-4498	62	25	.	.	PUNCT
ejpam-4498	63	1	the	the	DET
ejpam-4498	63	2	features	feature	NOUN
ejpam-4498	63	3	in	in	ADP
ejpam-4498	63	4	figure	figure	NOUN
ejpam-4498	63	5	3	3	NUM
ejpam-4498	63	6	can	can	AUX
ejpam-4498	63	7	be	be	AUX
ejpam-4498	63	8	expressed	express	VERB
ejpam-4498	63	9	as	as	ADP
ejpam-4498	63	10	given	give	VERB
ejpam-4498	63	11	below	below	ADV
ejpam-4498	63	12	.	.	PUNCT
ejpam-4498	64	1	(	(	PUNCT
ejpam-4498	64	2	1	1	NUM
ejpam-4498	64	3	+	+	SYM
ejpam-4498	64	4	2	2	NUM
ejpam-4498	64	5	+	+	NUM
ejpam-4498	64	6	3	3	NUM
ejpam-4498	64	7	+	+	SYM
ejpam-4498	64	8	4	4	NUM
ejpam-4498	64	9	+	+	SYM
ejpam-4498	64	10	5	5	NUM
ejpam-4498	64	11	)	)	PUNCT
ejpam-4498	64	12	·	·	PUNCT
ejpam-4498	64	13	5	5	NUM
ejpam-4498	64	14	=	=	SYM
ejpam-4498	64	15	(	(	PUNCT
ejpam-4498	64	16	1	1	NUM
ejpam-4498	64	17	·	·	SYM
ejpam-4498	64	18	5	5	NUM
ejpam-4498	64	19	+	+	CCONJ
ejpam-4498	64	20	2	2	NUM
ejpam-4498	64	21	·	·	SYM
ejpam-4498	64	22	4	4	NUM
ejpam-4498	64	23	+	+	SYM
ejpam-4498	64	24	3	3	NUM
ejpam-4498	64	25	·	·	SYM
ejpam-4498	64	26	3	3	NUM
ejpam-4498	65	1	+	+	CCONJ
ejpam-4498	65	2	4	4	NUM
ejpam-4498	65	3	·	·	SYM
ejpam-4498	65	4	2	2	NUM
ejpam-4498	65	5	+	+	SYM
ejpam-4498	65	6	5	5	NUM
ejpam-4498	65	7	·	·	SYM
ejpam-4498	65	8	1	1	NUM
ejpam-4498	65	9	)	)	PUNCT
ejpam-4498	66	1	+	+	NOUN
ejpam-4498	66	2	(	(	PUNCT
ejpam-4498	66	3	2	2	NUM
ejpam-4498	66	4	·	·	SYM
ejpam-4498	66	5	1	1	NUM
ejpam-4498	66	6	+	+	CCONJ
ejpam-4498	66	7	3	3	NUM
ejpam-4498	66	8	·	·	SYM
ejpam-4498	66	9	2	2	NUM
ejpam-4498	66	10	+	+	CCONJ
ejpam-4498	66	11	4	4	NUM
ejpam-4498	66	12	·	·	SYM
ejpam-4498	66	13	3	3	NUM
ejpam-4498	67	1	+	+	SYM
ejpam-4498	67	2	5	5	NUM
ejpam-4498	67	3	·	·	SYM
ejpam-4498	67	4	4	4	NUM
ejpam-4498	67	5	)	)	PUNCT
ejpam-4498	67	6	.	.	PUNCT
ejpam-4498	68	1	(	(	PUNCT
ejpam-4498	68	2	7	7	X
ejpam-4498	68	3	)	)	PUNCT
ejpam-4498	68	4	yukio	yukio	PROPN
ejpam-4498	68	5	kobayashi	kobayashi	PROPN
ejpam-4498	68	6	/	/	SYM
ejpam-4498	68	7	eur	eur	PROPN
ejpam-4498	68	8	.	.	PUNCT
ejpam-4498	69	1	j.	j.	PROPN
ejpam-4498	69	2	pure	pure	PROPN
ejpam-4498	69	3	appl	appl	PROPN
ejpam-4498	69	4	.	.	PROPN
ejpam-4498	69	5	math	math	PROPN
ejpam-4498	69	6	,	,	PUNCT
ejpam-4498	69	7	15	15	NUM
ejpam-4498	69	8	(	(	PUNCT
ejpam-4498	69	9	3	3	NUM
ejpam-4498	69	10	)	)	PUNCT
ejpam-4498	69	11	(	(	PUNCT
ejpam-4498	69	12	2022	2022	NUM
ejpam-4498	69	13	)	)	PUNCT
ejpam-4498	69	14	,	,	PUNCT
ejpam-4498	69	15	1237	1237	NUM
ejpam-4498	69	16	-	-	SYM
ejpam-4498	69	17	1253	1253	NUM
ejpam-4498	70	1	1240	1240	NUM
ejpam-4498	70	2	1	1	NUM
ejpam-4498	70	3	2	2	NUM
ejpam-4498	70	4	1	1	NUM
ejpam-4498	70	5	2	2	NUM
ejpam-4498	70	6	3	3	NUM
ejpam-4498	70	7	2	2	NUM
ejpam-4498	70	8	3	3	NUM
ejpam-4498	70	9	1	1	NUM
ejpam-4498	70	10	2	2	NUM
ejpam-4498	70	11	3	3	NUM
ejpam-4498	70	12	4	4	NUM
ejpam-4498	70	13	2	2	NUM
ejpam-4498	70	14	3	3	NUM
ejpam-4498	70	15	4	4	NUM
ejpam-4498	70	16	3	3	NUM
ejpam-4498	70	17	4	4	NUM
ejpam-4498	70	18	·	·	PUNCT
ejpam-4498	70	19	·	·	PUNCT
ejpam-4498	70	20	·	·	PUNCT
ejpam-4498	70	21	combination	combination	NOUN
ejpam-4498	70	22	no	no	INTJ
ejpam-4498	70	23	.	.	PUNCT
ejpam-4498	71	1	of	of	ADP
ejpam-4498	71	2	edges	edge	NOUN
ejpam-4498	71	3	triangular	triangular	NOUN
ejpam-4498	71	4	number	number	NOUN
ejpam-4498	71	5	2c2	2c2	NUM
ejpam-4498	71	6	3c2	3c2	NUM
ejpam-4498	71	7	4c2	4c2	NOUN
ejpam-4498	71	8	1	1	NUM
ejpam-4498	71	9	1	1	NUM
ejpam-4498	71	10	+	+	CCONJ
ejpam-4498	71	11	2	2	NUM
ejpam-4498	71	12	1	1	NUM
ejpam-4498	71	13	+	+	NUM
ejpam-4498	71	14	2	2	NUM
ejpam-4498	71	15	+	+	SYM
ejpam-4498	71	16	3	3	NUM
ejpam-4498	71	17	figure	figure	NOUN
ejpam-4498	71	18	1	1	NUM
ejpam-4498	71	19	:	:	PUNCT
ejpam-4498	71	20	pictorial	pictorial	ADJ
ejpam-4498	71	21	proof	proof	NOUN
ejpam-4498	71	22	of	of	ADP
ejpam-4498	71	23	the	the	DET
ejpam-4498	71	24	equivalence	equivalence	NOUN
ejpam-4498	71	25	of	of	ADP
ejpam-4498	71	26	(	(	PUNCT
ejpam-4498	71	27	1	1	NUM
ejpam-4498	71	28	)	)	PUNCT
ejpam-4498	71	29	,	,	PUNCT
ejpam-4498	71	30	(	(	PUNCT
ejpam-4498	71	31	2	2	NUM
ejpam-4498	71	32	)	)	PUNCT
ejpam-4498	71	33	,	,	PUNCT
ejpam-4498	71	34	(	(	PUNCT
ejpam-4498	71	35	3	3	NUM
ejpam-4498	71	36	)	)	PUNCT
ejpam-4498	71	37	,	,	PUNCT
ejpam-4498	71	38	and	and	CCONJ
ejpam-4498	71	39	(	(	PUNCT
ejpam-4498	71	40	4	4	NUM
ejpam-4498	71	41	)	)	PUNCT
ejpam-4498	71	42	.	.	PUNCT
ejpam-4498	72	1	the	the	DET
ejpam-4498	72	2	numerals	numeral	NOUN
ejpam-4498	72	3	in	in	ADP
ejpam-4498	72	4	the	the	DET
ejpam-4498	72	5	tree	tree	NOUN
ejpam-4498	72	6	diagrams	diagram	NOUN
ejpam-4498	72	7	indicate	indicate	VERB
ejpam-4498	72	8	the	the	DET
ejpam-4498	72	9	number	number	NOUN
ejpam-4498	72	10	of	of	ADP
ejpam-4498	72	11	cards	card	NOUN
ejpam-4498	72	12	.	.	PUNCT
ejpam-4498	73	1	1	1	NUM
ejpam-4498	74	1	+	+	NUM
ejpam-4498	74	2	2	2	NUM
ejpam-4498	74	3	+	+	NUM
ejpam-4498	74	4	3	3	NUM
ejpam-4498	74	5	+	+	SYM
ejpam-4498	74	6	4	4	NUM
ejpam-4498	74	7	+	+	SYM
ejpam-4498	74	8	5	5	NUM
ejpam-4498	74	9	1	1	NUM
ejpam-4498	74	10	+	+	NUM
ejpam-4498	74	11	2	2	NUM
ejpam-4498	74	12	+	+	NUM
ejpam-4498	74	13	3	3	NUM
ejpam-4498	74	14	+	+	SYM
ejpam-4498	74	15	4	4	NUM
ejpam-4498	74	16	+	+	SYM
ejpam-4498	74	17	5	5	NUM
ejpam-4498	74	18	1	1	NUM
ejpam-4498	74	19	+	+	NUM
ejpam-4498	74	20	2	2	NUM
ejpam-4498	74	21	+	+	NUM
ejpam-4498	74	22	3	3	NUM
ejpam-4498	74	23	+	+	SYM
ejpam-4498	74	24	4	4	NUM
ejpam-4498	74	25	+	+	SYM
ejpam-4498	74	26	5	5	NUM
ejpam-4498	74	27	1	1	NUM
ejpam-4498	74	28	+	+	NUM
ejpam-4498	74	29	2	2	NUM
ejpam-4498	74	30	+	+	NUM
ejpam-4498	74	31	3	3	NUM
ejpam-4498	74	32	+	+	SYM
ejpam-4498	74	33	4	4	NUM
ejpam-4498	74	34	+	+	SYM
ejpam-4498	74	35	5	5	NUM
ejpam-4498	74	36	1	1	NUM
ejpam-4498	74	37	+	+	NUM
ejpam-4498	74	38	2	2	NUM
ejpam-4498	74	39	+	+	NUM
ejpam-4498	74	40	3	3	NUM
ejpam-4498	74	41	+	+	SYM
ejpam-4498	74	42	4	4	NUM
ejpam-4498	74	43	+	+	SYM
ejpam-4498	74	44	5	5	NUM
ejpam-4498	74	45	figure	figure	NOUN
ejpam-4498	74	46	2	2	NUM
ejpam-4498	74	47	:	:	PUNCT
ejpam-4498	74	48	sequence	sequence	NOUN
ejpam-4498	74	49	of	of	ADP
ejpam-4498	74	50	five	five	NUM
ejpam-4498	74	51	equations	equation	NOUN
ejpam-4498	74	52	expressing	express	VERB
ejpam-4498	74	53	the	the	DET
ejpam-4498	74	54	sum	sum	NOUN
ejpam-4498	74	55	of	of	ADP
ejpam-4498	74	56	natural	natural	ADJ
ejpam-4498	74	57	numbers	number	NOUN
ejpam-4498	74	58	from	from	ADP
ejpam-4498	74	59	1	1	NUM
ejpam-4498	74	60	to	to	ADP
ejpam-4498	74	61	5	5	NUM
ejpam-4498	74	62	.	.	PUNCT
ejpam-4498	75	1	the	the	DET
ejpam-4498	75	2	first	first	ADJ
ejpam-4498	75	3	summation	summation	NOUN
ejpam-4498	75	4	1	1	NUM
ejpam-4498	75	5	·	·	SYM
ejpam-4498	75	6	5	5	NUM
ejpam-4498	75	7	+	+	SYM
ejpam-4498	75	8	2	2	NUM
ejpam-4498	75	9	·	·	SYM
ejpam-4498	75	10	4	4	NUM
ejpam-4498	75	11	+	+	SYM
ejpam-4498	75	12	3	3	NUM
ejpam-4498	75	13	·	·	SYM
ejpam-4498	75	14	3	3	NUM
ejpam-4498	75	15	+	+	SYM
ejpam-4498	75	16	4	4	NUM
ejpam-4498	75	17	·	·	SYM
ejpam-4498	75	18	2	2	NUM
ejpam-4498	75	19	+	+	NUM
ejpam-4498	75	20	5	5	NUM
ejpam-4498	75	21	·	·	SYM
ejpam-4498	75	22	1	1	NUM
ejpam-4498	75	23	and	and	CCONJ
ejpam-4498	75	24	the	the	DET
ejpam-4498	75	25	second	second	ADJ
ejpam-4498	75	26	2	2	NUM
ejpam-4498	75	27	·	·	SYM
ejpam-4498	75	28	1	1	NUM
ejpam-4498	75	29	+	+	SYM
ejpam-4498	75	30	3	3	NUM
ejpam-4498	75	31	·	·	SYM
ejpam-4498	75	32	2	2	NUM
ejpam-4498	75	33	+	+	NUM
ejpam-4498	75	34	4	4	NUM
ejpam-4498	75	35	·	·	SYM
ejpam-4498	75	36	3	3	NUM
ejpam-4498	75	37	+	+	SYM
ejpam-4498	75	38	5	5	NUM
ejpam-4498	75	39	·	·	SYM
ejpam-4498	75	40	4	4	NUM
ejpam-4498	75	41	are	be	AUX
ejpam-4498	75	42	the	the	DET
ejpam-4498	75	43	numbers	number	NOUN
ejpam-4498	75	44	in	in	ADP
ejpam-4498	75	45	the	the	DET
ejpam-4498	75	46	dark	dark	ADJ
ejpam-4498	75	47	grey	grey	NOUN
ejpam-4498	75	48	triangle	triangle	NOUN
ejpam-4498	75	49	and	and	CCONJ
ejpam-4498	75	50	in	in	ADP
ejpam-4498	75	51	the	the	DET
ejpam-4498	75	52	light	light	ADJ
ejpam-4498	75	53	grey	grey	ADJ
ejpam-4498	75	54	triangle	triangle	NOUN
ejpam-4498	75	55	in	in	ADP
ejpam-4498	75	56	figure	figure	NOUN
ejpam-4498	75	57	3	3	NUM
ejpam-4498	75	58	,	,	PUNCT
ejpam-4498	75	59	respectively	respectively	ADV
ejpam-4498	75	60	.	.	PUNCT
ejpam-4498	76	1	from	from	ADP
ejpam-4498	76	2	figure	figure	NOUN
ejpam-4498	76	3	4	4	NUM
ejpam-4498	76	4	,	,	PUNCT
ejpam-4498	76	5	we	we	PRON
ejpam-4498	76	6	can	can	AUX
ejpam-4498	76	7	express	express	VERB
ejpam-4498	76	8	the	the	DET
ejpam-4498	76	9	feature	feature	NOUN
ejpam-4498	76	10	as	as	SCONJ
ejpam-4498	76	11	follows	follow	VERB
ejpam-4498	76	12	.	.	PUNCT
ejpam-4498	77	1	(	(	PUNCT
ejpam-4498	77	2	1	1	NUM
ejpam-4498	77	3	+	+	SYM
ejpam-4498	77	4	2	2	NUM
ejpam-4498	77	5	+	+	NUM
ejpam-4498	77	6	3	3	NUM
ejpam-4498	77	7	+	+	SYM
ejpam-4498	77	8	4	4	NUM
ejpam-4498	77	9	+	+	SYM
ejpam-4498	77	10	5	5	NUM
ejpam-4498	77	11	)	)	PUNCT
ejpam-4498	77	12	·	·	PUNCT
ejpam-4498	78	1	5−	5−	NUM
ejpam-4498	78	2	(	(	PUNCT
ejpam-4498	78	3	1	1	NUM
ejpam-4498	78	4	+	+	NUM
ejpam-4498	78	5	2	2	NUM
ejpam-4498	78	6	+	+	NUM
ejpam-4498	78	7	3	3	NUM
ejpam-4498	78	8	+	+	SYM
ejpam-4498	78	9	4	4	NUM
ejpam-4498	78	10	+	+	SYM
ejpam-4498	78	11	5	5	NUM
ejpam-4498	78	12	)	)	PUNCT
ejpam-4498	78	13	=	=	SYM
ejpam-4498	78	14	(	(	PUNCT
ejpam-4498	78	15	1	1	NUM
ejpam-4498	78	16	·	·	SYM
ejpam-4498	78	17	4	4	NUM
ejpam-4498	78	18	+	+	SYM
ejpam-4498	78	19	2	2	NUM
ejpam-4498	78	20	·	·	SYM
ejpam-4498	78	21	3	3	NUM
ejpam-4498	78	22	+	+	CCONJ
ejpam-4498	78	23	3	3	NUM
ejpam-4498	78	24	·	·	SYM
ejpam-4498	78	25	2	2	NUM
ejpam-4498	78	26	+	+	CCONJ
ejpam-4498	78	27	4	4	NUM
ejpam-4498	78	28	·	·	SYM
ejpam-4498	78	29	1	1	NUM
ejpam-4498	78	30	)	)	PUNCT
ejpam-4498	78	31	+	+	NOUN
ejpam-4498	78	32	(	(	PUNCT
ejpam-4498	78	33	2	2	NUM
ejpam-4498	78	34	·	·	SYM
ejpam-4498	78	35	1	1	NUM
ejpam-4498	78	36	+	+	CCONJ
ejpam-4498	78	37	3	3	NUM
ejpam-4498	78	38	·	·	SYM
ejpam-4498	78	39	2	2	NUM
ejpam-4498	78	40	+	+	CCONJ
ejpam-4498	78	41	4	4	NUM
ejpam-4498	78	42	·	·	SYM
ejpam-4498	78	43	3	3	NUM
ejpam-4498	78	44	+	+	SYM
ejpam-4498	78	45	5	5	NUM
ejpam-4498	78	46	·	·	SYM
ejpam-4498	78	47	4	4	NUM
ejpam-4498	78	48	)	)	PUNCT
ejpam-4498	78	49	.	.	PUNCT
ejpam-4498	79	1	(	(	PUNCT
ejpam-4498	79	2	8)	8)	NUM
ejpam-4498	79	3	by	by	ADP
ejpam-4498	79	4	generalising	generalise	VERB
ejpam-4498	79	5	(	(	PUNCT
ejpam-4498	79	6	7	7	NUM
ejpam-4498	79	7	)	)	PUNCT
ejpam-4498	79	8	and	and	CCONJ
ejpam-4498	79	9	(	(	PUNCT
ejpam-4498	79	10	8)	8)	NUM
ejpam-4498	79	11	,	,	PUNCT
ejpam-4498	79	12	we	we	PRON
ejpam-4498	79	13	obtain	obtain	VERB
ejpam-4498	79	14	(	(	PUNCT
ejpam-4498	79	15	1	1	NUM
ejpam-4498	79	16	+	+	NUM
ejpam-4498	79	17	2	2	NUM
ejpam-4498	79	18	+	+	CCONJ
ejpam-4498	79	19	·	·	PUNCT
ejpam-4498	79	20	·	·	PUNCT
ejpam-4498	79	21	·	·	PUNCT
ejpam-4498	79	22	+	+	NUM
ejpam-4498	79	23	n	n	CCONJ
ejpam-4498	79	24	)	)	PUNCT
ejpam-4498	79	25	·	·	PUNCT
ejpam-4498	80	1	n	n	X
ejpam-4498	80	2	=	=	SYM
ejpam-4498	81	1	[	[	PUNCT
ejpam-4498	81	2	1	1	NUM
ejpam-4498	81	3	·	·	PUNCT
ejpam-4498	81	4	n+	n+	NUM
ejpam-4498	81	5	2	2	NUM
ejpam-4498	81	6	·	·	PUNCT
ejpam-4498	81	7	(	(	PUNCT
ejpam-4498	81	8	n−	n−	NOUN
ejpam-4498	81	9	1	1	NUM
ejpam-4498	81	10	)	)	PUNCT
ejpam-4498	81	11	+	+	CCONJ
ejpam-4498	81	12	·	·	PUNCT
ejpam-4498	81	13	·	·	PUNCT
ejpam-4498	81	14	·	·	PUNCT
ejpam-4498	81	15	+	+	NUM
ejpam-4498	81	16	n	n	CCONJ
ejpam-4498	81	17	·	·	PUNCT
ejpam-4498	81	18	1	1	NUM
ejpam-4498	81	19	]	]	PUNCT
ejpam-4498	81	20	+	+	PUNCT
ejpam-4498	81	21	[	[	X
ejpam-4498	81	22	2	2	NUM
ejpam-4498	81	23	·	·	SYM
ejpam-4498	81	24	1	1	NUM
ejpam-4498	82	1	+	+	CCONJ
ejpam-4498	82	2	3	3	NUM
ejpam-4498	82	3	·	·	SYM
ejpam-4498	82	4	2	2	NUM
ejpam-4498	82	5	+	+	CCONJ
ejpam-4498	82	6	·	·	PUNCT
ejpam-4498	82	7	·	·	PUNCT
ejpam-4498	82	8	·	·	PUNCT
ejpam-4498	83	1	+	+	NUM
ejpam-4498	83	2	n	n	CCONJ
ejpam-4498	83	3	·	·	PUNCT
ejpam-4498	83	4	(	(	PUNCT
ejpam-4498	83	5	n−	n−	NOUN
ejpam-4498	83	6	1	1	NUM
ejpam-4498	83	7	)	)	PUNCT
ejpam-4498	83	8	]	]	PUNCT
ejpam-4498	83	9	,	,	PUNCT
ejpam-4498	83	10	(	(	PUNCT
ejpam-4498	83	11	9	9	NUM
ejpam-4498	83	12	)	)	PUNCT
ejpam-4498	83	13	and	and	CCONJ
ejpam-4498	83	14	(	(	PUNCT
ejpam-4498	83	15	1	1	NUM
ejpam-4498	83	16	+	+	NUM
ejpam-4498	83	17	2	2	NUM
ejpam-4498	83	18	+	+	CCONJ
ejpam-4498	83	19	·	·	PUNCT
ejpam-4498	83	20	·	·	PUNCT
ejpam-4498	83	21	·	·	PUNCT
ejpam-4498	83	22	+	+	NUM
ejpam-4498	83	23	n	n	CCONJ
ejpam-4498	83	24	)	)	PUNCT
ejpam-4498	83	25	·	·	PUNCT
ejpam-4498	83	26	(	(	PUNCT
ejpam-4498	83	27	n−	n−	NOUN
ejpam-4498	83	28	1	1	NUM
ejpam-4498	83	29	)	)	PUNCT
ejpam-4498	83	30	yukio	yukio	PROPN
ejpam-4498	83	31	kobayashi	kobayashi	PROPN
ejpam-4498	83	32	/	/	SYM
ejpam-4498	83	33	eur	eur	PROPN
ejpam-4498	83	34	.	.	PUNCT
ejpam-4498	84	1	j.	j.	PROPN
ejpam-4498	84	2	pure	pure	PROPN
ejpam-4498	84	3	appl	appl	PROPN
ejpam-4498	84	4	.	.	PROPN
ejpam-4498	84	5	math	math	PROPN
ejpam-4498	84	6	,	,	PUNCT
ejpam-4498	84	7	15	15	NUM
ejpam-4498	84	8	(	(	PUNCT
ejpam-4498	84	9	3	3	NUM
ejpam-4498	84	10	)	)	PUNCT
ejpam-4498	84	11	(	(	PUNCT
ejpam-4498	84	12	2022	2022	NUM
ejpam-4498	84	13	)	)	PUNCT
ejpam-4498	84	14	,	,	PUNCT
ejpam-4498	84	15	1237	1237	NUM
ejpam-4498	84	16	-	-	SYM
ejpam-4498	84	17	1253	1253	NUM
ejpam-4498	84	18	1241	1241	NUM
ejpam-4498	84	19	1	1	NUM
ejpam-4498	85	1	+	+	NUM
ejpam-4498	85	2	2	2	NUM
ejpam-4498	85	3	+	+	NUM
ejpam-4498	85	4	3	3	NUM
ejpam-4498	85	5	+	+	SYM
ejpam-4498	85	6	4	4	NUM
ejpam-4498	85	7	+	+	SYM
ejpam-4498	85	8	5	5	NUM
ejpam-4498	85	9	1	1	NUM
ejpam-4498	85	10	+	+	NUM
ejpam-4498	85	11	2	2	NUM
ejpam-4498	85	12	+	+	NUM
ejpam-4498	85	13	3	3	NUM
ejpam-4498	85	14	+	+	SYM
ejpam-4498	85	15	4	4	NUM
ejpam-4498	85	16	+	+	SYM
ejpam-4498	85	17	5	5	NUM
ejpam-4498	85	18	1	1	NUM
ejpam-4498	85	19	+	+	NUM
ejpam-4498	85	20	2	2	NUM
ejpam-4498	85	21	+	+	NUM
ejpam-4498	85	22	3	3	NUM
ejpam-4498	85	23	+	+	SYM
ejpam-4498	85	24	4	4	NUM
ejpam-4498	85	25	+	+	SYM
ejpam-4498	85	26	5	5	NUM
ejpam-4498	85	27	1	1	NUM
ejpam-4498	85	28	+	+	NUM
ejpam-4498	85	29	2	2	NUM
ejpam-4498	85	30	+	+	NUM
ejpam-4498	85	31	3	3	NUM
ejpam-4498	85	32	+	+	SYM
ejpam-4498	85	33	4	4	NUM
ejpam-4498	85	34	+	+	SYM
ejpam-4498	85	35	5	5	NUM
ejpam-4498	85	36	1	1	NUM
ejpam-4498	85	37	+	+	NUM
ejpam-4498	85	38	2	2	NUM
ejpam-4498	85	39	+	+	NUM
ejpam-4498	85	40	3	3	NUM
ejpam-4498	85	41	+	+	SYM
ejpam-4498	85	42	4	4	NUM
ejpam-4498	85	43	+	+	SYM
ejpam-4498	85	44	5	5	NUM
ejpam-4498	85	45	figure	figure	NOUN
ejpam-4498	85	46	3	3	NUM
ejpam-4498	85	47	:	:	PUNCT
ejpam-4498	85	48	relationship	relationship	NOUN
ejpam-4498	85	49	between	between	ADP
ejpam-4498	85	50	the	the	DET
ejpam-4498	85	51	sums	sum	NOUN
ejpam-4498	85	52	of	of	ADP
ejpam-4498	85	53	natural	natural	ADJ
ejpam-4498	85	54	numbers	number	NOUN
ejpam-4498	85	55	,	,	PUNCT
ejpam-4498	85	56	triangular	triangular	NOUN
ejpam-4498	85	57	numbers	number	NOUN
ejpam-4498	85	58	,	,	PUNCT
ejpam-4498	85	59	and	and	CCONJ
ejpam-4498	85	60	the	the	DET
ejpam-4498	85	61	product	product	NOUN
ejpam-4498	85	62	of	of	ADP
ejpam-4498	85	63	consecutive	consecutive	ADJ
ejpam-4498	85	64	numbers	number	NOUN
ejpam-4498	85	65	.	.	PUNCT
ejpam-4498	86	1	1	1	NUM
ejpam-4498	87	1	+	+	NUM
ejpam-4498	87	2	2	2	NUM
ejpam-4498	87	3	+	+	NUM
ejpam-4498	87	4	3	3	NUM
ejpam-4498	87	5	+	+	SYM
ejpam-4498	87	6	4	4	NUM
ejpam-4498	87	7	+	+	SYM
ejpam-4498	87	8	5	5	NUM
ejpam-4498	87	9	1	1	NUM
ejpam-4498	87	10	+	+	NUM
ejpam-4498	87	11	2	2	NUM
ejpam-4498	87	12	+	+	NUM
ejpam-4498	87	13	3	3	NUM
ejpam-4498	87	14	+	+	SYM
ejpam-4498	87	15	4	4	NUM
ejpam-4498	87	16	+	+	SYM
ejpam-4498	87	17	5	5	NUM
ejpam-4498	87	18	1	1	NUM
ejpam-4498	87	19	+	+	NUM
ejpam-4498	87	20	2	2	NUM
ejpam-4498	87	21	+	+	NUM
ejpam-4498	87	22	3	3	NUM
ejpam-4498	87	23	+	+	SYM
ejpam-4498	87	24	4	4	NUM
ejpam-4498	87	25	+	+	SYM
ejpam-4498	87	26	5	5	NUM
ejpam-4498	87	27	1	1	NUM
ejpam-4498	87	28	+	+	NUM
ejpam-4498	87	29	2	2	NUM
ejpam-4498	87	30	+	+	NUM
ejpam-4498	87	31	3	3	NUM
ejpam-4498	87	32	+	+	SYM
ejpam-4498	87	33	4	4	NUM
ejpam-4498	87	34	+	+	SYM
ejpam-4498	87	35	5	5	NUM
ejpam-4498	87	36	1	1	NUM
ejpam-4498	87	37	+	+	NUM
ejpam-4498	87	38	2	2	NUM
ejpam-4498	87	39	+	+	NUM
ejpam-4498	87	40	3	3	NUM
ejpam-4498	87	41	+	+	SYM
ejpam-4498	87	42	4	4	NUM
ejpam-4498	87	43	+	+	SYM
ejpam-4498	87	44	5	5	NUM
ejpam-4498	87	45	figure	figure	NOUN
ejpam-4498	87	46	4	4	NUM
ejpam-4498	87	47	:	:	PUNCT
ejpam-4498	87	48	relationship	relationship	NOUN
ejpam-4498	87	49	between	between	ADP
ejpam-4498	87	50	the	the	DET
ejpam-4498	87	51	sums	sum	NOUN
ejpam-4498	87	52	of	of	ADP
ejpam-4498	87	53	natural	natural	ADJ
ejpam-4498	87	54	numbers	number	NOUN
ejpam-4498	87	55	,	,	PUNCT
ejpam-4498	87	56	triangular	triangular	NOUN
ejpam-4498	87	57	numbers	number	NOUN
ejpam-4498	87	58	,	,	PUNCT
ejpam-4498	87	59	and	and	CCONJ
ejpam-4498	87	60	the	the	DET
ejpam-4498	87	61	product	product	NOUN
ejpam-4498	87	62	of	of	ADP
ejpam-4498	87	63	consecutive	consecutive	ADJ
ejpam-4498	87	64	numbers	number	NOUN
ejpam-4498	87	65	.	.	PUNCT
ejpam-4498	88	1	=	=	PUNCT
ejpam-4498	89	1	[	[	X
ejpam-4498	89	2	1	1	NUM
ejpam-4498	89	3	·	·	PUNCT
ejpam-4498	89	4	(	(	PUNCT
ejpam-4498	89	5	n−	n−	NOUN
ejpam-4498	89	6	1	1	NUM
ejpam-4498	89	7	)	)	PUNCT
ejpam-4498	89	8	+	+	CCONJ
ejpam-4498	89	9	2	2	NUM
ejpam-4498	89	10	·	·	PUNCT
ejpam-4498	89	11	(	(	PUNCT
ejpam-4498	89	12	n−	n−	NOUN
ejpam-4498	89	13	2	2	NUM
ejpam-4498	89	14	)	)	PUNCT
ejpam-4498	89	15	+	+	CCONJ
ejpam-4498	89	16	·	·	PUNCT
ejpam-4498	89	17	·	·	PUNCT
ejpam-4498	89	18	·	·	PUNCT
ejpam-4498	89	19	+	+	CCONJ
ejpam-4498	89	20	(	(	PUNCT
ejpam-4498	89	21	n−	n−	NOUN
ejpam-4498	89	22	1	1	NUM
ejpam-4498	89	23	)	)	PUNCT
ejpam-4498	89	24	·	·	PUNCT
ejpam-4498	89	25	1	1	X
ejpam-4498	89	26	]	]	PUNCT
ejpam-4498	89	27	+	+	PUNCT
ejpam-4498	89	28	[	[	X
ejpam-4498	89	29	2	2	NUM
ejpam-4498	89	30	·	·	SYM
ejpam-4498	89	31	1	1	NUM
ejpam-4498	90	1	+	+	CCONJ
ejpam-4498	90	2	3	3	NUM
ejpam-4498	90	3	·	·	SYM
ejpam-4498	90	4	2	2	NUM
ejpam-4498	90	5	+	+	CCONJ
ejpam-4498	90	6	·	·	PUNCT
ejpam-4498	90	7	·	·	PUNCT
ejpam-4498	90	8	·	·	PUNCT
ejpam-4498	91	1	+	+	NUM
ejpam-4498	91	2	n	n	CCONJ
ejpam-4498	91	3	·	·	PUNCT
ejpam-4498	91	4	(	(	PUNCT
ejpam-4498	91	5	n−	n−	NOUN
ejpam-4498	91	6	1	1	NUM
ejpam-4498	91	7	)	)	PUNCT
ejpam-4498	91	8	]	]	PUNCT
ejpam-4498	91	9	.	.	PUNCT
ejpam-4498	92	1	(	(	PUNCT
ejpam-4498	92	2	10	10	NUM
ejpam-4498	92	3	)	)	PUNCT
ejpam-4498	92	4	considering	consider	VERB
ejpam-4498	92	5	(	(	PUNCT
ejpam-4498	92	6	4	4	NUM
ejpam-4498	92	7	)	)	PUNCT
ejpam-4498	92	8	,	,	PUNCT
ejpam-4498	92	9	equations	equation	NOUN
ejpam-4498	92	10	(	(	PUNCT
ejpam-4498	92	11	9	9	NUM
ejpam-4498	92	12	)	)	PUNCT
ejpam-4498	92	13	and	and	CCONJ
ejpam-4498	92	14	(	(	PUNCT
ejpam-4498	92	15	10	10	NUM
ejpam-4498	92	16	)	)	PUNCT
ejpam-4498	92	17	indicate	indicate	VERB
ejpam-4498	92	18	the	the	DET
ejpam-4498	92	19	relationship	relationship	NOUN
ejpam-4498	92	20	between	between	ADP
ejpam-4498	92	21	the	the	DET
ejpam-4498	92	22	sums	sum	NOUN
ejpam-4498	92	23	of	of	ADP
ejpam-4498	92	24	natural	natural	ADJ
ejpam-4498	92	25	numbers	number	NOUN
ejpam-4498	92	26	,	,	PUNCT
ejpam-4498	92	27	triangular	triangular	NOUN
ejpam-4498	92	28	numbers	number	NOUN
ejpam-4498	92	29	,	,	PUNCT
ejpam-4498	92	30	and	and	CCONJ
ejpam-4498	92	31	the	the	DET
ejpam-4498	92	32	product	product	NOUN
ejpam-4498	92	33	of	of	ADP
ejpam-4498	92	34	consecutive	consecutive	ADJ
ejpam-4498	92	35	numbers	number	NOUN
ejpam-4498	92	36	.	.	PUNCT
ejpam-4498	93	1	equations	equation	NOUN
ejpam-4498	93	2	(	(	PUNCT
ejpam-4498	93	3	1	1	NUM
ejpam-4498	93	4	)	)	PUNCT
ejpam-4498	93	5	and	and	CCONJ
ejpam-4498	93	6	(	(	PUNCT
ejpam-4498	93	7	9	9	X
ejpam-4498	93	8	)	)	PUNCT
ejpam-4498	93	9	can	can	AUX
ejpam-4498	93	10	be	be	AUX
ejpam-4498	93	11	used	use	VERB
ejpam-4498	93	12	to	to	PART
ejpam-4498	93	13	obtain	obtain	VERB
ejpam-4498	93	14	the	the	DET
ejpam-4498	93	15	sum	sum	NOUN
ejpam-4498	93	16	of	of	ADP
ejpam-4498	93	17	both	both	DET
ejpam-4498	93	18	triangular	triangular	NOUN
ejpam-4498	93	19	numbers	number	NOUN
ejpam-4498	93	20	and	and	CCONJ
ejpam-4498	93	21	the	the	DET
ejpam-4498	93	22	product	product	NOUN
ejpam-4498	93	23	of	of	ADP
ejpam-4498	93	24	consecutive	consecutive	ADJ
ejpam-4498	93	25	numbers	number	NOUN
ejpam-4498	93	26	.	.	PUNCT
ejpam-4498	94	1	from	from	ADP
ejpam-4498	94	2	(	(	PUNCT
ejpam-4498	94	3	1	1	NUM
ejpam-4498	94	4	)	)	PUNCT
ejpam-4498	94	5	,	,	PUNCT
ejpam-4498	94	6	1	1	NUM
ejpam-4498	94	7	·	·	PUNCT
ejpam-4498	94	8	n+	n+	NUM
ejpam-4498	94	9	2	2	NUM
ejpam-4498	94	10	·	·	PUNCT
ejpam-4498	94	11	(	(	PUNCT
ejpam-4498	94	12	n−	n−	NOUN
ejpam-4498	94	13	1	1	NUM
ejpam-4498	94	14	)	)	PUNCT
ejpam-4498	94	15	+	+	CCONJ
ejpam-4498	94	16	·	·	PUNCT
ejpam-4498	94	17	·	·	PUNCT
ejpam-4498	94	18	·	·	PUNCT
ejpam-4498	94	19	+	+	NUM
ejpam-4498	94	20	n	n	CCONJ
ejpam-4498	94	21	·	·	SYM
ejpam-4498	94	22	1	1	NUM
ejpam-4498	94	23	=	=	SYM
ejpam-4498	94	24	1	1	NUM
ejpam-4498	94	25	2	2	NUM
ejpam-4498	94	26	[	[	SYM
ejpam-4498	94	27	1	1	NUM
ejpam-4498	94	28	·	·	SYM
ejpam-4498	94	29	2	2	NUM
ejpam-4498	94	30	+	+	CCONJ
ejpam-4498	94	31	2	2	NUM
ejpam-4498	94	32	·	·	SYM
ejpam-4498	94	33	3	3	NUM
ejpam-4498	94	34	+	+	CCONJ
ejpam-4498	94	35	3	3	NUM
ejpam-4498	94	36	·	·	SYM
ejpam-4498	94	37	4	4	NUM
ejpam-4498	94	38	+	+	CCONJ
ejpam-4498	94	39	·	·	PUNCT
ejpam-4498	94	40	·	·	PUNCT
ejpam-4498	94	41	·	·	PUNCT
ejpam-4498	94	42	+	+	CCONJ
ejpam-4498	94	43	(	(	PUNCT
ejpam-4498	94	44	n−	n−	NOUN
ejpam-4498	94	45	1	1	NUM
ejpam-4498	94	46	)	)	PUNCT
ejpam-4498	94	47	·	·	PUNCT
ejpam-4498	94	48	n+	n+	X
ejpam-4498	94	49	n	n	X
ejpam-4498	94	50	·	·	PUNCT
ejpam-4498	94	51	(	(	PUNCT
ejpam-4498	94	52	n+	n+	NOUN
ejpam-4498	94	53	1	1	NUM
ejpam-4498	94	54	)	)	PUNCT
ejpam-4498	94	55	]	]	PUNCT
ejpam-4498	94	56	.	.	PUNCT
ejpam-4498	95	1	(	(	PUNCT
ejpam-4498	95	2	11	11	NUM
ejpam-4498	95	3	)	)	PUNCT
ejpam-4498	95	4	thus	thus	ADV
ejpam-4498	95	5	,	,	PUNCT
ejpam-4498	95	6	(	(	PUNCT
ejpam-4498	95	7	9	9	X
ejpam-4498	95	8	)	)	PUNCT
ejpam-4498	95	9	can	can	AUX
ejpam-4498	95	10	be	be	AUX
ejpam-4498	95	11	written	write	VERB
ejpam-4498	95	12	as	as	ADP
ejpam-4498	95	13	(	(	PUNCT
ejpam-4498	95	14	1	1	NUM
ejpam-4498	95	15	+	+	NUM
ejpam-4498	95	16	2	2	NUM
ejpam-4498	95	17	+	+	CCONJ
ejpam-4498	95	18	·	·	PUNCT
ejpam-4498	95	19	·	·	PUNCT
ejpam-4498	95	20	·	·	PUNCT
ejpam-4498	95	21	+	+	NUM
ejpam-4498	95	22	n	n	CCONJ
ejpam-4498	95	23	)	)	PUNCT
ejpam-4498	95	24	·	·	PUNCT
ejpam-4498	96	1	n	n	X
ejpam-4498	96	2	=	=	SYM
ejpam-4498	96	3	3	3	NUM
ejpam-4498	96	4	2	2	NUM
ejpam-4498	96	5	[	[	PUNCT
ejpam-4498	96	6	1	1	NUM
ejpam-4498	96	7	·	·	SYM
ejpam-4498	96	8	2	2	NUM
ejpam-4498	97	1	+	+	CCONJ
ejpam-4498	97	2	2	2	NUM
ejpam-4498	97	3	·	·	SYM
ejpam-4498	97	4	3	3	NUM
ejpam-4498	97	5	+	+	CCONJ
ejpam-4498	97	6	3	3	NUM
ejpam-4498	97	7	·	·	SYM
ejpam-4498	97	8	4	4	NUM
ejpam-4498	97	9	+	+	CCONJ
ejpam-4498	97	10	·	·	PUNCT
ejpam-4498	97	11	·	·	PUNCT
ejpam-4498	97	12	·	·	PUNCT
ejpam-4498	98	1	+	+	CCONJ
ejpam-4498	98	2	(	(	PUNCT
ejpam-4498	98	3	n−	n−	NOUN
ejpam-4498	98	4	1	1	NUM
ejpam-4498	98	5	)	)	PUNCT
ejpam-4498	98	6	·	·	PUNCT
ejpam-4498	98	7	n	n	CCONJ
ejpam-4498	98	8	]	]	PUNCT
ejpam-4498	98	9	+	+	CCONJ
ejpam-4498	98	10	1	1	NUM
ejpam-4498	98	11	2	2	NUM
ejpam-4498	98	12	n	n	NOUN
ejpam-4498	98	13	·	·	PUNCT
ejpam-4498	98	14	(	(	PUNCT
ejpam-4498	98	15	n+	n+	NOUN
ejpam-4498	98	16	1	1	NUM
ejpam-4498	98	17	)	)	PUNCT
ejpam-4498	98	18	.	.	PUNCT
ejpam-4498	99	1	(	(	PUNCT
ejpam-4498	99	2	12	12	NUM
ejpam-4498	99	3	)	)	PUNCT
ejpam-4498	99	4	by	by	ADP
ejpam-4498	99	5	performing	perform	VERB
ejpam-4498	99	6	slight	slight	ADJ
ejpam-4498	99	7	rearrangements	rearrangement	NOUN
ejpam-4498	99	8	,	,	PUNCT
ejpam-4498	99	9	we	we	PRON
ejpam-4498	99	10	obtain	obtain	VERB
ejpam-4498	99	11	1	1	NUM
ejpam-4498	99	12	·	·	SYM
ejpam-4498	99	13	2	2	NUM
ejpam-4498	99	14	+	+	CCONJ
ejpam-4498	99	15	2	2	NUM
ejpam-4498	99	16	·	·	SYM
ejpam-4498	99	17	3	3	NUM
ejpam-4498	99	18	+	+	CCONJ
ejpam-4498	99	19	3	3	NUM
ejpam-4498	99	20	·	·	SYM
ejpam-4498	99	21	4	4	NUM
ejpam-4498	99	22	+	+	CCONJ
ejpam-4498	99	23	·	·	PUNCT
ejpam-4498	99	24	·	·	PUNCT
ejpam-4498	99	25	·	·	PUNCT
ejpam-4498	100	1	+	+	CCONJ
ejpam-4498	100	2	(	(	PUNCT
ejpam-4498	100	3	n−	n−	NOUN
ejpam-4498	100	4	1	1	NUM
ejpam-4498	100	5	)	)	PUNCT
ejpam-4498	100	6	·	·	PUNCT
ejpam-4498	101	1	n	n	X
ejpam-4498	101	2	=	=	SYM
ejpam-4498	101	3	1	1	NUM
ejpam-4498	101	4	3	3	NUM
ejpam-4498	101	5	(	(	PUNCT
ejpam-4498	101	6	n−	n−	NOUN
ejpam-4498	101	7	1)n(n+	1)n(n+	PROPN
ejpam-4498	101	8	1	1	NUM
ejpam-4498	101	9	)	)	PUNCT
ejpam-4498	101	10	,	,	PUNCT
ejpam-4498	101	11	(	(	PUNCT
ejpam-4498	101	12	13	13	X
ejpam-4498	101	13	)	)	PUNCT
ejpam-4498	101	14	yukio	yukio	PROPN
ejpam-4498	101	15	kobayashi	kobayashi	PROPN
ejpam-4498	101	16	/	/	SYM
ejpam-4498	101	17	eur	eur	PROPN
ejpam-4498	101	18	.	.	PUNCT
ejpam-4498	102	1	j.	j.	PROPN
ejpam-4498	102	2	pure	pure	PROPN
ejpam-4498	102	3	appl	appl	PROPN
ejpam-4498	102	4	.	.	PROPN
ejpam-4498	102	5	math	math	PROPN
ejpam-4498	102	6	,	,	PUNCT
ejpam-4498	102	7	15	15	NUM
ejpam-4498	102	8	(	(	PUNCT
ejpam-4498	102	9	3	3	NUM
ejpam-4498	102	10	)	)	PUNCT
ejpam-4498	102	11	(	(	PUNCT
ejpam-4498	102	12	2022	2022	NUM
ejpam-4498	102	13	)	)	PUNCT
ejpam-4498	102	14	,	,	PUNCT
ejpam-4498	102	15	1237	1237	NUM
ejpam-4498	102	16	-	-	SYM
ejpam-4498	102	17	1253	1253	NUM
ejpam-4498	102	18	1242	1242	NUM
ejpam-4498	102	19	using	use	VERB
ejpam-4498	102	20	1	1	NUM
ejpam-4498	102	21	+	+	NUM
ejpam-4498	102	22	2	2	NUM
ejpam-4498	102	23	+	+	CCONJ
ejpam-4498	102	24	·	·	PUNCT
ejpam-4498	102	25	·	·	PUNCT
ejpam-4498	102	26	·	·	PUNCT
ejpam-4498	103	1	+	+	NUM
ejpam-4498	103	2	n	n	CCONJ
ejpam-4498	103	3	=	=	SYM
ejpam-4498	103	4	n	n	NOUN
ejpam-4498	103	5	·	·	PUNCT
ejpam-4498	103	6	(	(	PUNCT
ejpam-4498	103	7	n+	n+	X
ejpam-4498	103	8	1)/2	1)/2	X
ejpam-4498	103	9	.	.	PUNCT
ejpam-4498	103	10	similarly	similarly	ADV
ejpam-4498	103	11	,	,	PUNCT
ejpam-4498	103	12	from	from	ADP
ejpam-4498	103	13	equation	equation	NOUN
ejpam-4498	103	14	(	(	PUNCT
ejpam-4498	103	15	11	11	NUM
ejpam-4498	103	16	)	)	PUNCT
ejpam-4498	103	17	,	,	PUNCT
ejpam-4498	103	18	1	1	X
ejpam-4498	103	19	·	·	SYM
ejpam-4498	103	20	2	2	NUM
ejpam-4498	103	21	+	+	CCONJ
ejpam-4498	103	22	2	2	NUM
ejpam-4498	103	23	·	·	SYM
ejpam-4498	103	24	3	3	NUM
ejpam-4498	103	25	+	+	CCONJ
ejpam-4498	103	26	3	3	NUM
ejpam-4498	103	27	·	·	SYM
ejpam-4498	103	28	4	4	NUM
ejpam-4498	103	29	+	+	CCONJ
ejpam-4498	103	30	·	·	PUNCT
ejpam-4498	103	31	·	·	PUNCT
ejpam-4498	103	32	·	·	PUNCT
ejpam-4498	103	33	+	+	CCONJ
ejpam-4498	103	34	(	(	PUNCT
ejpam-4498	103	35	n−	n−	NOUN
ejpam-4498	103	36	1	1	NUM
ejpam-4498	103	37	)	)	PUNCT
ejpam-4498	103	38	·	·	PUNCT
ejpam-4498	103	39	n	n	X
ejpam-4498	103	40	=	=	SYM
ejpam-4498	103	41	2	2	NUM
ejpam-4498	103	42	[	[	SYM
ejpam-4498	103	43	1	1	NUM
ejpam-4498	103	44	·	·	PUNCT
ejpam-4498	103	45	n+	n+	NUM
ejpam-4498	103	46	2	2	NUM
ejpam-4498	103	47	·	·	PUNCT
ejpam-4498	103	48	(	(	PUNCT
ejpam-4498	103	49	n−	n−	NOUN
ejpam-4498	103	50	1	1	NUM
ejpam-4498	103	51	)	)	PUNCT
ejpam-4498	103	52	+	+	CCONJ
ejpam-4498	103	53	·	·	PUNCT
ejpam-4498	103	54	·	·	PUNCT
ejpam-4498	103	55	·	·	PUNCT
ejpam-4498	103	56	+	+	NUM
ejpam-4498	103	57	n	n	CCONJ
ejpam-4498	103	58	·	·	SYM
ejpam-4498	103	59	1]−	1]−	NUM
ejpam-4498	103	60	n	n	NOUN
ejpam-4498	103	61	·	·	PUNCT
ejpam-4498	103	62	(	(	PUNCT
ejpam-4498	103	63	n+	n+	NOUN
ejpam-4498	103	64	1	1	NUM
ejpam-4498	103	65	)	)	PUNCT
ejpam-4498	103	66	.	.	PUNCT
ejpam-4498	104	1	(	(	PUNCT
ejpam-4498	104	2	14	14	NUM
ejpam-4498	104	3	)	)	PUNCT
ejpam-4498	104	4	thus	thus	ADV
ejpam-4498	104	5	,	,	PUNCT
ejpam-4498	104	6	(	(	PUNCT
ejpam-4498	104	7	9	9	X
ejpam-4498	104	8	)	)	PUNCT
ejpam-4498	104	9	can	can	AUX
ejpam-4498	104	10	be	be	AUX
ejpam-4498	104	11	written	write	VERB
ejpam-4498	104	12	as	as	ADP
ejpam-4498	104	13	(	(	PUNCT
ejpam-4498	104	14	1	1	NUM
ejpam-4498	104	15	+	+	NUM
ejpam-4498	104	16	2	2	NUM
ejpam-4498	104	17	+	+	CCONJ
ejpam-4498	104	18	·	·	PUNCT
ejpam-4498	104	19	·	·	PUNCT
ejpam-4498	104	20	·	·	PUNCT
ejpam-4498	104	21	+	+	NUM
ejpam-4498	104	22	n	n	CCONJ
ejpam-4498	104	23	)	)	PUNCT
ejpam-4498	104	24	·	·	PUNCT
ejpam-4498	105	1	n	n	X
ejpam-4498	105	2	=	=	SYM
ejpam-4498	105	3	3	3	NUM
ejpam-4498	105	4	[	[	SYM
ejpam-4498	105	5	1	1	NUM
ejpam-4498	105	6	·	·	PUNCT
ejpam-4498	105	7	n+	n+	NUM
ejpam-4498	105	8	2	2	NUM
ejpam-4498	105	9	·	·	PUNCT
ejpam-4498	105	10	(	(	PUNCT
ejpam-4498	105	11	n−	n−	NOUN
ejpam-4498	105	12	1	1	NUM
ejpam-4498	105	13	)	)	PUNCT
ejpam-4498	105	14	+	+	CCONJ
ejpam-4498	105	15	·	·	PUNCT
ejpam-4498	105	16	·	·	PUNCT
ejpam-4498	105	17	·	·	PUNCT
ejpam-4498	105	18	+	+	NUM
ejpam-4498	105	19	n	n	CCONJ
ejpam-4498	105	20	·	·	SYM
ejpam-4498	105	21	1]−	1]−	NUM
ejpam-4498	105	22	n	n	NOUN
ejpam-4498	105	23	·	·	PUNCT
ejpam-4498	105	24	(	(	PUNCT
ejpam-4498	105	25	n+	n+	NOUN
ejpam-4498	105	26	1	1	NUM
ejpam-4498	105	27	)	)	PUNCT
ejpam-4498	105	28	.	.	PUNCT
ejpam-4498	106	1	(	(	PUNCT
ejpam-4498	106	2	15	15	NUM
ejpam-4498	106	3	)	)	PUNCT
ejpam-4498	106	4	by	by	ADP
ejpam-4498	106	5	performing	perform	VERB
ejpam-4498	106	6	some	some	DET
ejpam-4498	106	7	slight	slight	ADJ
ejpam-4498	106	8	rearrangements	rearrangement	NOUN
ejpam-4498	106	9	,	,	PUNCT
ejpam-4498	106	10	we	we	PRON
ejpam-4498	106	11	obtain	obtain	VERB
ejpam-4498	106	12	1	1	NUM
ejpam-4498	106	13	·	·	PUNCT
ejpam-4498	106	14	n+	n+	X
ejpam-4498	106	15	2	2	NUM
ejpam-4498	106	16	·	·	PUNCT
ejpam-4498	106	17	(	(	PUNCT
ejpam-4498	106	18	n−	n−	NOUN
ejpam-4498	106	19	1	1	NUM
ejpam-4498	106	20	)	)	PUNCT
ejpam-4498	106	21	+	+	CCONJ
ejpam-4498	106	22	·	·	PUNCT
ejpam-4498	106	23	·	·	PUNCT
ejpam-4498	106	24	·	·	PUNCT
ejpam-4498	106	25	+	+	NUM
ejpam-4498	106	26	n	n	CCONJ
ejpam-4498	106	27	·	·	SYM
ejpam-4498	106	28	1	1	NUM
ejpam-4498	106	29	=	=	SYM
ejpam-4498	106	30	1	1	NUM
ejpam-4498	106	31	6	6	NUM
ejpam-4498	106	32	n(n+	n(n+	NOUN
ejpam-4498	106	33	1)(n+	1)(n+	NUM
ejpam-4498	106	34	2	2	NUM
ejpam-4498	106	35	)	)	PUNCT
ejpam-4498	106	36	.	.	PUNCT
ejpam-4498	107	1	(	(	PUNCT
ejpam-4498	107	2	16	16	X
ejpam-4498	107	3	)	)	PUNCT
ejpam-4498	107	4	note	note	NOUN
ejpam-4498	107	5	that	that	SCONJ
ejpam-4498	107	6	equations	equation	NOUN
ejpam-4498	107	7	(	(	PUNCT
ejpam-4498	107	8	13	13	NUM
ejpam-4498	107	9	)	)	PUNCT
ejpam-4498	107	10	and	and	CCONJ
ejpam-4498	107	11	(	(	PUNCT
ejpam-4498	107	12	16	16	NUM
ejpam-4498	107	13	)	)	PUNCT
ejpam-4498	107	14	can	can	AUX
ejpam-4498	107	15	be	be	AUX
ejpam-4498	107	16	obtained	obtain	VERB
ejpam-4498	107	17	from	from	ADP
ejpam-4498	107	18	(	(	PUNCT
ejpam-4498	107	19	10	10	NUM
ejpam-4498	107	20	)	)	PUNCT
ejpam-4498	107	21	rather	rather	ADV
ejpam-4498	107	22	than	than	ADP
ejpam-4498	107	23	(	(	PUNCT
ejpam-4498	107	24	9	9	NUM
ejpam-4498	107	25	)	)	PUNCT
ejpam-4498	107	26	.	.	PUNCT
ejpam-4498	108	1	equation	equation	NOUN
ejpam-4498	108	2	(	(	PUNCT
ejpam-4498	108	3	16	16	NUM
ejpam-4498	108	4	)	)	PUNCT
ejpam-4498	108	5	can	can	AUX
ejpam-4498	108	6	be	be	AUX
ejpam-4498	108	7	rewritten	rewrite	VERB
ejpam-4498	108	8	in	in	ADP
ejpam-4498	108	9	two	two	NUM
ejpam-4498	108	10	ways	way	NOUN
ejpam-4498	108	11	.	.	PUNCT
ejpam-4498	109	1	the	the	DET
ejpam-4498	109	2	relationship	relationship	NOUN
ejpam-4498	109	3	between	between	ADP
ejpam-4498	109	4	the	the	DET
ejpam-4498	109	5	sum	sum	NOUN
ejpam-4498	109	6	of	of	ADP
ejpam-4498	109	7	the	the	DET
ejpam-4498	109	8	natural	natural	ADJ
ejpam-4498	109	9	numbers	number	NOUN
ejpam-4498	109	10	and	and	CCONJ
ejpam-4498	109	11	the	the	DET
ejpam-4498	109	12	sum	sum	NOUN
ejpam-4498	109	13	of	of	ADP
ejpam-4498	109	14	triangular	triangular	NOUN
ejpam-4498	109	15	numbers	number	NOUN
ejpam-4498	109	16	can	can	AUX
ejpam-4498	109	17	be	be	AUX
ejpam-4498	109	18	expressed	express	VERB
ejpam-4498	109	19	as	as	ADP
ejpam-4498	109	20	either	either	CCONJ
ejpam-4498	109	21	1	1	NUM
ejpam-4498	109	22	·	·	PUNCT
ejpam-4498	109	23	n+	n+	NUM
ejpam-4498	109	24	2	2	NUM
ejpam-4498	109	25	·	·	PUNCT
ejpam-4498	109	26	(	(	PUNCT
ejpam-4498	109	27	n−	n−	NOUN
ejpam-4498	109	28	1	1	NUM
ejpam-4498	109	29	)	)	PUNCT
ejpam-4498	109	30	+	+	CCONJ
ejpam-4498	109	31	·	·	PUNCT
ejpam-4498	109	32	·	·	PUNCT
ejpam-4498	109	33	·	·	PUNCT
ejpam-4498	110	1	+	+	NUM
ejpam-4498	110	2	n	n	CCONJ
ejpam-4498	110	3	·	·	SYM
ejpam-4498	110	4	1	1	NUM
ejpam-4498	110	5	=	=	SYM
ejpam-4498	110	6	1	1	NUM
ejpam-4498	110	7	3	3	NUM
ejpam-4498	110	8	(	(	PUNCT
ejpam-4498	110	9	n+	n+	NOUN
ejpam-4498	110	10	2	2	NUM
ejpam-4498	110	11	)	)	PUNCT
ejpam-4498	110	12	·	·	PUNCT
ejpam-4498	110	13	1	1	NUM
ejpam-4498	110	14	2	2	NUM
ejpam-4498	110	15	n(n+	n(n+	NUM
ejpam-4498	110	16	1	1	NUM
ejpam-4498	110	17	)	)	PUNCT
ejpam-4498	110	18	=	=	SYM
ejpam-4498	110	19	1	1	NUM
ejpam-4498	110	20	3	3	NUM
ejpam-4498	110	21	(	(	PUNCT
ejpam-4498	110	22	n+	n+	NOUN
ejpam-4498	110	23	2	2	NUM
ejpam-4498	110	24	)	)	PUNCT
ejpam-4498	110	25	n∑	n∑	NOUN
ejpam-4498	110	26	k=1	k=1	PUNCT
ejpam-4498	111	1	k	k	X
ejpam-4498	111	2	,	,	PUNCT
ejpam-4498	111	3	(	(	PUNCT
ejpam-4498	111	4	17	17	NUM
ejpam-4498	111	5	)	)	PUNCT
ejpam-4498	111	6	or	or	CCONJ
ejpam-4498	111	7	1	1	NUM
ejpam-4498	111	8	·	·	PUNCT
ejpam-4498	111	9	n+	n+	NUM
ejpam-4498	111	10	2	2	NUM
ejpam-4498	111	11	·	·	PUNCT
ejpam-4498	111	12	(	(	PUNCT
ejpam-4498	111	13	n−	n−	NOUN
ejpam-4498	111	14	1	1	NUM
ejpam-4498	111	15	)	)	PUNCT
ejpam-4498	111	16	+	+	CCONJ
ejpam-4498	111	17	·	·	PUNCT
ejpam-4498	111	18	·	·	PUNCT
ejpam-4498	111	19	·	·	PUNCT
ejpam-4498	111	20	+	+	NUM
ejpam-4498	111	21	n	n	CCONJ
ejpam-4498	111	22	·	·	SYM
ejpam-4498	111	23	1	1	NUM
ejpam-4498	111	24	=	=	SYM
ejpam-4498	111	25	1	1	NUM
ejpam-4498	111	26	3	3	NUM
ejpam-4498	111	27	n	n	PROPN
ejpam-4498	111	28	·	·	SYM
ejpam-4498	111	29	1	1	NUM
ejpam-4498	111	30	2	2	NUM
ejpam-4498	111	31	(	(	PUNCT
ejpam-4498	111	32	n+	n+	NUM
ejpam-4498	111	33	1)(n+	1)(n+	NUM
ejpam-4498	111	34	2	2	NUM
ejpam-4498	111	35	)	)	PUNCT
ejpam-4498	111	36	=	=	SYM
ejpam-4498	111	37	1	1	NUM
ejpam-4498	111	38	3	3	NUM
ejpam-4498	111	39	n	n	NOUN
ejpam-4498	111	40	n+1∑	n+1∑	PROPN
ejpam-4498	111	41	k=1	k=1	PROPN
ejpam-4498	111	42	k.	k.	PROPN
ejpam-4498	112	1	(	(	PUNCT
ejpam-4498	112	2	18	18	NUM
ejpam-4498	112	3	)	)	PUNCT
ejpam-4498	112	4	equations	equation	NOUN
ejpam-4498	112	5	(	(	PUNCT
ejpam-4498	112	6	17	17	NUM
ejpam-4498	112	7	)	)	PUNCT
ejpam-4498	112	8	and	and	CCONJ
ejpam-4498	112	9	(	(	PUNCT
ejpam-4498	112	10	18	18	NUM
ejpam-4498	112	11	)	)	PUNCT
ejpam-4498	112	12	can	can	AUX
ejpam-4498	112	13	also	also	ADV
ejpam-4498	112	14	be	be	AUX
ejpam-4498	112	15	explained	explain	VERB
ejpam-4498	112	16	in	in	ADP
ejpam-4498	112	17	terms	term	NOUN
ejpam-4498	112	18	of	of	ADP
ejpam-4498	112	19	elementary	elementary	ADJ
ejpam-4498	112	20	combinatorics	combinatoric	NOUN
ejpam-4498	112	21	,	,	PUNCT
ejpam-4498	112	22	as	as	SCONJ
ejpam-4498	112	23	shown	show	VERB
ejpam-4498	112	24	by	by	ADP
ejpam-4498	112	25	section	section	NOUN
ejpam-4498	112	26	6	6	NUM
ejpam-4498	112	27	.	.	NOUN
ejpam-4498	112	28	4	4	NUM
ejpam-4498	112	29	.	.	X
ejpam-4498	112	30	relationship	relationship	NOUN
ejpam-4498	112	31	between	between	ADP
ejpam-4498	112	32	the	the	DET
ejpam-4498	112	33	sums	sum	NOUN
ejpam-4498	112	34	of	of	ADP
ejpam-4498	112	35	natural	natural	ADJ
ejpam-4498	112	36	numbers	number	NOUN
ejpam-4498	112	37	,	,	PUNCT
ejpam-4498	112	38	triangular	triangular	NOUN
ejpam-4498	112	39	numbers	number	NOUN
ejpam-4498	112	40	,	,	PUNCT
ejpam-4498	112	41	and	and	CCONJ
ejpam-4498	112	42	square	square	ADJ
ejpam-4498	112	43	numbers	number	NOUN
ejpam-4498	112	44	the	the	DET
ejpam-4498	112	45	sequence	sequence	NOUN
ejpam-4498	112	46	of	of	ADP
ejpam-4498	112	47	equations	equation	NOUN
ejpam-4498	112	48	expressing	express	VERB
ejpam-4498	112	49	the	the	DET
ejpam-4498	112	50	sum	sum	NOUN
ejpam-4498	112	51	of	of	ADP
ejpam-4498	112	52	natural	natural	ADJ
ejpam-4498	112	53	numbers	number	NOUN
ejpam-4498	112	54	shown	show	VERB
ejpam-4498	112	55	in	in	ADP
ejpam-4498	112	56	figure	figure	NOUN
ejpam-4498	112	57	2	2	NUM
ejpam-4498	112	58	can	can	AUX
ejpam-4498	112	59	also	also	ADV
ejpam-4498	112	60	be	be	AUX
ejpam-4498	112	61	used	use	VERB
ejpam-4498	112	62	to	to	PART
ejpam-4498	112	63	determine	determine	VERB
ejpam-4498	112	64	the	the	DET
ejpam-4498	112	65	relationship	relationship	NOUN
ejpam-4498	112	66	between	between	ADP
ejpam-4498	112	67	the	the	DET
ejpam-4498	112	68	sums	sum	NOUN
ejpam-4498	112	69	of	of	ADP
ejpam-4498	112	70	natural	natural	ADJ
ejpam-4498	112	71	numbers	number	NOUN
ejpam-4498	112	72	,	,	PUNCT
ejpam-4498	112	73	triangular	triangular	NOUN
ejpam-4498	112	74	numbers	number	NOUN
ejpam-4498	112	75	,	,	PUNCT
ejpam-4498	112	76	and	and	CCONJ
ejpam-4498	112	77	square	square	ADJ
ejpam-4498	112	78	numbers	number	NOUN
ejpam-4498	112	79	.	.	PUNCT
ejpam-4498	113	1	the	the	DET
ejpam-4498	113	2	sum	sum	NOUN
ejpam-4498	113	3	of	of	ADP
ejpam-4498	113	4	natural	natural	ADJ
ejpam-4498	113	5	numbers	number	NOUN
ejpam-4498	113	6	shown	show	VERB
ejpam-4498	113	7	in	in	ADP
ejpam-4498	113	8	figure	figure	NOUN
ejpam-4498	113	9	2	2	NUM
ejpam-4498	113	10	yukio	yukio	PROPN
ejpam-4498	113	11	kobayashi	kobayashi	PROPN
ejpam-4498	113	12	/	/	SYM
ejpam-4498	113	13	eur	eur	PROPN
ejpam-4498	113	14	.	.	PUNCT
ejpam-4498	114	1	j.	j.	PROPN
ejpam-4498	114	2	pure	pure	PROPN
ejpam-4498	114	3	appl	appl	PROPN
ejpam-4498	114	4	.	.	PROPN
ejpam-4498	114	5	math	math	PROPN
ejpam-4498	114	6	,	,	PUNCT
ejpam-4498	114	7	15	15	NUM
ejpam-4498	114	8	(	(	PUNCT
ejpam-4498	114	9	3	3	NUM
ejpam-4498	114	10	)	)	PUNCT
ejpam-4498	114	11	(	(	PUNCT
ejpam-4498	114	12	2022	2022	NUM
ejpam-4498	114	13	)	)	PUNCT
ejpam-4498	114	14	,	,	PUNCT
ejpam-4498	114	15	1237	1237	NUM
ejpam-4498	114	16	-	-	SYM
ejpam-4498	114	17	1253	1253	NUM
ejpam-4498	114	18	1243	1243	NUM
ejpam-4498	114	19	can	can	AUX
ejpam-4498	114	20	be	be	AUX
ejpam-4498	114	21	expressed	express	VERB
ejpam-4498	114	22	differently	differently	ADV
ejpam-4498	114	23	from	from	ADP
ejpam-4498	114	24	that	that	PRON
ejpam-4498	114	25	in	in	ADP
ejpam-4498	114	26	figures	figure	NOUN
ejpam-4498	114	27	3	3	NUM
ejpam-4498	114	28	and	and	CCONJ
ejpam-4498	114	29	4	4	NUM
ejpam-4498	114	30	.	.	X
ejpam-4498	114	31	from	from	ADP
ejpam-4498	114	32	figure	figure	NOUN
ejpam-4498	114	33	5	5	NUM
ejpam-4498	114	34	,	,	PUNCT
ejpam-4498	114	35	we	we	PRON
ejpam-4498	114	36	can	can	AUX
ejpam-4498	114	37	express	express	VERB
ejpam-4498	114	38	the	the	DET
ejpam-4498	114	39	feature	feature	NOUN
ejpam-4498	114	40	as	as	ADP
ejpam-4498	114	41	(	(	PUNCT
ejpam-4498	114	42	1	1	NUM
ejpam-4498	115	1	+	+	SYM
ejpam-4498	115	2	2	2	NUM
ejpam-4498	115	3	+	+	NUM
ejpam-4498	115	4	3	3	NUM
ejpam-4498	115	5	+	+	SYM
ejpam-4498	115	6	4	4	NUM
ejpam-4498	115	7	+	+	SYM
ejpam-4498	115	8	5	5	NUM
ejpam-4498	115	9	)	)	PUNCT
ejpam-4498	115	10	·	·	PUNCT
ejpam-4498	116	1	5	5	NUM
ejpam-4498	116	2	=	=	SYM
ejpam-4498	116	3	(	(	PUNCT
ejpam-4498	116	4	1	1	NUM
ejpam-4498	116	5	·	·	SYM
ejpam-4498	116	6	5	5	NUM
ejpam-4498	116	7	+	+	CCONJ
ejpam-4498	116	8	2	2	NUM
ejpam-4498	116	9	·	·	SYM
ejpam-4498	116	10	4	4	NUM
ejpam-4498	116	11	+	+	SYM
ejpam-4498	116	12	3	3	NUM
ejpam-4498	116	13	·	·	SYM
ejpam-4498	116	14	3	3	NUM
ejpam-4498	116	15	+	+	CCONJ
ejpam-4498	116	16	4	4	NUM
ejpam-4498	116	17	·	·	SYM
ejpam-4498	116	18	2	2	NUM
ejpam-4498	116	19	+	+	SYM
ejpam-4498	116	20	5	5	NUM
ejpam-4498	116	21	·	·	SYM
ejpam-4498	116	22	1	1	NUM
ejpam-4498	116	23	)	)	PUNCT
ejpam-4498	116	24	+	+	NOUN
ejpam-4498	116	25	(	(	PUNCT
ejpam-4498	116	26	1	1	NUM
ejpam-4498	116	27	·	·	SYM
ejpam-4498	116	28	1	1	NUM
ejpam-4498	116	29	+	+	CCONJ
ejpam-4498	116	30	2	2	NUM
ejpam-4498	116	31	·	·	SYM
ejpam-4498	116	32	2	2	NUM
ejpam-4498	116	33	+	+	CCONJ
ejpam-4498	116	34	3	3	NUM
ejpam-4498	116	35	·	·	SYM
ejpam-4498	116	36	3	3	NUM
ejpam-4498	116	37	+	+	CCONJ
ejpam-4498	116	38	4	4	NUM
ejpam-4498	116	39	·	·	SYM
ejpam-4498	116	40	4	4	NUM
ejpam-4498	116	41	+	+	SYM
ejpam-4498	116	42	5	5	NUM
ejpam-4498	116	43	·	·	SYM
ejpam-4498	116	44	5	5	NUM
ejpam-4498	116	45	)	)	PUNCT
ejpam-4498	116	46	−(1	−(1	NOUN
ejpam-4498	116	47	+	+	CCONJ
ejpam-4498	116	48	2	2	NUM
ejpam-4498	116	49	+	+	NUM
ejpam-4498	116	50	3	3	NUM
ejpam-4498	116	51	+	+	SYM
ejpam-4498	116	52	4	4	NUM
ejpam-4498	116	53	+	+	SYM
ejpam-4498	116	54	5	5	NUM
ejpam-4498	116	55	)	)	PUNCT
ejpam-4498	116	56	.	.	PUNCT
ejpam-4498	117	1	(	(	PUNCT
ejpam-4498	117	2	19	19	NUM
ejpam-4498	117	3	)	)	PUNCT
ejpam-4498	117	4	by	by	ADP
ejpam-4498	117	5	performing	perform	VERB
ejpam-4498	117	6	some	some	DET
ejpam-4498	117	7	slight	slight	ADJ
ejpam-4498	117	8	rearrangements	rearrangement	NOUN
ejpam-4498	117	9	,	,	PUNCT
ejpam-4498	117	10	we	we	PRON
ejpam-4498	117	11	obtain	obtain	VERB
ejpam-4498	117	12	1	1	NUM
ejpam-4498	117	13	·	·	SYM
ejpam-4498	117	14	5	5	NUM
ejpam-4498	117	15	+	+	CCONJ
ejpam-4498	117	16	2	2	NUM
ejpam-4498	117	17	·	·	SYM
ejpam-4498	117	18	4	4	NUM
ejpam-4498	118	1	+	+	SYM
ejpam-4498	118	2	3	3	NUM
ejpam-4498	118	3	·	·	SYM
ejpam-4498	118	4	3	3	NUM
ejpam-4498	118	5	+	+	CCONJ
ejpam-4498	118	6	4	4	NUM
ejpam-4498	118	7	·	·	SYM
ejpam-4498	118	8	2	2	NUM
ejpam-4498	119	1	+	+	CCONJ
ejpam-4498	119	2	5	5	NUM
ejpam-4498	119	3	·	·	SYM
ejpam-4498	119	4	1	1	NUM
ejpam-4498	119	5	=	=	SYM
ejpam-4498	119	6	(	(	PUNCT
ejpam-4498	119	7	1	1	NUM
ejpam-4498	119	8	+	+	NUM
ejpam-4498	119	9	2	2	NUM
ejpam-4498	119	10	+	+	NUM
ejpam-4498	119	11	3	3	NUM
ejpam-4498	119	12	+	+	SYM
ejpam-4498	119	13	4	4	NUM
ejpam-4498	119	14	+	+	SYM
ejpam-4498	119	15	5	5	NUM
ejpam-4498	119	16	)	)	PUNCT
ejpam-4498	119	17	·	·	PUNCT
ejpam-4498	120	1	6	6	NUM
ejpam-4498	120	2	−(12	−(12	NOUN
ejpam-4498	120	3	+	+	CCONJ
ejpam-4498	120	4	22	22	NUM
ejpam-4498	120	5	+	+	NUM
ejpam-4498	120	6	32	32	NUM
ejpam-4498	120	7	+	+	NUM
ejpam-4498	120	8	42	42	NUM
ejpam-4498	120	9	+	+	NUM
ejpam-4498	120	10	52	52	NUM
ejpam-4498	120	11	)	)	PUNCT
ejpam-4498	120	12	.	.	PUNCT
ejpam-4498	121	1	(	(	PUNCT
ejpam-4498	121	2	20	20	NUM
ejpam-4498	121	3	)	)	PUNCT
ejpam-4498	121	4	equation	equation	NOUN
ejpam-4498	121	5	(	(	PUNCT
ejpam-4498	121	6	20	20	NUM
ejpam-4498	121	7	)	)	PUNCT
ejpam-4498	121	8	defines	define	VERB
ejpam-4498	121	9	the	the	DET
ejpam-4498	121	10	relationship	relationship	NOUN
ejpam-4498	121	11	between	between	ADP
ejpam-4498	121	12	the	the	DET
ejpam-4498	121	13	sums	sum	NOUN
ejpam-4498	121	14	of	of	ADP
ejpam-4498	121	15	natural	natural	ADJ
ejpam-4498	121	16	numbers	number	NOUN
ejpam-4498	121	17	,	,	PUNCT
ejpam-4498	121	18	square	square	ADJ
ejpam-4498	121	19	numbers	number	NOUN
ejpam-4498	121	20	,	,	PUNCT
ejpam-4498	121	21	and	and	CCONJ
ejpam-4498	121	22	triangular	triangular	NOUN
ejpam-4498	121	23	numbers	number	NOUN
ejpam-4498	121	24	.	.	PUNCT
ejpam-4498	122	1	1	1	NUM
ejpam-4498	123	1	+	+	NUM
ejpam-4498	123	2	2	2	NUM
ejpam-4498	123	3	+	+	NUM
ejpam-4498	123	4	3	3	NUM
ejpam-4498	123	5	+	+	SYM
ejpam-4498	123	6	4	4	NUM
ejpam-4498	123	7	+	+	SYM
ejpam-4498	123	8	5	5	NUM
ejpam-4498	123	9	1	1	NUM
ejpam-4498	123	10	+	+	NUM
ejpam-4498	123	11	2	2	NUM
ejpam-4498	123	12	+	+	NUM
ejpam-4498	123	13	3	3	NUM
ejpam-4498	123	14	+	+	SYM
ejpam-4498	123	15	4	4	NUM
ejpam-4498	123	16	+	+	SYM
ejpam-4498	123	17	5	5	NUM
ejpam-4498	123	18	1	1	NUM
ejpam-4498	123	19	+	+	NUM
ejpam-4498	123	20	2	2	NUM
ejpam-4498	123	21	+	+	NUM
ejpam-4498	123	22	3	3	NUM
ejpam-4498	123	23	+	+	SYM
ejpam-4498	123	24	4	4	NUM
ejpam-4498	123	25	+	+	SYM
ejpam-4498	123	26	5	5	NUM
ejpam-4498	123	27	1	1	NUM
ejpam-4498	123	28	+	+	NUM
ejpam-4498	123	29	2	2	NUM
ejpam-4498	123	30	+	+	NUM
ejpam-4498	123	31	3	3	NUM
ejpam-4498	123	32	+	+	SYM
ejpam-4498	123	33	4	4	NUM
ejpam-4498	123	34	+	+	SYM
ejpam-4498	123	35	5	5	NUM
ejpam-4498	123	36	1	1	NUM
ejpam-4498	123	37	+	+	NUM
ejpam-4498	123	38	2	2	NUM
ejpam-4498	123	39	+	+	NUM
ejpam-4498	123	40	3	3	NUM
ejpam-4498	123	41	+	+	SYM
ejpam-4498	123	42	4	4	NUM
ejpam-4498	123	43	+	+	SYM
ejpam-4498	123	44	5	5	NUM
ejpam-4498	123	45	figure	figure	NOUN
ejpam-4498	123	46	5	5	NUM
ejpam-4498	123	47	:	:	PUNCT
ejpam-4498	123	48	relationship	relationship	NOUN
ejpam-4498	123	49	between	between	ADP
ejpam-4498	123	50	the	the	DET
ejpam-4498	123	51	sums	sum	NOUN
ejpam-4498	123	52	of	of	ADP
ejpam-4498	123	53	natural	natural	ADJ
ejpam-4498	123	54	numbers	number	NOUN
ejpam-4498	123	55	,	,	PUNCT
ejpam-4498	123	56	square	square	ADJ
ejpam-4498	123	57	numbers	number	NOUN
ejpam-4498	123	58	,	,	PUNCT
ejpam-4498	123	59	and	and	CCONJ
ejpam-4498	123	60	triangular	triangular	NOUN
ejpam-4498	123	61	numbers	number	NOUN
ejpam-4498	123	62	.	.	PUNCT
ejpam-4498	124	1	equations	equation	NOUN
ejpam-4498	124	2	(	(	PUNCT
ejpam-4498	124	3	20	20	NUM
ejpam-4498	124	4	)	)	PUNCT
ejpam-4498	124	5	and	and	CCONJ
ejpam-4498	124	6	(	(	PUNCT
ejpam-4498	124	7	17	17	NUM
ejpam-4498	124	8	)	)	PUNCT
ejpam-4498	124	9	can	can	AUX
ejpam-4498	124	10	be	be	AUX
ejpam-4498	124	11	used	use	VERB
ejpam-4498	124	12	to	to	PART
ejpam-4498	124	13	find	find	VERB
ejpam-4498	124	14	the	the	DET
ejpam-4498	124	15	sum	sum	NOUN
ejpam-4498	124	16	of	of	ADP
ejpam-4498	124	17	square	square	ADJ
ejpam-4498	124	18	numbers	number	NOUN
ejpam-4498	124	19	.	.	PUNCT
ejpam-4498	125	1	by	by	ADP
ejpam-4498	125	2	generalising	generalise	VERB
ejpam-4498	125	3	(	(	PUNCT
ejpam-4498	125	4	20	20	NUM
ejpam-4498	125	5	)	)	PUNCT
ejpam-4498	125	6	,	,	PUNCT
ejpam-4498	125	7	we	we	PRON
ejpam-4498	125	8	obtain	obtain	VERB
ejpam-4498	125	9	12	12	NUM
ejpam-4498	125	10	+	+	CCONJ
ejpam-4498	125	11	22	22	NUM
ejpam-4498	125	12	+	+	CCONJ
ejpam-4498	125	13	·	·	PUNCT
ejpam-4498	125	14	·	·	PUNCT
ejpam-4498	125	15	·	·	PUNCT
ejpam-4498	125	16	+	+	NUM
ejpam-4498	125	17	n2	n2	NOUN
ejpam-4498	125	18	=	=	SYM
ejpam-4498	125	19	(	(	PUNCT
ejpam-4498	125	20	1	1	NUM
ejpam-4498	125	21	+	+	NUM
ejpam-4498	125	22	2	2	NUM
ejpam-4498	125	23	+	+	CCONJ
ejpam-4498	125	24	·	·	PUNCT
ejpam-4498	125	25	·	·	PUNCT
ejpam-4498	125	26	·	·	PUNCT
ejpam-4498	125	27	+	+	NUM
ejpam-4498	125	28	n	n	CCONJ
ejpam-4498	125	29	)	)	PUNCT
ejpam-4498	125	30	·	·	PUNCT
ejpam-4498	125	31	(	(	PUNCT
ejpam-4498	125	32	n+	n+	NOUN
ejpam-4498	125	33	1	1	X
ejpam-4498	125	34	)	)	PUNCT
ejpam-4498	125	35	−[1	−[1	NOUN
ejpam-4498	125	36	·	·	PUNCT
ejpam-4498	125	37	n+	n+	X
ejpam-4498	125	38	2	2	NUM
ejpam-4498	125	39	·	·	PUNCT
ejpam-4498	125	40	(	(	PUNCT
ejpam-4498	125	41	n−	n−	NOUN
ejpam-4498	125	42	1	1	NUM
ejpam-4498	125	43	)	)	PUNCT
ejpam-4498	125	44	+	+	CCONJ
ejpam-4498	125	45	·	·	PUNCT
ejpam-4498	125	46	·	·	PUNCT
ejpam-4498	125	47	·	·	PUNCT
ejpam-4498	126	1	+	+	NUM
ejpam-4498	126	2	n	n	CCONJ
ejpam-4498	126	3	·	·	PUNCT
ejpam-4498	126	4	1	1	NUM
ejpam-4498	126	5	]	]	PUNCT
ejpam-4498	126	6	=	=	PUNCT
ejpam-4498	127	1	[	[	X
ejpam-4498	127	2	1	1	NUM
ejpam-4498	127	3	2	2	NUM
ejpam-4498	127	4	n(n+	n(n+	NUM
ejpam-4498	127	5	1	1	NUM
ejpam-4498	127	6	)	)	PUNCT
ejpam-4498	127	7	]	]	PUNCT
ejpam-4498	127	8	(	(	PUNCT
ejpam-4498	127	9	n+	n+	NUM
ejpam-4498	127	10	1)−	1)−	NUM
ejpam-4498	127	11	1	1	NUM
ejpam-4498	127	12	3	3	NUM
ejpam-4498	127	13	(	(	PUNCT
ejpam-4498	127	14	n+	n+	NOUN
ejpam-4498	127	15	2	2	NUM
ejpam-4498	127	16	)	)	PUNCT
ejpam-4498	127	17	·	·	PUNCT
ejpam-4498	127	18	1	1	NUM
ejpam-4498	127	19	2	2	NUM
ejpam-4498	127	20	n(n+	n(n+	NUM
ejpam-4498	127	21	1	1	NUM
ejpam-4498	127	22	)	)	PUNCT
ejpam-4498	127	23	=	=	SYM
ejpam-4498	127	24	1	1	NUM
ejpam-4498	127	25	6	6	NUM
ejpam-4498	127	26	n(n+	n(n+	NOUN
ejpam-4498	127	27	1)(2n+	1)(2n+	NUM
ejpam-4498	127	28	1	1	NUM
ejpam-4498	127	29	)	)	PUNCT
ejpam-4498	127	30	.	.	PUNCT
ejpam-4498	128	1	(	(	PUNCT
ejpam-4498	128	2	21	21	NUM
ejpam-4498	128	3	)	)	PUNCT
ejpam-4498	128	4	equation	equation	NOUN
ejpam-4498	128	5	(	(	PUNCT
ejpam-4498	128	6	21	21	NUM
ejpam-4498	128	7	)	)	PUNCT
ejpam-4498	128	8	can	can	AUX
ejpam-4498	128	9	be	be	AUX
ejpam-4498	128	10	rewritten	rewrite	VERB
ejpam-4498	128	11	to	to	PART
ejpam-4498	128	12	express	express	VERB
ejpam-4498	128	13	the	the	DET
ejpam-4498	128	14	relationship	relationship	NOUN
ejpam-4498	128	15	between	between	ADP
ejpam-4498	128	16	the	the	DET
ejpam-4498	128	17	sums	sum	NOUN
ejpam-4498	128	18	of	of	ADP
ejpam-4498	128	19	natural	natural	ADJ
ejpam-4498	128	20	numbers	number	NOUN
ejpam-4498	128	21	and	and	CCONJ
ejpam-4498	128	22	square	square	ADJ
ejpam-4498	128	23	numbers	number	NOUN
ejpam-4498	128	24	,	,	PUNCT
ejpam-4498	128	25	as	as	SCONJ
ejpam-4498	128	26	given	give	VERB
ejpam-4498	128	27	below	below	ADV
ejpam-4498	128	28	.	.	PUNCT
ejpam-4498	129	1	12	12	NUM
ejpam-4498	130	1	+	+	NUM
ejpam-4498	130	2	22	22	NUM
ejpam-4498	130	3	+	+	CCONJ
ejpam-4498	130	4	·	·	PUNCT
ejpam-4498	130	5	·	·	PUNCT
ejpam-4498	130	6	·	·	PUNCT
ejpam-4498	130	7	+	+	NUM
ejpam-4498	130	8	n2	n2	ADJ
ejpam-4498	130	9	=	=	SYM
ejpam-4498	130	10	1	1	NUM
ejpam-4498	130	11	3	3	NUM
ejpam-4498	130	12	(	(	PUNCT
ejpam-4498	130	13	2n+	2n+	NUM
ejpam-4498	130	14	1	1	NUM
ejpam-4498	130	15	)	)	PUNCT
ejpam-4498	130	16	n∑	n∑	NOUN
ejpam-4498	131	1	k=1	k=1	PROPN
ejpam-4498	131	2	k.	k.	PROPN
ejpam-4498	132	1	(	(	PUNCT
ejpam-4498	132	2	22	22	NUM
ejpam-4498	132	3	)	)	PUNCT
ejpam-4498	132	4	yukio	yukio	PROPN
ejpam-4498	132	5	kobayashi	kobayashi	PROPN
ejpam-4498	132	6	/	/	SYM
ejpam-4498	132	7	eur	eur	PROPN
ejpam-4498	132	8	.	.	PUNCT
ejpam-4498	133	1	j.	j.	PROPN
ejpam-4498	133	2	pure	pure	PROPN
ejpam-4498	133	3	appl	appl	PROPN
ejpam-4498	133	4	.	.	PROPN
ejpam-4498	133	5	math	math	PROPN
ejpam-4498	133	6	,	,	PUNCT
ejpam-4498	133	7	15	15	NUM
ejpam-4498	133	8	(	(	PUNCT
ejpam-4498	133	9	3	3	NUM
ejpam-4498	133	10	)	)	PUNCT
ejpam-4498	133	11	(	(	PUNCT
ejpam-4498	133	12	2022	2022	NUM
ejpam-4498	133	13	)	)	PUNCT
ejpam-4498	133	14	,	,	PUNCT
ejpam-4498	133	15	1237	1237	NUM
ejpam-4498	133	16	-	-	SYM
ejpam-4498	133	17	1253	1253	NUM
ejpam-4498	133	18	1244	1244	NUM
ejpam-4498	133	19	5	5	NUM
ejpam-4498	133	20	.	.	PUNCT
ejpam-4498	134	1	view	view	NOUN
ejpam-4498	134	2	of	of	ADP
ejpam-4498	134	3	integral	integral	ADJ
ejpam-4498	134	4	representations	representation	NOUN
ejpam-4498	134	5	the	the	DET
ejpam-4498	134	6	sum	sum	NOUN
ejpam-4498	134	7	of	of	ADP
ejpam-4498	134	8	triangular	triangular	NOUN
ejpam-4498	134	9	numbers	number	NOUN
ejpam-4498	134	10	can	can	AUX
ejpam-4498	134	11	be	be	AUX
ejpam-4498	134	12	obtained	obtain	VERB
ejpam-4498	134	13	using	use	VERB
ejpam-4498	134	14	a	a	DET
ejpam-4498	134	15	heuristic	heuristic	ADJ
ejpam-4498	134	16	method	method	NOUN
ejpam-4498	134	17	,	,	PUNCT
ejpam-4498	134	18	as	as	SCONJ
ejpam-4498	134	19	shown	show	VERB
ejpam-4498	134	20	in	in	ADP
ejpam-4498	134	21	section	section	NOUN
ejpam-4498	134	22	3	3	NUM
ejpam-4498	134	23	.	.	PUNCT
ejpam-4498	135	1	the	the	DET
ejpam-4498	135	2	sum	sum	NOUN
ejpam-4498	135	3	of	of	ADP
ejpam-4498	135	4	triangular	triangular	NOUN
ejpam-4498	135	5	numbers	number	NOUN
ejpam-4498	135	6	may	may	AUX
ejpam-4498	135	7	appear	appear	VERB
ejpam-4498	135	8	to	to	PART
ejpam-4498	135	9	have	have	AUX
ejpam-4498	135	10	been	be	AUX
ejpam-4498	135	11	found	find	VERB
ejpam-4498	135	12	easily	easily	ADV
ejpam-4498	135	13	using	use	VERB
ejpam-4498	135	14	a	a	DET
ejpam-4498	135	15	character	character	NOUN
ejpam-4498	135	16	expression	expression	NOUN
ejpam-4498	135	17	n∑	n∑	NOUN
ejpam-4498	135	18	k=1	k=1	PUNCT
ejpam-4498	136	1	k(n	k(n	X
ejpam-4498	136	2	+	+	CCONJ
ejpam-4498	136	3	1	1	NUM
ejpam-4498	136	4	−	−	NOUN
ejpam-4498	136	5	k	k	NOUN
ejpam-4498	136	6	)	)	PUNCT
ejpam-4498	136	7	,	,	PUNCT
ejpam-4498	136	8	instead	instead	ADV
ejpam-4498	136	9	of	of	ADP
ejpam-4498	136	10	a	a	DET
ejpam-4498	136	11	numerical	numerical	ADJ
ejpam-4498	136	12	expression	expression	NOUN
ejpam-4498	136	13	;	;	PUNCT
ejpam-4498	136	14	however	however	ADV
ejpam-4498	136	15	,	,	PUNCT
ejpam-4498	136	16	this	this	PRON
ejpam-4498	136	17	is	be	AUX
ejpam-4498	136	18	not	not	PART
ejpam-4498	136	19	always	always	ADV
ejpam-4498	136	20	the	the	DET
ejpam-4498	136	21	case	case	NOUN
ejpam-4498	136	22	.	.	PUNCT
ejpam-4498	137	1	the	the	DET
ejpam-4498	137	2	following	follow	VERB
ejpam-4498	137	3	expression	expression	NOUN
ejpam-4498	137	4	indicates	indicate	VERB
ejpam-4498	137	5	the	the	DET
ejpam-4498	137	6	relationship	relationship	NOUN
ejpam-4498	137	7	between	between	ADP
ejpam-4498	137	8	the	the	DET
ejpam-4498	137	9	sums	sum	NOUN
ejpam-4498	137	10	of	of	ADP
ejpam-4498	137	11	triangular	triangular	NOUN
ejpam-4498	137	12	numbers	number	NOUN
ejpam-4498	137	13	and	and	CCONJ
ejpam-4498	137	14	square	square	ADJ
ejpam-4498	137	15	numbers	number	NOUN
ejpam-4498	137	16	.	.	PUNCT
ejpam-4498	138	1	n∑	n∑	INTJ
ejpam-4498	138	2	k=1	k=1	PUNCT
ejpam-4498	139	1	k(n+	k(n+	PROPN
ejpam-4498	139	2	1−	1−	NUM
ejpam-4498	140	1	k	k	NOUN
ejpam-4498	140	2	)	)	PUNCT
ejpam-4498	140	3	=	=	SYM
ejpam-4498	140	4	(	(	PUNCT
ejpam-4498	140	5	n+	n+	NOUN
ejpam-4498	140	6	1	1	NUM
ejpam-4498	140	7	)	)	PUNCT
ejpam-4498	140	8	n∑	n∑	NOUN
ejpam-4498	140	9	k=1	k=1	PUNCT
ejpam-4498	141	1	k	k	X
ejpam-4498	142	1	−	−	PROPN
ejpam-4498	142	2	n∑	n∑	INTJ
ejpam-4498	142	3	k=1	k=1	PROPN
ejpam-4498	142	4	k2	k2	PROPN
ejpam-4498	142	5	.	.	PUNCT
ejpam-4498	143	1	(	(	PUNCT
ejpam-4498	143	2	23	23	NUM
ejpam-4498	143	3	)	)	PUNCT
ejpam-4498	143	4	if	if	SCONJ
ejpam-4498	143	5	students	student	NOUN
ejpam-4498	143	6	are	be	AUX
ejpam-4498	143	7	unfamiliar	unfamiliar	ADJ
ejpam-4498	143	8	with	with	ADP
ejpam-4498	143	9	(	(	PUNCT
ejpam-4498	143	10	21	21	NUM
ejpam-4498	143	11	)	)	PUNCT
ejpam-4498	143	12	or	or	CCONJ
ejpam-4498	143	13	(	(	PUNCT
ejpam-4498	143	14	22	22	NUM
ejpam-4498	143	15	)	)	PUNCT
ejpam-4498	143	16	,	,	PUNCT
ejpam-4498	143	17	they	they	PRON
ejpam-4498	143	18	should	should	AUX
ejpam-4498	143	19	consider	consider	VERB
ejpam-4498	143	20	how	how	SCONJ
ejpam-4498	143	21	to	to	PART
ejpam-4498	143	22	find	find	VERB
ejpam-4498	143	23	the	the	DET
ejpam-4498	143	24	sum	sum	NOUN
ejpam-4498	143	25	of	of	ADP
ejpam-4498	143	26	triangular	triangular	NOUN
ejpam-4498	143	27	numbers	number	NOUN
ejpam-4498	143	28	(	(	PUNCT
ejpam-4498	143	29	23	23	NUM
ejpam-4498	143	30	)	)	PUNCT
ejpam-4498	143	31	without	without	ADP
ejpam-4498	143	32	these	these	DET
ejpam-4498	143	33	expression	expression	NOUN
ejpam-4498	143	34	.	.	PUNCT
ejpam-4498	144	1	to	to	PART
ejpam-4498	144	2	calculate	calculate	VERB
ejpam-4498	145	1	n∑	n∑	PROPN
ejpam-4498	145	2	k=1	k=1	PROPN
ejpam-4498	146	1	k(n+	k(n+	PROPN
ejpam-4498	146	2	1−	1−	NUM
ejpam-4498	147	1	k	k	NOUN
ejpam-4498	147	2	)	)	PUNCT
ejpam-4498	147	3	in	in	ADP
ejpam-4498	147	4	a	a	DET
ejpam-4498	147	5	way	way	NOUN
ejpam-4498	147	6	that	that	PRON
ejpam-4498	147	7	does	do	AUX
ejpam-4498	147	8	not	not	PART
ejpam-4498	147	9	rely	rely	VERB
ejpam-4498	147	10	on	on	ADP
ejpam-4498	147	11	(	(	PUNCT
ejpam-4498	147	12	21	21	NUM
ejpam-4498	147	13	)	)	PUNCT
ejpam-4498	147	14	or	or	CCONJ
ejpam-4498	147	15	(	(	PUNCT
ejpam-4498	147	16	22	22	NUM
ejpam-4498	147	17	)	)	PUNCT
ejpam-4498	147	18	,	,	PUNCT
ejpam-4498	147	19	it	it	PRON
ejpam-4498	147	20	is	be	AUX
ejpam-4498	147	21	convenient	convenient	ADJ
ejpam-4498	147	22	to	to	PART
ejpam-4498	147	23	note	note	VERB
ejpam-4498	147	24	the	the	DET
ejpam-4498	147	25	integral	integral	ADJ
ejpam-4498	147	26	representations	representation	NOUN
ejpam-4498	147	27	[	[	X
ejpam-4498	147	28	5	5	NUM
ejpam-4498	147	29	]	]	PUNCT
ejpam-4498	147	30	.	.	PUNCT
ejpam-4498	148	1	instructors	instructor	NOUN
ejpam-4498	148	2	should	should	AUX
ejpam-4498	148	3	encourage	encourage	VERB
ejpam-4498	148	4	students	student	NOUN
ejpam-4498	148	5	to	to	PART
ejpam-4498	148	6	realise	realise	VERB
ejpam-4498	148	7	n∑	n∑	NOUN
ejpam-4498	148	8	k=1	k=1	PUNCT
ejpam-4498	149	1	k	k	X
ejpam-4498	150	1	=	=	PUNCT
ejpam-4498	150	2	n∑	n∑	PROPN
ejpam-4498	150	3	k=1	k=1	PROPN
ejpam-4498	151	1	∫	∫	PROPN
ejpam-4498	151	2	k	k	PROPN
ejpam-4498	151	3	0	0	NUM
ejpam-4498	151	4	dx	dx	PROPN
ejpam-4498	151	5	,	,	PUNCT
ejpam-4498	151	6	(	(	PUNCT
ejpam-4498	151	7	24	24	NUM
ejpam-4498	151	8	)	)	PUNCT
ejpam-4498	151	9	and	and	CCONJ
ejpam-4498	151	10	n∑	n∑	NOUN
ejpam-4498	151	11	k=1	k=1	PROPN
ejpam-4498	152	1	k2	k2	PROPN
ejpam-4498	152	2	=	=	PROPN
ejpam-4498	152	3	2	2	NUM
ejpam-4498	152	4	n∑	n∑	NOUN
ejpam-4498	152	5	k=1	k=1	PROPN
ejpam-4498	152	6	∫	∫	PROPN
ejpam-4498	153	1	k	k	PROPN
ejpam-4498	153	2	0	0	PROPN
ejpam-4498	153	3	xdx	xdx	PROPN
ejpam-4498	153	4	.	.	PUNCT
ejpam-4498	154	1	(	(	PUNCT
ejpam-4498	154	2	25	25	NUM
ejpam-4498	154	3	)	)	PUNCT
ejpam-4498	154	4	using	use	VERB
ejpam-4498	154	5	(	(	PUNCT
ejpam-4498	154	6	24	24	NUM
ejpam-4498	154	7	)	)	PUNCT
ejpam-4498	154	8	and	and	CCONJ
ejpam-4498	154	9	(	(	PUNCT
ejpam-4498	154	10	25	25	NUM
ejpam-4498	154	11	)	)	PUNCT
ejpam-4498	154	12	,	,	PUNCT
ejpam-4498	154	13	n∑	n∑	NOUN
ejpam-4498	154	14	k=1	k=1	PROPN
ejpam-4498	154	15	k(n+	k(n+	PROPN
ejpam-4498	154	16	1−	1−	NUM
ejpam-4498	154	17	k	k	NOUN
ejpam-4498	154	18	)	)	PUNCT
ejpam-4498	154	19	=	=	SYM
ejpam-4498	155	1	n∑	n∑	NOUN
ejpam-4498	155	2	k=1	k=1	PROPN
ejpam-4498	156	1	∫	∫	PROPN
ejpam-4498	157	1	k	k	NOUN
ejpam-4498	157	2	0	0	PUNCT
ejpam-4498	157	3	(	(	PUNCT
ejpam-4498	157	4	n+	n+	NUM
ejpam-4498	157	5	1−	1−	NUM
ejpam-4498	157	6	2x)dx	2x)dx	NUM
ejpam-4498	157	7	=	=	SYM
ejpam-4498	157	8	n	n	NUM
ejpam-4498	157	9	∫	∫	NOUN
ejpam-4498	157	10	1	1	NUM
ejpam-4498	157	11	0	0	NUM
ejpam-4498	157	12	(	(	PUNCT
ejpam-4498	157	13	n+	n+	NUM
ejpam-4498	157	14	1−	1−	NUM
ejpam-4498	157	15	2x)dx+	2x)dx+	NUM
ejpam-4498	157	16	(	(	PUNCT
ejpam-4498	157	17	n−	n−	NOUN
ejpam-4498	157	18	1	1	NUM
ejpam-4498	157	19	)	)	PUNCT
ejpam-4498	157	20	∫	∫	PROPN
ejpam-4498	157	21	2	2	NUM
ejpam-4498	157	22	1	1	NUM
ejpam-4498	157	23	(	(	PUNCT
ejpam-4498	157	24	n+	n+	NUM
ejpam-4498	157	25	1−	1−	NUM
ejpam-4498	157	26	2x)dx+	2x)dx+	NUM
ejpam-4498	157	27	·	·	PUNCT
ejpam-4498	157	28	·	·	PUNCT
ejpam-4498	157	29	·	·	PUNCT
ejpam-4498	158	1	+	+	PUNCT
ejpam-4498	158	2	[	[	X
ejpam-4498	158	3	n−	n−	NOUN
ejpam-4498	158	4	(	(	PUNCT
ejpam-4498	158	5	k	k	NOUN
ejpam-4498	158	6	−	−	PROPN
ejpam-4498	158	7	1	1	NUM
ejpam-4498	158	8	)	)	PUNCT
ejpam-4498	158	9	]	]	PUNCT
ejpam-4498	159	1	∫	∫	PROPN
ejpam-4498	159	2	k	k	PROPN
ejpam-4498	159	3	k−1	k−1	PROPN
ejpam-4498	159	4	(	(	PUNCT
ejpam-4498	159	5	n+	n+	PROPN
ejpam-4498	159	6	1−	1−	NUM
ejpam-4498	159	7	2x)dx+	2x)dx+	NUM
ejpam-4498	159	8	·	·	PUNCT
ejpam-4498	159	9	·	·	PUNCT
ejpam-4498	159	10	·	·	PUNCT
ejpam-4498	159	11	+	+	NUM
ejpam-4498	159	12	∫	∫	PROPN
ejpam-4498	159	13	n	n	CCONJ
ejpam-4498	159	14	n−1	n−1	PROPN
ejpam-4498	159	15	(	(	PUNCT
ejpam-4498	159	16	n+	n+	PROPN
ejpam-4498	159	17	1−	1−	NUM
ejpam-4498	159	18	2x)dx	2x)dx	NUM
ejpam-4498	159	19	=	=	SYM
ejpam-4498	159	20	n∑	n∑	NOUN
ejpam-4498	159	21	k=1	k=1	X
ejpam-4498	159	22	(	(	PUNCT
ejpam-4498	159	23	n+	n+	NUM
ejpam-4498	159	24	1−	1−	NUM
ejpam-4498	159	25	k	k	NOUN
ejpam-4498	159	26	)	)	PUNCT
ejpam-4498	159	27	∫	∫	PROPN
ejpam-4498	160	1	k	k	PROPN
ejpam-4498	160	2	k−1	k−1	PROPN
ejpam-4498	160	3	(	(	PUNCT
ejpam-4498	160	4	n+	n+	PROPN
ejpam-4498	160	5	1−	1−	NUM
ejpam-4498	160	6	2x)dx	2x)dx	NUM
ejpam-4498	160	7	=	=	SYM
ejpam-4498	161	1	n∑	n∑	NOUN
ejpam-4498	161	2	k=1	k=1	X
ejpam-4498	161	3	(	(	PUNCT
ejpam-4498	161	4	n+	n+	NUM
ejpam-4498	161	5	1−	1−	NUM
ejpam-4498	161	6	k	k	NOUN
ejpam-4498	161	7	)	)	PUNCT
ejpam-4498	161	8	[	[	PUNCT
ejpam-4498	161	9	(	(	PUNCT
ejpam-4498	161	10	n+	n+	NUM
ejpam-4498	161	11	1)x−	1)x−	NUM
ejpam-4498	161	12	x2	x2	NOUN
ejpam-4498	162	1	]	]	X
ejpam-4498	162	2	k	k	PROPN
ejpam-4498	162	3	k−1	k−1	PROPN
ejpam-4498	162	4	yukio	yukio	PROPN
ejpam-4498	162	5	kobayashi	kobayashi	PROPN
ejpam-4498	162	6	/	/	SYM
ejpam-4498	162	7	eur	eur	PROPN
ejpam-4498	162	8	.	.	PUNCT
ejpam-4498	163	1	j.	j.	PROPN
ejpam-4498	163	2	pure	pure	PROPN
ejpam-4498	163	3	appl	appl	PROPN
ejpam-4498	163	4	.	.	PROPN
ejpam-4498	163	5	math	math	PROPN
ejpam-4498	163	6	,	,	PUNCT
ejpam-4498	163	7	15	15	NUM
ejpam-4498	163	8	(	(	PUNCT
ejpam-4498	163	9	3	3	NUM
ejpam-4498	163	10	)	)	PUNCT
ejpam-4498	163	11	(	(	PUNCT
ejpam-4498	163	12	2022	2022	NUM
ejpam-4498	163	13	)	)	PUNCT
ejpam-4498	163	14	,	,	PUNCT
ejpam-4498	163	15	1237	1237	NUM
ejpam-4498	163	16	-	-	SYM
ejpam-4498	163	17	1253	1253	NUM
ejpam-4498	163	18	1245	1245	NUM
ejpam-4498	163	19	=	=	SYM
ejpam-4498	164	1	n∑	n∑	NOUN
ejpam-4498	164	2	k=1	k=1	X
ejpam-4498	164	3	(	(	PUNCT
ejpam-4498	164	4	n+	n+	NUM
ejpam-4498	164	5	1−	1−	NUM
ejpam-4498	164	6	k)[(n+	k)[(n+	NOUN
ejpam-4498	164	7	1)k	1)k	NUM
ejpam-4498	164	8	−	−	PROPN
ejpam-4498	164	9	k2	k2	PROPN
ejpam-4498	164	10	−	−	PROPN
ejpam-4498	164	11	(	(	PUNCT
ejpam-4498	164	12	n+	n+	NOUN
ejpam-4498	164	13	1)(k	1)(k	NUM
ejpam-4498	164	14	−	−	PROPN
ejpam-4498	164	15	1	1	NUM
ejpam-4498	164	16	)	)	PUNCT
ejpam-4498	164	17	+	+	CCONJ
ejpam-4498	164	18	(	(	PUNCT
ejpam-4498	164	19	k	k	X
ejpam-4498	164	20	−	−	PROPN
ejpam-4498	164	21	1)2	1)2	NUM
ejpam-4498	164	22	]	]	X
ejpam-4498	165	1	=	=	PUNCT
ejpam-4498	165	2	n∑	n∑	NOUN
ejpam-4498	165	3	k=1	k=1	X
ejpam-4498	166	1	(	(	PUNCT
ejpam-4498	166	2	n+	n+	NUM
ejpam-4498	166	3	1−	1−	NUM
ejpam-4498	166	4	k)(n−	k)(n−	PROPN
ejpam-4498	166	5	2k	2k	NOUN
ejpam-4498	166	6	+	+	CCONJ
ejpam-4498	166	7	2	2	X
ejpam-4498	166	8	)	)	PUNCT
ejpam-4498	166	9	=	=	SYM
ejpam-4498	166	10	(	(	PUNCT
ejpam-4498	166	11	n+	n+	NOUN
ejpam-4498	166	12	2	2	NUM
ejpam-4498	166	13	)	)	PUNCT
ejpam-4498	166	14	n∑	n∑	NOUN
ejpam-4498	166	15	k=1	k=1	X
ejpam-4498	167	1	(	(	PUNCT
ejpam-4498	167	2	n+	n+	NUM
ejpam-4498	167	3	1−	1−	NUM
ejpam-4498	167	4	k)−	k)−	PROPN
ejpam-4498	167	5	2	2	NUM
ejpam-4498	168	1	n∑	n∑	NOUN
ejpam-4498	168	2	k=1	k=1	PROPN
ejpam-4498	168	3	k(n+	k(n+	PROPN
ejpam-4498	168	4	1−	1−	NUM
ejpam-4498	168	5	k	k	NOUN
ejpam-4498	168	6	)	)	PUNCT
ejpam-4498	168	7	,	,	PUNCT
ejpam-4498	168	8	(	(	PUNCT
ejpam-4498	168	9	26	26	NUM
ejpam-4498	168	10	)	)	PUNCT
ejpam-4498	168	11	and	and	CCONJ
ejpam-4498	168	12	by	by	ADP
ejpam-4498	168	13	rearranging	rearrange	VERB
ejpam-4498	168	14	,	,	PUNCT
ejpam-4498	168	15	we	we	PRON
ejpam-4498	168	16	obtain	obtain	VERB
ejpam-4498	168	17	3	3	NUM
ejpam-4498	168	18	n∑	n∑	NOUN
ejpam-4498	168	19	k=1	k=1	PROPN
ejpam-4498	168	20	k(n+	k(n+	PROPN
ejpam-4498	168	21	1−	1−	NUM
ejpam-4498	168	22	k	k	NOUN
ejpam-4498	168	23	)	)	PUNCT
ejpam-4498	168	24	=	=	SYM
ejpam-4498	168	25	(	(	PUNCT
ejpam-4498	168	26	n+	n+	NOUN
ejpam-4498	168	27	2	2	NUM
ejpam-4498	168	28	)	)	PUNCT
ejpam-4498	168	29	n∑	n∑	NOUN
ejpam-4498	168	30	k=1	k=1	X
ejpam-4498	169	1	(	(	PUNCT
ejpam-4498	169	2	n+	n+	NUM
ejpam-4498	169	3	1−	1−	NUM
ejpam-4498	169	4	k	k	NOUN
ejpam-4498	169	5	)	)	PUNCT
ejpam-4498	169	6	=	=	SYM
ejpam-4498	169	7	(	(	PUNCT
ejpam-4498	169	8	n+	n+	NUM
ejpam-4498	169	9	2)(n+	2)(n+	NUM
ejpam-4498	169	10	1	1	NUM
ejpam-4498	169	11	)	)	PUNCT
ejpam-4498	169	12	n∑	n∑	NOUN
ejpam-4498	170	1	k=1	k=1	PROPN
ejpam-4498	171	1	1−	1−	NUM
ejpam-4498	171	2	(	(	PUNCT
ejpam-4498	171	3	n+	n+	NOUN
ejpam-4498	171	4	2	2	NUM
ejpam-4498	171	5	)	)	PUNCT
ejpam-4498	171	6	n∑	n∑	NOUN
ejpam-4498	171	7	k=1	k=1	PUNCT
ejpam-4498	172	1	k	k	X
ejpam-4498	172	2	=	=	PUNCT
ejpam-4498	172	3	(	(	PUNCT
ejpam-4498	172	4	n+	n+	NUM
ejpam-4498	172	5	2)(n+	2)(n+	NUM
ejpam-4498	172	6	1)n−	1)n−	NUM
ejpam-4498	172	7	(	(	PUNCT
ejpam-4498	172	8	n+	n+	NOUN
ejpam-4498	172	9	2	2	NUM
ejpam-4498	172	10	)	)	PUNCT
ejpam-4498	172	11	·	·	PUNCT
ejpam-4498	172	12	1	1	NUM
ejpam-4498	172	13	2	2	NUM
ejpam-4498	172	14	(	(	PUNCT
ejpam-4498	172	15	n+	n+	X
ejpam-4498	172	16	1)n	1)n	X
ejpam-4498	172	17	=	=	SYM
ejpam-4498	172	18	1	1	NUM
ejpam-4498	172	19	2	2	NUM
ejpam-4498	172	20	n(n+	n(n+	NOUN
ejpam-4498	172	21	1)(n+	1)(n+	NUM
ejpam-4498	172	22	2	2	NUM
ejpam-4498	172	23	)	)	PUNCT
ejpam-4498	172	24	.	.	PUNCT
ejpam-4498	173	1	(	(	PUNCT
ejpam-4498	173	2	27	27	NUM
ejpam-4498	173	3	)	)	PUNCT
ejpam-4498	173	4	thus	thus	ADV
ejpam-4498	173	5	,	,	PUNCT
ejpam-4498	173	6	we	we	PRON
ejpam-4498	173	7	obtain	obtain	VERB
ejpam-4498	174	1	n∑	n∑	PROPN
ejpam-4498	174	2	k=1	k=1	PROPN
ejpam-4498	174	3	k(n+	k(n+	PROPN
ejpam-4498	175	1	1−	1−	NUM
ejpam-4498	175	2	k	k	NOUN
ejpam-4498	175	3	)	)	PUNCT
ejpam-4498	175	4	=	=	SYM
ejpam-4498	175	5	1	1	NUM
ejpam-4498	175	6	6	6	NUM
ejpam-4498	175	7	n(n+	n(n+	NOUN
ejpam-4498	175	8	1)(n+	1)(n+	NUM
ejpam-4498	175	9	2	2	NUM
ejpam-4498	175	10	)	)	PUNCT
ejpam-4498	175	11	.	.	PUNCT
ejpam-4498	176	1	(	(	PUNCT
ejpam-4498	176	2	28	28	NUM
ejpam-4498	176	3	)	)	PUNCT
ejpam-4498	176	4	this	this	DET
ejpam-4498	176	5	approach	approach	NOUN
ejpam-4498	176	6	uses	use	VERB
ejpam-4498	176	7	the	the	DET
ejpam-4498	176	8	definite	definite	ADJ
ejpam-4498	176	9	integral	integral	ADJ
ejpam-4498	176	10	to	to	PART
ejpam-4498	176	11	calculate	calculate	VERB
ejpam-4498	176	12	the	the	DET
ejpam-4498	176	13	series	series	NOUN
ejpam-4498	176	14	.	.	PUNCT
ejpam-4498	177	1	from	from	ADP
ejpam-4498	177	2	a	a	DET
ejpam-4498	177	3	pedagogical	pedagogical	ADJ
ejpam-4498	177	4	viewpoint	viewpoint	NOUN
ejpam-4498	177	5	,	,	PUNCT
ejpam-4498	177	6	instructors	instructor	NOUN
ejpam-4498	177	7	can	can	AUX
ejpam-4498	177	8	apply	apply	VERB
ejpam-4498	177	9	this	this	DET
ejpam-4498	177	10	material	material	NOUN
ejpam-4498	177	11	as	as	ADP
ejpam-4498	177	12	a	a	DET
ejpam-4498	177	13	useful	useful	ADJ
ejpam-4498	177	14	exercise	exercise	NOUN
ejpam-4498	177	15	towards	towards	ADP
ejpam-4498	177	16	integrated	integrated	ADJ
ejpam-4498	177	17	learning	learning	NOUN
ejpam-4498	177	18	of	of	ADP
ejpam-4498	177	19	integrals	integral	NOUN
ejpam-4498	177	20	and	and	CCONJ
ejpam-4498	177	21	series	series	NOUN
ejpam-4498	177	22	.	.	PUNCT
ejpam-4498	178	1	we	we	PRON
ejpam-4498	178	2	can	can	AUX
ejpam-4498	178	3	also	also	ADV
ejpam-4498	178	4	determine	determine	VERB
ejpam-4498	178	5	that	that	PRON
ejpam-4498	179	1	n∑	n∑	PROPN
ejpam-4498	179	2	k=1	k=1	VERB
ejpam-4498	180	1	kℓ	kℓ	NOUN
ejpam-4498	180	2	using	use	VERB
ejpam-4498	180	3	integral	integral	ADJ
ejpam-4498	180	4	representations	representation	NOUN
ejpam-4498	180	5	,	,	PUNCT
ejpam-4498	180	6	where	where	SCONJ
ejpam-4498	180	7	ℓ	ℓ	PROPN
ejpam-4498	180	8	is	be	AUX
ejpam-4498	180	9	a	a	DET
ejpam-4498	180	10	positive	positive	ADJ
ejpam-4498	180	11	integer	integer	NOUN
ejpam-4498	180	12	,	,	PUNCT
ejpam-4498	180	13	as	as	SCONJ
ejpam-4498	180	14	shown	show	VERB
ejpam-4498	180	15	in	in	ADP
ejpam-4498	180	16	a	a	DET
ejpam-4498	180	17	previous	previous	ADJ
ejpam-4498	180	18	study	study	NOUN
ejpam-4498	180	19	[	[	X
ejpam-4498	180	20	5	5	NUM
ejpam-4498	180	21	]	]	PUNCT
ejpam-4498	180	22	.	.	PUNCT
ejpam-4498	181	1	6	6	X
ejpam-4498	181	2	.	.	X
ejpam-4498	181	3	combinatoric	combinatoric	ADJ
ejpam-4498	181	4	sums	sum	NOUN
ejpam-4498	181	5	of	of	ADP
ejpam-4498	181	6	triangular	triangular	NOUN
ejpam-4498	181	7	numbers	number	NOUN
ejpam-4498	181	8	and	and	CCONJ
ejpam-4498	181	9	square	square	ADJ
ejpam-4498	181	10	numbers	number	NOUN
ejpam-4498	181	11	6.1	6.1	NUM
ejpam-4498	181	12	.	.	PUNCT
ejpam-4498	182	1	triangular	triangular	NOUN
ejpam-4498	182	2	numbers	number	NOUN
ejpam-4498	182	3	the	the	DET
ejpam-4498	182	4	relationship	relationship	NOUN
ejpam-4498	182	5	between	between	ADP
ejpam-4498	182	6	the	the	DET
ejpam-4498	182	7	sum	sum	NOUN
ejpam-4498	182	8	of	of	ADP
ejpam-4498	182	9	natural	natural	ADJ
ejpam-4498	182	10	numbers	number	NOUN
ejpam-4498	182	11	and	and	CCONJ
ejpam-4498	182	12	triangular	triangular	NOUN
ejpam-4498	182	13	numbers	number	NOUN
ejpam-4498	182	14	expressed	express	VERB
ejpam-4498	182	15	by	by	ADP
ejpam-4498	182	16	(	(	PUNCT
ejpam-4498	182	17	17	17	NUM
ejpam-4498	182	18	)	)	PUNCT
ejpam-4498	182	19	and	and	CCONJ
ejpam-4498	182	20	(	(	PUNCT
ejpam-4498	182	21	18	18	NUM
ejpam-4498	182	22	)	)	PUNCT
ejpam-4498	182	23	,	,	PUNCT
ejpam-4498	182	24	respectively	respectively	ADV
ejpam-4498	182	25	,	,	PUNCT
ejpam-4498	182	26	indicates	indicate	VERB
ejpam-4498	182	27	the	the	DET
ejpam-4498	182	28	number	number	NOUN
ejpam-4498	182	29	of	of	ADP
ejpam-4498	182	30	combinations	combination	NOUN
ejpam-4498	182	31	,	,	PUNCT
ejpam-4498	182	32	n+2c3	n+2c3	PROPN
ejpam-4498	182	33	.	.	PUNCT
ejpam-4498	183	1	instructors	instructor	NOUN
ejpam-4498	183	2	should	should	AUX
ejpam-4498	183	3	ask	ask	VERB
ejpam-4498	183	4	students	student	NOUN
ejpam-4498	183	5	to	to	PART
ejpam-4498	183	6	consider	consider	VERB
ejpam-4498	183	7	why	why	SCONJ
ejpam-4498	183	8	the	the	DET
ejpam-4498	183	9	sum	sum	NOUN
ejpam-4498	183	10	of	of	ADP
ejpam-4498	183	11	triangular	triangular	NOUN
ejpam-4498	183	12	numbers	number	NOUN
ejpam-4498	183	13	is	be	AUX
ejpam-4498	183	14	equal	equal	ADJ
ejpam-4498	183	15	to	to	ADP
ejpam-4498	183	16	n+2c3	n+2c3	PROPN
ejpam-4498	183	17	.	.	PUNCT
ejpam-4498	184	1	students	student	NOUN
ejpam-4498	184	2	can	can	AUX
ejpam-4498	184	3	be	be	AUX
ejpam-4498	184	4	convinced	convince	VERB
ejpam-4498	184	5	of	of	ADP
ejpam-4498	184	6	this	this	DET
ejpam-4498	184	7	equivalence	equivalence	NOUN
ejpam-4498	184	8	by	by	ADP
ejpam-4498	184	9	drawing	draw	VERB
ejpam-4498	184	10	tree	tree	NOUN
ejpam-4498	184	11	diagrams	diagram	NOUN
ejpam-4498	184	12	for	for	ADP
ejpam-4498	184	13	n+2c3	n+2c3	PROPN
ejpam-4498	184	14	.	.	PUNCT
ejpam-4498	185	1	drawing	draw	VERB
ejpam-4498	185	2	diagrams	diagram	NOUN
ejpam-4498	185	3	in	in	ADP
ejpam-4498	185	4	connection	connection	NOUN
ejpam-4498	185	5	with	with	ADP
ejpam-4498	185	6	the	the	DET
ejpam-4498	185	7	following	follow	VERB
ejpam-4498	185	8	expansion	expansion	NOUN
ejpam-4498	185	9	of	of	ADP
ejpam-4498	185	10	a	a	DET
ejpam-4498	185	11	polynomial	polynomial	NOUN
ejpam-4498	185	12	is	be	AUX
ejpam-4498	185	13	preferable	preferable	ADJ
ejpam-4498	185	14	.	.	PUNCT
ejpam-4498	186	1	yukio	yukio	PROPN
ejpam-4498	186	2	kobayashi	kobayashi	PROPN
ejpam-4498	186	3	/	/	SYM
ejpam-4498	186	4	eur	eur	PROPN
ejpam-4498	186	5	.	.	PUNCT
ejpam-4498	187	1	j.	j.	PROPN
ejpam-4498	187	2	pure	pure	PROPN
ejpam-4498	187	3	appl	appl	PROPN
ejpam-4498	187	4	.	.	PROPN
ejpam-4498	187	5	math	math	PROPN
ejpam-4498	187	6	,	,	PUNCT
ejpam-4498	187	7	15	15	NUM
ejpam-4498	187	8	(	(	PUNCT
ejpam-4498	187	9	3	3	NUM
ejpam-4498	187	10	)	)	PUNCT
ejpam-4498	187	11	(	(	PUNCT
ejpam-4498	187	12	2022	2022	NUM
ejpam-4498	187	13	)	)	PUNCT
ejpam-4498	187	14	,	,	PUNCT
ejpam-4498	187	15	1237	1237	NUM
ejpam-4498	187	16	-	-	SYM
ejpam-4498	187	17	1253	1253	NUM
ejpam-4498	187	18	1246	1246	NUM
ejpam-4498	187	19	(	(	PUNCT
ejpam-4498	187	20	a+	a+	PUNCT
ejpam-4498	187	21	b)n+2	b)n+2	PROPN
ejpam-4498	187	22	=	=	PUNCT
ejpam-4498	187	23	term	term	NOUN
ejpam-4498	187	24	1︷	1︷	NUM
ejpam-4498	187	25	︸︸	︸︸	NUM
ejpam-4498	187	26	︷	︷	PUNCT
ejpam-4498	187	27	(	(	PUNCT
ejpam-4498	187	28	a+	a+	PUNCT
ejpam-4498	187	29	b	b	NOUN
ejpam-4498	187	30	)	)	PUNCT
ejpam-4498	187	31	term	term	NOUN
ejpam-4498	187	32	2︷	2︷	NUM
ejpam-4498	187	33	︸︸	︸︸	NUM
ejpam-4498	187	34	︷	︷	PUNCT
ejpam-4498	187	35	(	(	PUNCT
ejpam-4498	187	36	a+	a+	PUNCT
ejpam-4498	187	37	b	b	NOUN
ejpam-4498	187	38	)	)	PUNCT
ejpam-4498	187	39	·	·	PUNCT
ejpam-4498	187	40	·	·	PUNCT
ejpam-4498	187	41	·	·	PUNCT
ejpam-4498	188	1	term	term	NOUN
ejpam-4498	188	2	n+2︷	n+2︷	ADV
ejpam-4498	188	3	︸︸	︸︸	PUNCT
ejpam-4498	188	4	︷	︷	PUNCT
ejpam-4498	188	5	(	(	PUNCT
ejpam-4498	188	6	a+	a+	PUNCT
ejpam-4498	188	7	b	b	X
ejpam-4498	188	8	)	)	PUNCT
ejpam-4498	188	9	=	=	NOUN
ejpam-4498	188	10	n+2c0a	n+2c0a	ADP
ejpam-4498	188	11	n+2b0	n+2b0	PROPN
ejpam-4498	188	12	+	+	CCONJ
ejpam-4498	188	13	n+2c1a	n+2c1a	PRON
ejpam-4498	188	14	n+1b1	n+1b1	PROPN
ejpam-4498	188	15	+	+	CCONJ
ejpam-4498	188	16	n+2c2a	n+2c2a	NUM
ejpam-4498	188	17	nb2	nb2	NOUN
ejpam-4498	188	18	+	+	CCONJ
ejpam-4498	188	19	n+2c3a	n+2c3a	NUM
ejpam-4498	188	20	n−1b3	n−1b3	NOUN
ejpam-4498	188	21	+	+	X
ejpam-4498	188	22	·	·	PUNCT
ejpam-4498	188	23	·	·	PUNCT
ejpam-4498	188	24	·	·	PUNCT
ejpam-4498	188	25	+	+	NUM
ejpam-4498	188	26	n+2cn+2a	n+2cn+2a	VERB
ejpam-4498	188	27	0bn+2	0bn+2	NOUN
ejpam-4498	188	28	.	.	PUNCT
ejpam-4498	189	1	(	(	PUNCT
ejpam-4498	189	2	29	29	NUM
ejpam-4498	189	3	)	)	PUNCT
ejpam-4498	189	4	the	the	DET
ejpam-4498	189	5	combination	combination	NOUN
ejpam-4498	189	6	of	of	ADP
ejpam-4498	189	7	(	(	PUNCT
ejpam-4498	189	8	a+b)s	a+b)	VERB
ejpam-4498	189	9	required	require	VERB
ejpam-4498	189	10	to	to	PART
ejpam-4498	189	11	choose	choose	VERB
ejpam-4498	189	12	(	(	PUNCT
ejpam-4498	189	13	n−1	n−1	PROPN
ejpam-4498	189	14	)	)	PUNCT
ejpam-4498	189	15	as	as	ADP
ejpam-4498	189	16	and	and	CCONJ
ejpam-4498	189	17	three	three	NUM
ejpam-4498	189	18	bs	bs	NOUN
ejpam-4498	189	19	can	can	AUX
ejpam-4498	189	20	be	be	AUX
ejpam-4498	189	21	represented	represent	VERB
ejpam-4498	189	22	by	by	ADP
ejpam-4498	189	23	n+2c3	n+2c3	PROPN
ejpam-4498	189	24	in	in	ADP
ejpam-4498	189	25	(	(	PUNCT
ejpam-4498	189	26	29	29	NUM
ejpam-4498	189	27	)	)	PUNCT
ejpam-4498	189	28	.	.	PUNCT
ejpam-4498	190	1	1	1	NUM
ejpam-4498	190	2	2	2	NUM
ejpam-4498	190	3	3	3	NUM
ejpam-4498	190	4	n+	n+	ADP
ejpam-4498	190	5	2	2	NUM
ejpam-4498	190	6	...	...	SYM
ejpam-4498	190	7	3	3	NUM
ejpam-4498	190	8	4	4	NUM
ejpam-4498	190	9	n+	n+	SYM
ejpam-4498	190	10	2	2	NUM
ejpam-4498	190	11	...	...	PUNCT
ejpam-4498	190	12	n+	n+	ADP
ejpam-4498	190	13	1	1	NUM
ejpam-4498	190	14	n+	n+	SYM
ejpam-4498	190	15	2	2	NUM
ejpam-4498	190	16	...	...	PUNCT
ejpam-4498	190	17	...	...	PUNCT
ejpam-4498	191	1	2	2	NUM
ejpam-4498	191	2	3	3	NUM
ejpam-4498	191	3	4	4	NUM
ejpam-4498	191	4	n+	n+	SYM
ejpam-4498	191	5	2	2	NUM
ejpam-4498	191	6	...	...	SYM
ejpam-4498	191	7	4	4	NUM
ejpam-4498	191	8	5	5	NUM
ejpam-4498	191	9	n+	n+	SYM
ejpam-4498	191	10	2	2	NUM
ejpam-4498	191	11	...	...	PUNCT
ejpam-4498	191	12	n+	n+	ADP
ejpam-4498	191	13	1	1	NUM
ejpam-4498	191	14	n+	n+	SYM
ejpam-4498	191	15	2	2	NUM
ejpam-4498	191	16	...	...	PUNCT
ejpam-4498	191	17	...	...	PUNCT
ejpam-4498	192	1	n	n	X
ejpam-4498	192	2	n+	n+	ADP
ejpam-4498	192	3	1	1	NUM
ejpam-4498	192	4	n+	n+	SYM
ejpam-4498	192	5	2	2	NUM
ejpam-4498	192	6	·	·	PUNCT
ejpam-4498	192	7	·	·	PUNCT
ejpam-4498	192	8	·	·	PUNCT
ejpam-4498	193	1	no	no	INTJ
ejpam-4498	193	2	.	.	PUNCT
ejpam-4498	193	3	of	of	ADP
ejpam-4498	193	4	edges	edge	NOUN
ejpam-4498	193	5	1	1	NUM
ejpam-4498	194	1	+	+	NUM
ejpam-4498	194	2	2	2	NUM
ejpam-4498	194	3	+	+	CCONJ
ejpam-4498	194	4	·	·	PUNCT
ejpam-4498	194	5	·	·	PUNCT
ejpam-4498	194	6	·	·	PUNCT
ejpam-4498	194	7	+	+	SYM
ejpam-4498	194	8	n	n	CCONJ
ejpam-4498	194	9	1	1	NUM
ejpam-4498	194	10	+	+	NUM
ejpam-4498	194	11	2	2	NUM
ejpam-4498	194	12	+	+	CCONJ
ejpam-4498	194	13	·	·	PUNCT
ejpam-4498	194	14	·	·	PUNCT
ejpam-4498	194	15	·	·	PUNCT
ejpam-4498	194	16	+	+	CCONJ
ejpam-4498	194	17	(	(	PUNCT
ejpam-4498	194	18	n−	n−	NOUN
ejpam-4498	194	19	1	1	NUM
ejpam-4498	194	20	)	)	PUNCT
ejpam-4498	194	21	1	1	NUM
ejpam-4498	194	22	figure	figure	NOUN
ejpam-4498	194	23	6	6	NUM
ejpam-4498	194	24	:	:	PUNCT
ejpam-4498	194	25	tree	tree	NOUN
ejpam-4498	194	26	diagrams	diagram	NOUN
ejpam-4498	194	27	for	for	ADP
ejpam-4498	194	28	n+2c3	n+2c3	PROPN
ejpam-4498	194	29	.	.	PUNCT
ejpam-4498	195	1	the	the	DET
ejpam-4498	195	2	numerals	numeral	NOUN
ejpam-4498	195	3	indicate	indicate	VERB
ejpam-4498	195	4	the	the	DET
ejpam-4498	195	5	number	number	NOUN
ejpam-4498	195	6	of	of	ADP
ejpam-4498	195	7	(	(	PUNCT
ejpam-4498	195	8	a+	a+	PRON
ejpam-4498	195	9	b	b	NOUN
ejpam-4498	195	10	)	)	PUNCT
ejpam-4498	195	11	terms	term	NOUN
ejpam-4498	195	12	in	in	ADP
ejpam-4498	195	13	equation	equation	NOUN
ejpam-4498	195	14	(	(	PUNCT
ejpam-4498	195	15	29	29	NUM
ejpam-4498	195	16	)	)	PUNCT
ejpam-4498	195	17	.	.	PUNCT
ejpam-4498	196	1	among	among	ADP
ejpam-4498	196	2	the	the	DET
ejpam-4498	196	3	four	four	NUM
ejpam-4498	196	4	representations	representation	NOUN
ejpam-4498	196	5	of	of	ADP
ejpam-4498	196	6	triangular	triangular	NOUN
ejpam-4498	196	7	numbers	number	NOUN
ejpam-4498	196	8	,	,	PUNCT
ejpam-4498	196	9	(	(	PUNCT
ejpam-4498	196	10	1	1	NUM
ejpam-4498	196	11	)	)	PUNCT
ejpam-4498	196	12	,	,	PUNCT
ejpam-4498	196	13	(	(	PUNCT
ejpam-4498	196	14	2	2	NUM
ejpam-4498	196	15	)	)	PUNCT
ejpam-4498	196	16	,	,	PUNCT
ejpam-4498	196	17	(	(	PUNCT
ejpam-4498	196	18	3	3	NUM
ejpam-4498	196	19	)	)	PUNCT
ejpam-4498	196	20	,	,	PUNCT
ejpam-4498	196	21	and	and	CCONJ
ejpam-4498	196	22	(	(	PUNCT
ejpam-4498	196	23	4	4	NUM
ejpam-4498	196	24	)	)	PUNCT
ejpam-4498	196	25	,	,	PUNCT
ejpam-4498	196	26	(	(	PUNCT
ejpam-4498	196	27	3	3	X
ejpam-4498	196	28	)	)	PUNCT
ejpam-4498	196	29	is	be	AUX
ejpam-4498	196	30	appropriate	appropriate	ADJ
ejpam-4498	196	31	to	to	PART
ejpam-4498	196	32	compare	compare	VERB
ejpam-4498	196	33	the	the	DET
ejpam-4498	196	34	sum	sum	NOUN
ejpam-4498	196	35	of	of	ADP
ejpam-4498	196	36	triangular	triangular	NOUN
ejpam-4498	196	37	numbers	number	NOUN
ejpam-4498	196	38	with	with	ADP
ejpam-4498	196	39	n+2c3	n+2c3	PROPN
ejpam-4498	196	40	.	.	PUNCT
ejpam-4498	197	1	the	the	DET
ejpam-4498	197	2	diagrams	diagram	NOUN
ejpam-4498	197	3	in	in	ADP
ejpam-4498	197	4	figure	figure	NOUN
ejpam-4498	197	5	6	6	NUM
ejpam-4498	197	6	indicate	indicate	VERB
ejpam-4498	197	7	the	the	DET
ejpam-4498	197	8	sum	sum	NOUN
ejpam-4498	197	9	of	of	ADP
ejpam-4498	197	10	triangular	triangular	NOUN
ejpam-4498	197	11	numbers	number	NOUN
ejpam-4498	197	12	expressed	express	VERB
ejpam-4498	197	13	in	in	ADP
ejpam-4498	197	14	equation	equation	NOUN
ejpam-4498	197	15	(	(	PUNCT
ejpam-4498	197	16	3	3	NUM
ejpam-4498	197	17	)	)	PUNCT
ejpam-4498	197	18	.	.	PUNCT
ejpam-4498	198	1	counting	count	VERB
ejpam-4498	198	2	the	the	DET
ejpam-4498	198	3	number	number	NOUN
ejpam-4498	198	4	of	of	ADP
ejpam-4498	198	5	edges	edge	NOUN
ejpam-4498	198	6	is	be	AUX
ejpam-4498	198	7	another	another	DET
ejpam-4498	198	8	heuristic	heuristic	ADJ
ejpam-4498	198	9	method	method	NOUN
ejpam-4498	198	10	.	.	PUNCT
ejpam-4498	199	1	from	from	ADP
ejpam-4498	199	2	(	(	PUNCT
ejpam-4498	199	3	2	2	NUM
ejpam-4498	199	4	)	)	PUNCT
ejpam-4498	199	5	and	and	CCONJ
ejpam-4498	199	6	(	(	PUNCT
ejpam-4498	199	7	3	3	NUM
ejpam-4498	199	8	)	)	PUNCT
ejpam-4498	199	9	,	,	PUNCT
ejpam-4498	199	10	we	we	PRON
ejpam-4498	199	11	can	can	AUX
ejpam-4498	199	12	determine	determine	VERB
ejpam-4498	199	13	the	the	DET
ejpam-4498	199	14	relationship	relationship	NOUN
ejpam-4498	199	15	n∑	n∑	INTJ
ejpam-4498	200	1	k=1	k=1	PROPN
ejpam-4498	201	1	k+1c2	k+1c2	PROPN
ejpam-4498	201	2	=	=	PROPN
ejpam-4498	201	3	n+2c3	n+2c3	PROPN
ejpam-4498	201	4	,	,	PUNCT
ejpam-4498	201	5	(	(	PUNCT
ejpam-4498	201	6	30	30	NUM
ejpam-4498	201	7	)	)	PUNCT
ejpam-4498	201	8	which	which	PRON
ejpam-4498	201	9	indicates	indicate	VERB
ejpam-4498	201	10	the	the	DET
ejpam-4498	201	11	total	total	ADJ
ejpam-4498	201	12	number	number	NOUN
ejpam-4498	201	13	of	of	ADP
ejpam-4498	201	14	edges	edge	NOUN
ejpam-4498	201	15	in	in	ADP
ejpam-4498	201	16	the	the	DET
ejpam-4498	201	17	tree	tree	NOUN
ejpam-4498	201	18	diagrams	diagram	NOUN
ejpam-4498	201	19	in	in	ADP
ejpam-4498	201	20	figures	figure	NOUN
ejpam-4498	201	21	1	1	NUM
ejpam-4498	201	22	and	and	CCONJ
ejpam-4498	201	23	6	6	NUM
ejpam-4498	201	24	.	.	PUNCT
ejpam-4498	201	25	to	to	PART
ejpam-4498	201	26	summarise	summarise	VERB
ejpam-4498	201	27	,	,	PUNCT
ejpam-4498	201	28	the	the	DET
ejpam-4498	201	29	sum	sum	NOUN
ejpam-4498	201	30	of	of	ADP
ejpam-4498	201	31	triangular	triangular	NOUN
ejpam-4498	201	32	numbers	number	NOUN
ejpam-4498	201	33	can	can	AUX
ejpam-4498	201	34	be	be	AUX
ejpam-4498	201	35	expressed	express	VERB
ejpam-4498	201	36	in	in	ADP
ejpam-4498	201	37	various	various	ADJ
ejpam-4498	201	38	ways	way	NOUN
ejpam-4498	201	39	,	,	PUNCT
ejpam-4498	201	40	as	as	ADP
ejpam-4498	201	41	yukio	yukio	PROPN
ejpam-4498	201	42	kobayashi	kobayashi	PROPN
ejpam-4498	201	43	/	/	SYM
ejpam-4498	201	44	eur	eur	PROPN
ejpam-4498	201	45	.	.	PUNCT
ejpam-4498	202	1	j.	j.	PROPN
ejpam-4498	202	2	pure	pure	PROPN
ejpam-4498	202	3	appl	appl	PROPN
ejpam-4498	202	4	.	.	PROPN
ejpam-4498	202	5	math	math	PROPN
ejpam-4498	202	6	,	,	PUNCT
ejpam-4498	202	7	15	15	NUM
ejpam-4498	202	8	(	(	PUNCT
ejpam-4498	202	9	3	3	NUM
ejpam-4498	202	10	)	)	PUNCT
ejpam-4498	202	11	(	(	PUNCT
ejpam-4498	202	12	2022	2022	NUM
ejpam-4498	202	13	)	)	PUNCT
ejpam-4498	202	14	,	,	PUNCT
ejpam-4498	202	15	1237	1237	NUM
ejpam-4498	202	16	-	-	SYM
ejpam-4498	202	17	1253	1253	NUM
ejpam-4498	202	18	1247	1247	NUM
ejpam-4498	202	19	follows	follow	VERB
ejpam-4498	202	20	.	.	PUNCT
ejpam-4498	203	1	n∑	n∑	PUNCT
ejpam-4498	204	1	ℓ=1	ℓ=1	ADP
ejpam-4498	204	2	ℓ∑	ℓ∑	INTJ
ejpam-4498	204	3	k=1	k=1	X
ejpam-4498	204	4	k	k	X
ejpam-4498	205	1	=	=	PUNCT
ejpam-4498	205	2			NUM
ejpam-4498	205	3	1	1	NUM
ejpam-4498	205	4	3	3	NUM
ejpam-4498	205	5	(	(	PUNCT
ejpam-4498	205	6	n+	n+	NOUN
ejpam-4498	205	7	2)n+1c2	2)n+1c2	NUM
ejpam-4498	205	8	,	,	PUNCT
ejpam-4498	205	9	1	1	NUM
ejpam-4498	205	10	·	·	PUNCT
ejpam-4498	205	11	n+	n+	NUM
ejpam-4498	205	12	2	2	NUM
ejpam-4498	205	13	·	·	PUNCT
ejpam-4498	205	14	(	(	PUNCT
ejpam-4498	205	15	n−	n−	NOUN
ejpam-4498	205	16	1	1	NUM
ejpam-4498	205	17	)	)	PUNCT
ejpam-4498	205	18	+	+	CCONJ
ejpam-4498	205	19	3	3	NUM
ejpam-4498	205	20	·	·	PUNCT
ejpam-4498	205	21	(	(	PUNCT
ejpam-4498	205	22	n−	n−	NOUN
ejpam-4498	205	23	2	2	NUM
ejpam-4498	205	24	)	)	PUNCT
ejpam-4498	205	25	+	+	CCONJ
ejpam-4498	205	26	·	·	PUNCT
ejpam-4498	205	27	·	·	PUNCT
ejpam-4498	205	28	·	·	PUNCT
ejpam-4498	205	29	+	+	CCONJ
ejpam-4498	205	30	k	k	X
ejpam-4498	205	31	·	·	PUNCT
ejpam-4498	206	1	[	[	X
ejpam-4498	206	2	n−	n−	NOUN
ejpam-4498	206	3	(	(	PUNCT
ejpam-4498	206	4	k	k	NOUN
ejpam-4498	206	5	−	−	PROPN
ejpam-4498	206	6	1	1	NUM
ejpam-4498	206	7	)	)	PUNCT
ejpam-4498	206	8	]	]	PUNCT
ejpam-4498	206	9	+	+	CCONJ
ejpam-4498	206	10	·	·	PUNCT
ejpam-4498	206	11	·	·	PUNCT
ejpam-4498	206	12	·	·	PUNCT
ejpam-4498	206	13	+	+	NUM
ejpam-4498	206	14	n	n	CCONJ
ejpam-4498	206	15	·	·	SYM
ejpam-4498	206	16	1	1	NUM
ejpam-4498	206	17	,	,	PUNCT
ejpam-4498	206	18	n∑	n∑	NOUN
ejpam-4498	206	19	k=1	k=1	PROPN
ejpam-4498	206	20	k(n+	k(n+	PROPN
ejpam-4498	206	21	1−	1−	NUM
ejpam-4498	206	22	k	k	NOUN
ejpam-4498	206	23	)	)	PUNCT
ejpam-4498	206	24	,	,	PUNCT
ejpam-4498	206	25	(	(	PUNCT
ejpam-4498	206	26	n+	n+	NOUN
ejpam-4498	206	27	1	1	NUM
ejpam-4498	206	28	)	)	PUNCT
ejpam-4498	206	29	n∑	n∑	NOUN
ejpam-4498	206	30	k=1	k=1	PUNCT
ejpam-4498	207	1	k	k	X
ejpam-4498	208	1	−	−	PROPN
ejpam-4498	208	2	n∑	n∑	INTJ
ejpam-4498	208	3	k=1	k=1	PROPN
ejpam-4498	208	4	k2	k2	PROPN
ejpam-4498	208	5	,	,	PUNCT
ejpam-4498	208	6	(	(	PUNCT
ejpam-4498	208	7	n+	n+	X
ejpam-4498	208	8	1)n+1c2	1)n+1c2	NUM
ejpam-4498	208	9	−	−	NOUN
ejpam-4498	208	10	1	1	NUM
ejpam-4498	208	11	3	3	NUM
ejpam-4498	208	12	(	(	PUNCT
ejpam-4498	208	13	2n+	2n+	NUM
ejpam-4498	208	14	1)n+1c2	1)n+1c2	NUM
ejpam-4498	208	15	,	,	PUNCT
ejpam-4498	208	16	n+2c3	n+2c3	PROPN
ejpam-4498	208	17	,	,	PUNCT
ejpam-4498	208	18	n∑	n∑	NOUN
ejpam-4498	208	19	k=1	k=1	PROPN
ejpam-4498	208	20	k+1c2	k+1c2	PROPN
ejpam-4498	208	21	.	.	PUNCT
ejpam-4498	209	1	these	these	DET
ejpam-4498	209	2	equations	equation	NOUN
ejpam-4498	209	3	can	can	AUX
ejpam-4498	209	4	also	also	ADV
ejpam-4498	209	5	be	be	AUX
ejpam-4498	209	6	interpreted	interpret	VERB
ejpam-4498	209	7	in	in	ADP
ejpam-4498	209	8	various	various	ADJ
ejpam-4498	209	9	ways	way	NOUN
ejpam-4498	209	10	to	to	PART
ejpam-4498	209	11	express	express	VERB
ejpam-4498	209	12	n+2c3	n+2c3	PROPN
ejpam-4498	209	13	.	.	PUNCT
ejpam-4498	210	1	6.2	6.2	NUM
ejpam-4498	210	2	.	.	PUNCT
ejpam-4498	211	1	square	square	ADJ
ejpam-4498	211	2	numbers	number	NOUN
ejpam-4498	211	3	the	the	DET
ejpam-4498	211	4	sum	sum	NOUN
ejpam-4498	211	5	of	of	ADP
ejpam-4498	211	6	natural	natural	ADJ
ejpam-4498	211	7	numbers	number	NOUN
ejpam-4498	211	8	and	and	CCONJ
ejpam-4498	211	9	of	of	ADP
ejpam-4498	211	10	triangular	triangular	NOUN
ejpam-4498	211	11	numbers	number	NOUN
ejpam-4498	211	12	can	can	AUX
ejpam-4498	211	13	be	be	AUX
ejpam-4498	211	14	explained	explain	VERB
ejpam-4498	211	15	in	in	ADP
ejpam-4498	211	16	terms	term	NOUN
ejpam-4498	211	17	of	of	ADP
ejpam-4498	211	18	combinatorics	combinatoric	NOUN
ejpam-4498	211	19	;	;	PUNCT
ejpam-4498	211	20	thus	thus	ADV
ejpam-4498	211	21	,	,	PUNCT
ejpam-4498	211	22	(	(	PUNCT
ejpam-4498	211	23	20	20	NUM
ejpam-4498	211	24	)	)	PUNCT
ejpam-4498	211	25	suggests	suggest	VERB
ejpam-4498	211	26	that	that	SCONJ
ejpam-4498	211	27	the	the	DET
ejpam-4498	211	28	sum	sum	NOUN
ejpam-4498	211	29	of	of	ADP
ejpam-4498	211	30	square	square	ADJ
ejpam-4498	211	31	numbers	number	NOUN
ejpam-4498	211	32	can	can	AUX
ejpam-4498	211	33	also	also	ADV
ejpam-4498	211	34	be	be	AUX
ejpam-4498	211	35	implied	imply	VERB
ejpam-4498	211	36	by	by	ADP
ejpam-4498	211	37	combinatorics	combinatoric	NOUN
ejpam-4498	211	38	,	,	PUNCT
ejpam-4498	211	39	and	and	CCONJ
ejpam-4498	211	40	a	a	DET
ejpam-4498	211	41	diagram	diagram	NOUN
ejpam-4498	211	42	corresponding	correspond	VERB
ejpam-4498	211	43	to	to	PART
ejpam-4498	211	44	figure	figure	VERB
ejpam-4498	211	45	1	1	NUM
ejpam-4498	211	46	can	can	AUX
ejpam-4498	211	47	be	be	AUX
ejpam-4498	211	48	drawn	draw	VERB
ejpam-4498	211	49	as	as	SCONJ
ejpam-4498	211	50	shown	show	VERB
ejpam-4498	211	51	in	in	ADP
ejpam-4498	211	52	figure	figure	NOUN
ejpam-4498	211	53	7	7	NUM
ejpam-4498	211	54	.	.	PUNCT
ejpam-4498	211	55	instructors	instructor	NOUN
ejpam-4498	211	56	should	should	AUX
ejpam-4498	211	57	begin	begin	VERB
ejpam-4498	211	58	by	by	ADP
ejpam-4498	211	59	asking	ask	VERB
ejpam-4498	211	60	students	student	NOUN
ejpam-4498	211	61	about	about	ADP
ejpam-4498	211	62	the	the	DET
ejpam-4498	211	63	relationship	relationship	NOUN
ejpam-4498	211	64	between	between	ADP
ejpam-4498	211	65	triangular	triangular	NOUN
ejpam-4498	211	66	and	and	CCONJ
ejpam-4498	211	67	square	square	ADJ
ejpam-4498	211	68	numbers	number	NOUN
ejpam-4498	211	69	.	.	PUNCT
ejpam-4498	212	1	1	1	NUM
ejpam-4498	212	2	1	1	NUM
ejpam-4498	212	3	1	1	NUM
ejpam-4498	212	4	1	1	NUM
ejpam-4498	212	5	2	2	NUM
ejpam-4498	212	6	2	2	NUM
ejpam-4498	212	7	1	1	NUM
ejpam-4498	212	8	2	2	NUM
ejpam-4498	212	9	1	1	NUM
ejpam-4498	212	10	1	1	NUM
ejpam-4498	212	11	2	2	NUM
ejpam-4498	212	12	3	3	NUM
ejpam-4498	212	13	2	2	NUM
ejpam-4498	212	14	1	1	NUM
ejpam-4498	212	15	2	2	NUM
ejpam-4498	212	16	3	3	NUM
ejpam-4498	212	17	3	3	NUM
ejpam-4498	212	18	1	1	NUM
ejpam-4498	212	19	2	2	NUM
ejpam-4498	212	20	3	3	NUM
ejpam-4498	212	21	·	·	PUNCT
ejpam-4498	212	22	·	·	PUNCT
ejpam-4498	212	23	·	·	PUNCT
ejpam-4498	213	1	no	no	INTJ
ejpam-4498	213	2	.	.	PUNCT
ejpam-4498	213	3	of	of	ADP
ejpam-4498	213	4	edges	edge	NOUN
ejpam-4498	213	5	square	square	ADJ
ejpam-4498	213	6	number	number	NOUN
ejpam-4498	213	7	1×	1×	NUM
ejpam-4498	213	8	1	1	NUM
ejpam-4498	213	9	2×	2×	NUM
ejpam-4498	213	10	2	2	NUM
ejpam-4498	213	11	3×	3×	NUM
ejpam-4498	213	12	3	3	NUM
ejpam-4498	213	13	figure	figure	NOUN
ejpam-4498	213	14	7	7	NUM
ejpam-4498	213	15	:	:	PUNCT
ejpam-4498	213	16	pictorial	pictorial	ADJ
ejpam-4498	213	17	proof	proof	NOUN
ejpam-4498	213	18	of	of	ADP
ejpam-4498	213	19	the	the	DET
ejpam-4498	213	20	relationship	relationship	NOUN
ejpam-4498	213	21	between	between	ADP
ejpam-4498	213	22	triangular	triangular	NOUN
ejpam-4498	213	23	and	and	CCONJ
ejpam-4498	213	24	square	square	ADJ
ejpam-4498	213	25	numbers	number	NOUN
ejpam-4498	213	26	.	.	PUNCT
ejpam-4498	214	1	the	the	DET
ejpam-4498	214	2	numerals	numeral	NOUN
ejpam-4498	214	3	in	in	ADP
ejpam-4498	214	4	the	the	DET
ejpam-4498	214	5	tree	tree	NOUN
ejpam-4498	214	6	diagrams	diagram	NOUN
ejpam-4498	214	7	indicate	indicate	VERB
ejpam-4498	214	8	the	the	DET
ejpam-4498	214	9	number	number	NOUN
ejpam-4498	214	10	of	of	ADP
ejpam-4498	214	11	cards	card	NOUN
ejpam-4498	214	12	and	and	CCONJ
ejpam-4498	214	13	the	the	DET
ejpam-4498	214	14	edges	edge	NOUN
ejpam-4498	214	15	represent	represent	VERB
ejpam-4498	214	16	how	how	SCONJ
ejpam-4498	214	17	the	the	DET
ejpam-4498	214	18	two	two	NUM
ejpam-4498	214	19	numbered	numbered	ADJ
ejpam-4498	214	20	cards	card	NOUN
ejpam-4498	214	21	are	be	AUX
ejpam-4498	214	22	assembled	assemble	VERB
ejpam-4498	214	23	.	.	PUNCT
ejpam-4498	215	1	to	to	PART
ejpam-4498	215	2	show	show	VERB
ejpam-4498	215	3	that	that	SCONJ
ejpam-4498	215	4	square	square	ADJ
ejpam-4498	215	5	numbers	number	NOUN
ejpam-4498	215	6	are	be	AUX
ejpam-4498	215	7	the	the	DET
ejpam-4498	215	8	sum	sum	NOUN
ejpam-4498	215	9	of	of	ADP
ejpam-4498	215	10	two	two	NUM
ejpam-4498	215	11	consecutive	consecutive	ADJ
ejpam-4498	215	12	triangular	triangular	NOUN
ejpam-4498	215	13	numbers	number	NOUN
ejpam-4498	215	14	,	,	PUNCT
ejpam-4498	215	15	square	square	ADJ
ejpam-4498	215	16	numbers	number	NOUN
ejpam-4498	215	17	are	be	AUX
ejpam-4498	215	18	represented	represent	VERB
ejpam-4498	215	19	by	by	ADP
ejpam-4498	215	20	open	open	ADJ
ejpam-4498	215	21	or	or	CCONJ
ejpam-4498	215	22	filled	fill	VERB
ejpam-4498	215	23	circles	circle	NOUN
ejpam-4498	215	24	.	.	PUNCT
ejpam-4498	216	1	yukio	yukio	PROPN
ejpam-4498	216	2	kobayashi	kobayashi	PROPN
ejpam-4498	216	3	/	/	SYM
ejpam-4498	216	4	eur	eur	PROPN
ejpam-4498	216	5	.	.	PUNCT
ejpam-4498	217	1	j.	j.	PROPN
ejpam-4498	217	2	pure	pure	PROPN
ejpam-4498	217	3	appl	appl	PROPN
ejpam-4498	217	4	.	.	PROPN
ejpam-4498	217	5	math	math	PROPN
ejpam-4498	217	6	,	,	PUNCT
ejpam-4498	217	7	15	15	NUM
ejpam-4498	217	8	(	(	PUNCT
ejpam-4498	217	9	3	3	NUM
ejpam-4498	217	10	)	)	PUNCT
ejpam-4498	217	11	(	(	PUNCT
ejpam-4498	217	12	2022	2022	NUM
ejpam-4498	217	13	)	)	PUNCT
ejpam-4498	217	14	,	,	PUNCT
ejpam-4498	217	15	1237	1237	NUM
ejpam-4498	217	16	-	-	SYM
ejpam-4498	217	17	1253	1253	NUM
ejpam-4498	217	18	1248	1248	NUM
ejpam-4498	217	19	students	student	NOUN
ejpam-4498	217	20	should	should	AUX
ejpam-4498	217	21	notice	notice	VERB
ejpam-4498	217	22	the	the	DET
ejpam-4498	217	23	following	follow	VERB
ejpam-4498	217	24	additional	additional	ADJ
ejpam-4498	217	25	formulae	formulae	NOUN
ejpam-4498	217	26	from	from	ADP
ejpam-4498	217	27	figure	figure	NOUN
ejpam-4498	217	28	7	7	NUM
ejpam-4498	217	29	using	use	VERB
ejpam-4498	217	30	a	a	DET
ejpam-4498	217	31	heuristic	heuristic	ADJ
ejpam-4498	217	32	method	method	NOUN
ejpam-4498	217	33	.	.	PUNCT
ejpam-4498	218	1	equation	equation	NOUN
ejpam-4498	218	2	(	(	PUNCT
ejpam-4498	218	3	31	31	NUM
ejpam-4498	218	4	)	)	PUNCT
ejpam-4498	218	5	indicates	indicate	VERB
ejpam-4498	218	6	that	that	SCONJ
ejpam-4498	218	7	square	square	ADJ
ejpam-4498	218	8	numbers	number	NOUN
ejpam-4498	218	9	are	be	AUX
ejpam-4498	218	10	the	the	DET
ejpam-4498	218	11	sum	sum	NOUN
ejpam-4498	218	12	of	of	ADP
ejpam-4498	218	13	two	two	NUM
ejpam-4498	218	14	consecutive	consecutive	ADJ
ejpam-4498	218	15	triangular	triangular	NOUN
ejpam-4498	218	16	numbers	number	NOUN
ejpam-4498	218	17	.	.	PUNCT
ejpam-4498	219	1	12	12	NUM
ejpam-4498	219	2	=	=	SYM
ejpam-4498	219	3	1	1	NUM
ejpam-4498	219	4	.	.	NOUN
ejpam-4498	219	5	22	22	NUM
ejpam-4498	219	6	=	=	SYM
ejpam-4498	219	7	1	1	NUM
ejpam-4498	219	8	+	+	CCONJ
ejpam-4498	219	9	(	(	PUNCT
ejpam-4498	219	10	1	1	NUM
ejpam-4498	219	11	+	+	NUM
ejpam-4498	219	12	2	2	NUM
ejpam-4498	219	13	)	)	PUNCT
ejpam-4498	219	14	.	.	PUNCT
ejpam-4498	220	1	32	32	NUM
ejpam-4498	220	2	=	=	SYM
ejpam-4498	220	3	(	(	PUNCT
ejpam-4498	220	4	1	1	NUM
ejpam-4498	220	5	+	+	NUM
ejpam-4498	220	6	2	2	NUM
ejpam-4498	220	7	)	)	PUNCT
ejpam-4498	220	8	+	+	CCONJ
ejpam-4498	220	9	(	(	PUNCT
ejpam-4498	220	10	1	1	NUM
ejpam-4498	220	11	+	+	NUM
ejpam-4498	220	12	2	2	NUM
ejpam-4498	220	13	+	+	NUM
ejpam-4498	220	14	3	3	NUM
ejpam-4498	220	15	)	)	PUNCT
ejpam-4498	220	16	.	.	PUNCT
ejpam-4498	220	17	·	·	PUNCT
ejpam-4498	220	18	·	·	PUNCT
ejpam-4498	220	19	·	·	PUNCT
ejpam-4498	221	1			NOUN
ejpam-4498	221	2	(	(	PUNCT
ejpam-4498	221	3	31	31	NUM
ejpam-4498	221	4	)	)	PUNCT
ejpam-4498	221	5	the	the	DET
ejpam-4498	221	6	sum	sum	NOUN
ejpam-4498	221	7	of	of	ADP
ejpam-4498	221	8	square	square	ADJ
ejpam-4498	221	9	numbers	number	NOUN
ejpam-4498	221	10	can	can	AUX
ejpam-4498	221	11	be	be	AUX
ejpam-4498	221	12	written	write	VERB
ejpam-4498	221	13	as	as	ADP
ejpam-4498	221	14	the	the	DET
ejpam-4498	221	15	sum	sum	NOUN
ejpam-4498	221	16	of	of	ADP
ejpam-4498	221	17	triangular	triangular	NOUN
ejpam-4498	221	18	numbers	number	NOUN
ejpam-4498	221	19	by	by	ADP
ejpam-4498	221	20	counting	count	VERB
ejpam-4498	221	21	the	the	DET
ejpam-4498	221	22	number	number	NOUN
ejpam-4498	221	23	of	of	ADP
ejpam-4498	221	24	points	point	NOUN
ejpam-4498	221	25	drawn	draw	VERB
ejpam-4498	221	26	in	in	ADP
ejpam-4498	221	27	figure	figure	NOUN
ejpam-4498	221	28	7	7	NUM
ejpam-4498	221	29	,	,	PUNCT
ejpam-4498	221	30	as	as	SCONJ
ejpam-4498	221	31	shown	show	VERB
ejpam-4498	221	32	in	in	ADP
ejpam-4498	221	33	figure	figure	NOUN
ejpam-4498	221	34	8	8	NUM
ejpam-4498	221	35	.	.	NOUN
ejpam-4498	221	36	1	1	NUM
ejpam-4498	221	37	1	1	NUM
ejpam-4498	221	38	+	+	NUM
ejpam-4498	221	39	2	2	NUM
ejpam-4498	221	40	1	1	NUM
ejpam-4498	221	41	1	1	NUM
ejpam-4498	221	42	+	+	CCONJ
ejpam-4498	221	43	2	2	NUM
ejpam-4498	221	44	1	1	NUM
ejpam-4498	221	45	+	+	NUM
ejpam-4498	221	46	2	2	NUM
ejpam-4498	221	47	+	+	SYM
ejpam-4498	221	48	3	3	NUM
ejpam-4498	221	49	figure	figure	NOUN
ejpam-4498	221	50	8	8	NUM
ejpam-4498	221	51	:	:	PUNCT
ejpam-4498	221	52	visual	visual	ADJ
ejpam-4498	221	53	interpretation	interpretation	NOUN
ejpam-4498	221	54	of	of	ADP
ejpam-4498	221	55	the	the	DET
ejpam-4498	221	56	sum	sum	NOUN
ejpam-4498	221	57	of	of	ADP
ejpam-4498	221	58	square	square	ADJ
ejpam-4498	221	59	numbers	number	NOUN
ejpam-4498	221	60	.	.	PUNCT
ejpam-4498	222	1	generating	generate	VERB
ejpam-4498	222	2	12	12	NUM
ejpam-4498	222	3	+	+	CCONJ
ejpam-4498	222	4	22	22	NUM
ejpam-4498	223	1	+	+	CCONJ
ejpam-4498	223	2	32	32	NUM
ejpam-4498	223	3	=	=	SYM
ejpam-4498	223	4	2	2	NUM
ejpam-4498	223	5	·	·	PUNCT
ejpam-4498	224	1	[	[	X
ejpam-4498	224	2	1	1	NUM
ejpam-4498	224	3	+	+	CCONJ
ejpam-4498	224	4	(	(	PUNCT
ejpam-4498	224	5	1	1	NUM
ejpam-4498	224	6	+	+	NUM
ejpam-4498	224	7	2	2	NUM
ejpam-4498	224	8	)	)	PUNCT
ejpam-4498	224	9	]	]	PUNCT
ejpam-4498	225	1	+	+	CCONJ
ejpam-4498	225	2	(	(	PUNCT
ejpam-4498	225	3	1	1	NUM
ejpam-4498	225	4	+	+	NUM
ejpam-4498	225	5	2	2	NUM
ejpam-4498	225	6	+	+	NUM
ejpam-4498	225	7	3	3	NUM
ejpam-4498	225	8	)	)	PUNCT
ejpam-4498	225	9	,	,	PUNCT
ejpam-4498	225	10	(	(	PUNCT
ejpam-4498	225	11	32	32	NUM
ejpam-4498	225	12	)	)	PUNCT
ejpam-4498	225	13	we	we	PRON
ejpam-4498	225	14	obtain	obtain	VERB
ejpam-4498	225	15	12	12	NUM
ejpam-4498	225	16	+	+	CCONJ
ejpam-4498	225	17	22	22	NUM
ejpam-4498	225	18	+	+	CCONJ
ejpam-4498	225	19	·	·	PUNCT
ejpam-4498	225	20	·	·	PUNCT
ejpam-4498	225	21	·	·	PUNCT
ejpam-4498	225	22	+	+	NUM
ejpam-4498	225	23	n2	n2	ADJ
ejpam-4498	225	24	=	=	SYM
ejpam-4498	225	25	2	2	NUM
ejpam-4498	225	26	·	·	PUNCT
ejpam-4498	225	27	{	{	PUNCT
ejpam-4498	225	28	1	1	NUM
ejpam-4498	225	29	+	+	CCONJ
ejpam-4498	225	30	(	(	PUNCT
ejpam-4498	225	31	1	1	NUM
ejpam-4498	225	32	+	+	NUM
ejpam-4498	225	33	2	2	NUM
ejpam-4498	225	34	)	)	PUNCT
ejpam-4498	225	35	+	+	NUM
ejpam-4498	225	36	·	·	PUNCT
ejpam-4498	225	37	·	·	PUNCT
ejpam-4498	225	38	·	·	PUNCT
ejpam-4498	226	1	+	+	PUNCT
ejpam-4498	227	1	[	[	X
ejpam-4498	227	2	1	1	NUM
ejpam-4498	227	3	+	+	NUM
ejpam-4498	227	4	2	2	NUM
ejpam-4498	227	5	+	+	CCONJ
ejpam-4498	227	6	·	·	PUNCT
ejpam-4498	227	7	·	·	PUNCT
ejpam-4498	227	8	·	·	PUNCT
ejpam-4498	227	9	+	+	CCONJ
ejpam-4498	227	10	(	(	PUNCT
ejpam-4498	227	11	n−	n−	NOUN
ejpam-4498	227	12	1	1	NUM
ejpam-4498	227	13	)	)	PUNCT
ejpam-4498	227	14	]	]	PUNCT
ejpam-4498	227	15	}	}	PUNCT
ejpam-4498	227	16	+	+	ADJ
ejpam-4498	227	17	(	(	PUNCT
ejpam-4498	227	18	1	1	NUM
ejpam-4498	227	19	+	+	NUM
ejpam-4498	227	20	2	2	NUM
ejpam-4498	227	21	+	+	CCONJ
ejpam-4498	227	22	·	·	PUNCT
ejpam-4498	227	23	·	·	PUNCT
ejpam-4498	227	24	·	·	PUNCT
ejpam-4498	227	25	+	+	NUM
ejpam-4498	227	26	n	n	CCONJ
ejpam-4498	227	27	)	)	PUNCT
ejpam-4498	227	28	=	=	SYM
ejpam-4498	227	29	2n+2c3	2n+2c3	NUM
ejpam-4498	227	30	−	−	PROPN
ejpam-4498	227	31	n+1c2	n+1c2	PROPN
ejpam-4498	227	32	.	.	PUNCT
ejpam-4498	228	1	(	(	PUNCT
ejpam-4498	228	2	33	33	NUM
ejpam-4498	228	3	)	)	PUNCT
ejpam-4498	228	4	using	use	VERB
ejpam-4498	228	5	the	the	DET
ejpam-4498	228	6	expression	expression	NOUN
ejpam-4498	228	7	for	for	ADP
ejpam-4498	228	8	the	the	DET
ejpam-4498	228	9	sum	sum	NOUN
ejpam-4498	228	10	of	of	ADP
ejpam-4498	228	11	triangular	triangular	NOUN
ejpam-4498	228	12	numbers	number	NOUN
ejpam-4498	228	13	in	in	ADP
ejpam-4498	228	14	section	section	NOUN
ejpam-4498	228	15	6.1	6.1	NUM
ejpam-4498	228	16	,	,	PUNCT
ejpam-4498	228	17	instructors	instructor	NOUN
ejpam-4498	228	18	should	should	AUX
ejpam-4498	228	19	have	have	VERB
ejpam-4498	228	20	students	student	NOUN
ejpam-4498	228	21	verify	verify	VERB
ejpam-4498	228	22	that	that	SCONJ
ejpam-4498	228	23	(	(	PUNCT
ejpam-4498	228	24	33	33	NUM
ejpam-4498	228	25	)	)	PUNCT
ejpam-4498	228	26	agrees	agree	VERB
ejpam-4498	228	27	with	with	ADP
ejpam-4498	228	28	1	1	NUM
ejpam-4498	228	29	6	6	NUM
ejpam-4498	228	30	n(n+	n(n+	NOUN
ejpam-4498	228	31	1)(2n+	1)(2n+	NUM
ejpam-4498	228	32	1	1	NUM
ejpam-4498	228	33	)	)	PUNCT
ejpam-4498	228	34	.	.	PUNCT
ejpam-4498	229	1	considering	consider	VERB
ejpam-4498	229	2	table	table	NOUN
ejpam-4498	229	3	1	1	NUM
ejpam-4498	229	4	,	,	PUNCT
ejpam-4498	229	5	the	the	DET
ejpam-4498	229	6	sum	sum	NOUN
ejpam-4498	229	7	of	of	ADP
ejpam-4498	229	8	triangular	triangular	NOUN
ejpam-4498	229	9	numbers	number	NOUN
ejpam-4498	229	10	can	can	AUX
ejpam-4498	229	11	be	be	AUX
ejpam-4498	229	12	regarded	regard	VERB
ejpam-4498	229	13	as	as	ADP
ejpam-4498	229	14	an	an	DET
ejpam-4498	229	15	extended	extended	ADJ
ejpam-4498	229	16	version	version	NOUN
ejpam-4498	229	17	of	of	ADP
ejpam-4498	229	18	the	the	DET
ejpam-4498	229	19	sum	sum	NOUN
ejpam-4498	229	20	of	of	ADP
ejpam-4498	229	21	natural	natural	ADJ
ejpam-4498	229	22	numbers	number	NOUN
ejpam-4498	229	23	,	,	PUNCT
ejpam-4498	229	24	and	and	CCONJ
ejpam-4498	229	25	the	the	DET
ejpam-4498	229	26	sum	sum	NOUN
ejpam-4498	229	27	of	of	ADP
ejpam-4498	229	28	square	square	ADJ
ejpam-4498	229	29	numbers	number	NOUN
ejpam-4498	229	30	is	be	AUX
ejpam-4498	229	31	related	relate	VERB
ejpam-4498	229	32	to	to	ADP
ejpam-4498	229	33	the	the	DET
ejpam-4498	229	34	sum	sum	NOUN
ejpam-4498	229	35	of	of	ADP
ejpam-4498	229	36	both	both	DET
ejpam-4498	229	37	triangular	triangular	NOUN
ejpam-4498	229	38	numbers	number	NOUN
ejpam-4498	229	39	and	and	CCONJ
ejpam-4498	229	40	natural	natural	ADJ
ejpam-4498	229	41	numbers	number	NOUN
ejpam-4498	229	42	.	.	PUNCT
ejpam-4498	230	1	table	table	NOUN
ejpam-4498	230	2	1	1	NUM
ejpam-4498	230	3	.	.	PUNCT
ejpam-4498	230	4	comparison	comparison	NOUN
ejpam-4498	230	5	between	between	ADP
ejpam-4498	230	6	the	the	DET
ejpam-4498	230	7	sums	sum	NOUN
ejpam-4498	230	8	of	of	ADP
ejpam-4498	230	9	triangular	triangular	NOUN
ejpam-4498	230	10	numbers	number	NOUN
ejpam-4498	230	11	,	,	PUNCT
ejpam-4498	230	12	natural	natural	ADJ
ejpam-4498	230	13	numbers	number	NOUN
ejpam-4498	230	14	,	,	PUNCT
ejpam-4498	230	15	and	and	CCONJ
ejpam-4498	230	16	square	square	ADJ
ejpam-4498	230	17	numbers	number	NOUN
ejpam-4498	230	18	.	.	PUNCT
ejpam-4498	231	1	1	1	NUM
ejpam-4498	232	1	+	+	NUM
ejpam-4498	232	2	2	2	NUM
ejpam-4498	232	3	+	+	CCONJ
ejpam-4498	232	4	·	·	PUNCT
ejpam-4498	232	5	·	·	PUNCT
ejpam-4498	232	6	·	·	PUNCT
ejpam-4498	232	7	+	+	NUM
ejpam-4498	232	8	n	n	PROPN
ejpam-4498	232	9	=	=	SYM
ejpam-4498	232	10	n+1c2	n+1c2	PROPN
ejpam-4498	232	11	.	.	PROPN
ejpam-4498	232	12	1	1	NUM
ejpam-4498	232	13	·	·	PUNCT
ejpam-4498	232	14	n+	n+	NUM
ejpam-4498	233	1	2	2	NUM
ejpam-4498	233	2	·	·	PUNCT
ejpam-4498	233	3	(	(	PUNCT
ejpam-4498	233	4	n−	n−	NOUN
ejpam-4498	233	5	1	1	NUM
ejpam-4498	233	6	)	)	PUNCT
ejpam-4498	233	7	+	+	CCONJ
ejpam-4498	233	8	·	·	PUNCT
ejpam-4498	233	9	·	·	PUNCT
ejpam-4498	233	10	·	·	PUNCT
ejpam-4498	233	11	+	+	NUM
ejpam-4498	233	12	n	n	CCONJ
ejpam-4498	233	13	·	·	SYM
ejpam-4498	233	14	1	1	NUM
ejpam-4498	233	15	=	=	SYM
ejpam-4498	233	16	n+2c3	n+2c3	PROPN
ejpam-4498	233	17	.	.	PUNCT
ejpam-4498	234	1	12	12	NUM
ejpam-4498	235	1	+	+	NUM
ejpam-4498	235	2	22	22	NUM
ejpam-4498	235	3	+	+	CCONJ
ejpam-4498	235	4	·	·	PUNCT
ejpam-4498	235	5	·	·	PUNCT
ejpam-4498	235	6	·	·	PUNCT
ejpam-4498	235	7	+	+	NUM
ejpam-4498	235	8	n2	n2	ADJ
ejpam-4498	235	9	=	=	SYM
ejpam-4498	235	10	2	2	NUM
ejpam-4498	235	11	n+1c3	n+1c3	PROPN
ejpam-4498	235	12	+	+	CCONJ
ejpam-4498	235	13	n+1c2	n+1c2	PROPN
ejpam-4498	235	14	.	.	PUNCT
ejpam-4498	235	15	note	note	VERB
ejpam-4498	235	16	the	the	DET
ejpam-4498	235	17	difference	difference	NOUN
ejpam-4498	235	18	between	between	ADP
ejpam-4498	235	19	n+2c3	n+2c3	PROPN
ejpam-4498	235	20	and	and	CCONJ
ejpam-4498	235	21	n+1c3	n+1c3	PROPN
ejpam-4498	235	22	.	.	PUNCT
ejpam-4498	236	1	yukio	yukio	PROPN
ejpam-4498	236	2	kobayashi	kobayashi	PROPN
ejpam-4498	236	3	/	/	SYM
ejpam-4498	236	4	eur	eur	PROPN
ejpam-4498	236	5	.	.	PUNCT
ejpam-4498	237	1	j.	j.	PROPN
ejpam-4498	237	2	pure	pure	PROPN
ejpam-4498	237	3	appl	appl	PROPN
ejpam-4498	237	4	.	.	PROPN
ejpam-4498	237	5	math	math	PROPN
ejpam-4498	237	6	,	,	PUNCT
ejpam-4498	237	7	15	15	NUM
ejpam-4498	237	8	(	(	PUNCT
ejpam-4498	237	9	3	3	NUM
ejpam-4498	237	10	)	)	PUNCT
ejpam-4498	237	11	(	(	PUNCT
ejpam-4498	237	12	2022	2022	NUM
ejpam-4498	237	13	)	)	PUNCT
ejpam-4498	237	14	,	,	PUNCT
ejpam-4498	237	15	1237	1237	NUM
ejpam-4498	237	16	-	-	SYM
ejpam-4498	237	17	1253	1253	NUM
ejpam-4498	237	18	1249	1249	NUM
ejpam-4498	237	19	7	7	NUM
ejpam-4498	237	20	.	.	PUNCT
ejpam-4498	237	21	related	relate	VERB
ejpam-4498	237	22	problems	problem	NOUN
ejpam-4498	237	23	triangular	triangular	NOUN
ejpam-4498	237	24	numbers	number	NOUN
ejpam-4498	237	25	are	be	AUX
ejpam-4498	237	26	fundamental	fundamental	ADJ
ejpam-4498	237	27	numbers	number	NOUN
ejpam-4498	237	28	in	in	ADP
ejpam-4498	237	29	the	the	DET
ejpam-4498	237	30	field	field	NOUN
ejpam-4498	237	31	of	of	ADP
ejpam-4498	237	32	combinations	combination	NOUN
ejpam-4498	237	33	.	.	PUNCT
ejpam-4498	238	1	the	the	DET
ejpam-4498	238	2	combination	combination	NOUN
ejpam-4498	238	3	nck	nck	NOUN
ejpam-4498	238	4	can	can	AUX
ejpam-4498	238	5	be	be	AUX
ejpam-4498	238	6	expressed	express	VERB
ejpam-4498	238	7	using	use	VERB
ejpam-4498	238	8	triangular	triangular	NOUN
ejpam-4498	238	9	numbers	number	NOUN
ejpam-4498	238	10	,	,	PUNCT
ejpam-4498	238	11	as	as	SCONJ
ejpam-4498	238	12	shown	show	VERB
ejpam-4498	238	13	in	in	ADP
ejpam-4498	238	14	table	table	NOUN
ejpam-4498	238	15	2	2	NUM
ejpam-4498	238	16	.	.	PUNCT
ejpam-4498	239	1	the	the	DET
ejpam-4498	239	2	additional	additional	ADJ
ejpam-4498	239	3	formulae	formulae	NOUN
ejpam-4498	239	4	for	for	ADP
ejpam-4498	239	5	nck	nck	NOUN
ejpam-4498	239	6	indicate	indicate	VERB
ejpam-4498	239	7	the	the	DET
ejpam-4498	239	8	total	total	ADJ
ejpam-4498	239	9	number	number	NOUN
ejpam-4498	239	10	of	of	ADP
ejpam-4498	239	11	edges	edge	NOUN
ejpam-4498	239	12	in	in	ADP
ejpam-4498	239	13	the	the	DET
ejpam-4498	239	14	tree	tree	NOUN
ejpam-4498	239	15	diagrams	diagram	NOUN
ejpam-4498	239	16	for	for	ADP
ejpam-4498	239	17	nck	nck	NOUN
ejpam-4498	239	18	.	.	PUNCT
ejpam-4498	240	1	as	as	SCONJ
ejpam-4498	240	2	shown	show	VERB
ejpam-4498	240	3	in	in	ADP
ejpam-4498	240	4	sections	section	NOUN
ejpam-4498	240	5	2	2	NUM
ejpam-4498	240	6	and	and	CCONJ
ejpam-4498	240	7	6.1	6.1	NUM
ejpam-4498	240	8	,	,	PUNCT
ejpam-4498	240	9	nc2s	nc2s	PROPN
ejpam-4498	240	10	are	be	AUX
ejpam-4498	240	11	triangular	triangular	NOUN
ejpam-4498	240	12	numbers	number	NOUN
ejpam-4498	240	13	and	and	CCONJ
ejpam-4498	240	14	nc3s	nc3s	NOUN
ejpam-4498	240	15	are	be	AUX
ejpam-4498	240	16	the	the	DET
ejpam-4498	240	17	sums	sum	NOUN
ejpam-4498	240	18	of	of	ADP
ejpam-4498	240	19	triangular	triangular	NOUN
ejpam-4498	240	20	numbers	number	NOUN
ejpam-4498	240	21	.	.	PUNCT
ejpam-4498	241	1	instructors	instructor	NOUN
ejpam-4498	241	2	should	should	AUX
ejpam-4498	241	3	invite	invite	VERB
ejpam-4498	241	4	students	student	NOUN
ejpam-4498	241	5	discover	discover	VERB
ejpam-4498	241	6	the	the	DET
ejpam-4498	241	7	relationship	relationship	NOUN
ejpam-4498	241	8	between	between	ADP
ejpam-4498	241	9	ncks	nck	NOUN
ejpam-4498	241	10	and	and	CCONJ
ejpam-4498	241	11	the	the	DET
ejpam-4498	241	12	sum	sum	NOUN
ejpam-4498	241	13	of	of	ADP
ejpam-4498	241	14	triangular	triangular	NOUN
ejpam-4498	241	15	numbers	number	NOUN
ejpam-4498	241	16	using	use	VERB
ejpam-4498	241	17	a	a	DET
ejpam-4498	241	18	heuristic	heuristic	ADJ
ejpam-4498	241	19	method	method	NOUN
ejpam-4498	241	20	,	,	PUNCT
ejpam-4498	241	21	as	as	SCONJ
ejpam-4498	241	22	shown	show	VERB
ejpam-4498	241	23	in	in	ADP
ejpam-4498	241	24	table	table	NOUN
ejpam-4498	241	25	2	2	NUM
ejpam-4498	241	26	.	.	PUNCT
ejpam-4498	242	1	for	for	ADP
ejpam-4498	242	2	example	example	NOUN
ejpam-4498	242	3	,	,	PUNCT
ejpam-4498	242	4	for	for	ADP
ejpam-4498	242	5	nc4	nc4	NOUN
ejpam-4498	242	6	,	,	PUNCT
ejpam-4498	242	7	6c4	6c4	NUM
ejpam-4498	242	8	=	=	SYM
ejpam-4498	242	9	1	1	NUM
ejpam-4498	242	10	+	+	NUM
ejpam-4498	242	11	4	4	NUM
ejpam-4498	242	12	+	+	SYM
ejpam-4498	242	13	10	10	NUM
ejpam-4498	242	14	=	=	SYM
ejpam-4498	242	15	1	1	NUM
ejpam-4498	242	16	+	+	CCONJ
ejpam-4498	242	17	(	(	PUNCT
ejpam-4498	242	18	1	1	NUM
ejpam-4498	242	19	+	+	NUM
ejpam-4498	242	20	3	3	NUM
ejpam-4498	242	21	)	)	PUNCT
ejpam-4498	242	22	+	+	CCONJ
ejpam-4498	242	23	(	(	PUNCT
ejpam-4498	242	24	1	1	NUM
ejpam-4498	242	25	+	+	NUM
ejpam-4498	242	26	3	3	NUM
ejpam-4498	242	27	+	+	NUM
ejpam-4498	242	28	6	6	NUM
ejpam-4498	242	29	)	)	PUNCT
ejpam-4498	242	30	=	=	SYM
ejpam-4498	242	31	1	1	NUM
ejpam-4498	243	1	+	+	CCONJ
ejpam-4498	243	2	[	[	PUNCT
ejpam-4498	243	3	1	1	NUM
ejpam-4498	243	4	+	+	CCONJ
ejpam-4498	243	5	(	(	PUNCT
ejpam-4498	243	6	1	1	NUM
ejpam-4498	243	7	+	+	NUM
ejpam-4498	243	8	2	2	NUM
ejpam-4498	243	9	)	)	PUNCT
ejpam-4498	243	10	]	]	PUNCT
ejpam-4498	244	1	+	+	CCONJ
ejpam-4498	244	2	[	[	X
ejpam-4498	244	3	1	1	NUM
ejpam-4498	244	4	+	+	CCONJ
ejpam-4498	244	5	(	(	PUNCT
ejpam-4498	244	6	1	1	NUM
ejpam-4498	244	7	+	+	NUM
ejpam-4498	244	8	2	2	NUM
ejpam-4498	244	9	)	)	PUNCT
ejpam-4498	244	10	+	+	CCONJ
ejpam-4498	244	11	(	(	PUNCT
ejpam-4498	244	12	1	1	NUM
ejpam-4498	244	13	+	+	NUM
ejpam-4498	244	14	2	2	NUM
ejpam-4498	244	15	+	+	NUM
ejpam-4498	244	16	3	3	NUM
ejpam-4498	244	17	)	)	PUNCT
ejpam-4498	244	18	]	]	PUNCT
ejpam-4498	244	19	=	=	PUNCT
ejpam-4498	244	20	3c3	3c3	NUM
ejpam-4498	244	21	+	+	NUM
ejpam-4498	244	22	4c3	4c3	NUM
ejpam-4498	245	1	+	+	CCONJ
ejpam-4498	245	2	5c3	5c3	NUM
ejpam-4498	245	3	,	,	PUNCT
ejpam-4498	245	4	(	(	PUNCT
ejpam-4498	245	5	34	34	NUM
ejpam-4498	245	6	)	)	PUNCT
ejpam-4498	245	7	which	which	PRON
ejpam-4498	245	8	is	be	AUX
ejpam-4498	245	9	also	also	ADV
ejpam-4498	245	10	the	the	DET
ejpam-4498	245	11	sum	sum	NOUN
ejpam-4498	245	12	of	of	ADP
ejpam-4498	245	13	the	the	DET
ejpam-4498	245	14	sums	sum	NOUN
ejpam-4498	245	15	of	of	ADP
ejpam-4498	245	16	triangular	triangular	NOUN
ejpam-4498	245	17	numbers	number	NOUN
ejpam-4498	245	18	.	.	PUNCT
ejpam-4498	246	1	for	for	ADP
ejpam-4498	246	2	example	example	NOUN
ejpam-4498	246	3	,	,	PUNCT
ejpam-4498	246	4	for	for	ADP
ejpam-4498	246	5	nc5	nc5	NOUN
ejpam-4498	246	6	,	,	PUNCT
ejpam-4498	246	7	8c5	8c5	NUM
ejpam-4498	246	8	=	=	SYM
ejpam-4498	246	9	1	1	NUM
ejpam-4498	246	10	+	+	NUM
ejpam-4498	246	11	5	5	NUM
ejpam-4498	246	12	+	+	SYM
ejpam-4498	246	13	15	15	NUM
ejpam-4498	246	14	+	+	NUM
ejpam-4498	246	15	35	35	NUM
ejpam-4498	246	16	=	=	SYM
ejpam-4498	246	17	1	1	NUM
ejpam-4498	246	18	+	+	CCONJ
ejpam-4498	246	19	(	(	PUNCT
ejpam-4498	246	20	1	1	NUM
ejpam-4498	246	21	+	+	CCONJ
ejpam-4498	246	22	4	4	NUM
ejpam-4498	246	23	)	)	PUNCT
ejpam-4498	246	24	+	+	CCONJ
ejpam-4498	246	25	(	(	PUNCT
ejpam-4498	246	26	1	1	NUM
ejpam-4498	246	27	+	+	SYM
ejpam-4498	246	28	4	4	NUM
ejpam-4498	246	29	+	+	SYM
ejpam-4498	246	30	10	10	NUM
ejpam-4498	246	31	)	)	PUNCT
ejpam-4498	246	32	+	+	CCONJ
ejpam-4498	246	33	(	(	PUNCT
ejpam-4498	246	34	1	1	NUM
ejpam-4498	246	35	+	+	SYM
ejpam-4498	246	36	4	4	NUM
ejpam-4498	246	37	+	+	SYM
ejpam-4498	246	38	10	10	NUM
ejpam-4498	246	39	+	+	NUM
ejpam-4498	246	40	20	20	NUM
ejpam-4498	246	41	)	)	PUNCT
ejpam-4498	246	42	=	=	SYM
ejpam-4498	246	43	1	1	NUM
ejpam-4498	247	1	+	+	CCONJ
ejpam-4498	247	2	[	[	PUNCT
ejpam-4498	247	3	1	1	NUM
ejpam-4498	247	4	+	+	CCONJ
ejpam-4498	247	5	(	(	PUNCT
ejpam-4498	247	6	1	1	NUM
ejpam-4498	247	7	+	+	NUM
ejpam-4498	247	8	3	3	NUM
ejpam-4498	247	9	)	)	PUNCT
ejpam-4498	247	10	]	]	PUNCT
ejpam-4498	248	1	+	+	CCONJ
ejpam-4498	248	2	[	[	X
ejpam-4498	248	3	1	1	NUM
ejpam-4498	248	4	+	+	CCONJ
ejpam-4498	248	5	(	(	PUNCT
ejpam-4498	248	6	1	1	NUM
ejpam-4498	248	7	+	+	NUM
ejpam-4498	248	8	3	3	NUM
ejpam-4498	248	9	)	)	PUNCT
ejpam-4498	248	10	+	+	CCONJ
ejpam-4498	248	11	(	(	PUNCT
ejpam-4498	248	12	1	1	NUM
ejpam-4498	248	13	+	+	NUM
ejpam-4498	248	14	3	3	NUM
ejpam-4498	248	15	+	+	NUM
ejpam-4498	248	16	6	6	NUM
ejpam-4498	248	17	)	)	PUNCT
ejpam-4498	248	18	]	]	PUNCT
ejpam-4498	249	1	+	+	CCONJ
ejpam-4498	249	2	[	[	X
ejpam-4498	249	3	1	1	NUM
ejpam-4498	249	4	+	+	CCONJ
ejpam-4498	249	5	(	(	PUNCT
ejpam-4498	249	6	1	1	NUM
ejpam-4498	249	7	+	+	NUM
ejpam-4498	249	8	3	3	NUM
ejpam-4498	249	9	)	)	PUNCT
ejpam-4498	249	10	+	+	CCONJ
ejpam-4498	249	11	(	(	PUNCT
ejpam-4498	249	12	1	1	NUM
ejpam-4498	249	13	+	+	NUM
ejpam-4498	249	14	3	3	NUM
ejpam-4498	249	15	+	+	CCONJ
ejpam-4498	249	16	6	6	NUM
ejpam-4498	249	17	)	)	PUNCT
ejpam-4498	249	18	+	+	CCONJ
ejpam-4498	249	19	(	(	PUNCT
ejpam-4498	249	20	1	1	NUM
ejpam-4498	249	21	+	+	NUM
ejpam-4498	249	22	3	3	NUM
ejpam-4498	249	23	+	+	SYM
ejpam-4498	249	24	6	6	NUM
ejpam-4498	249	25	+	+	SYM
ejpam-4498	249	26	10	10	NUM
ejpam-4498	249	27	)	)	PUNCT
ejpam-4498	249	28	]	]	PUNCT
ejpam-4498	250	1	=	=	SYM
ejpam-4498	250	2	4c4	4c4	NUM
ejpam-4498	251	1	+	+	CCONJ
ejpam-4498	251	2	5c4	5c4	NUM
ejpam-4498	251	3	+	+	CCONJ
ejpam-4498	251	4	6c4	6c4	NUM
ejpam-4498	251	5	+	+	NUM
ejpam-4498	251	6	7c4	7c4	NUM
ejpam-4498	251	7	=	=	SYM
ejpam-4498	251	8	3c3	3c3	NUM
ejpam-4498	252	1	+	+	CCONJ
ejpam-4498	252	2	(	(	PUNCT
ejpam-4498	252	3	3c3	3c3	NUM
ejpam-4498	252	4	+	+	NUM
ejpam-4498	252	5	4c3	4c3	NUM
ejpam-4498	252	6	)	)	PUNCT
ejpam-4498	253	1	+	+	CCONJ
ejpam-4498	253	2	(	(	PUNCT
ejpam-4498	253	3	3c3	3c3	NUM
ejpam-4498	253	4	+	+	NUM
ejpam-4498	253	5	4c3	4c3	NUM
ejpam-4498	253	6	+	+	CCONJ
ejpam-4498	253	7	5c3	5c3	NUM
ejpam-4498	253	8	)	)	PUNCT
ejpam-4498	254	1	+	+	CCONJ
ejpam-4498	254	2	(	(	PUNCT
ejpam-4498	254	3	3c3	3c3	NUM
ejpam-4498	254	4	+	+	NUM
ejpam-4498	254	5	4c3	4c3	NUM
ejpam-4498	255	1	+	+	CCONJ
ejpam-4498	255	2	5c3	5c3	NUM
ejpam-4498	255	3	+	+	CCONJ
ejpam-4498	255	4	6c3	6c3	NUM
ejpam-4498	255	5	)	)	PUNCT
ejpam-4498	255	6	,	,	PUNCT
ejpam-4498	255	7	(	(	PUNCT
ejpam-4498	255	8	35	35	NUM
ejpam-4498	255	9	)	)	PUNCT
ejpam-4498	255	10	which	which	PRON
ejpam-4498	255	11	is	be	AUX
ejpam-4498	255	12	also	also	ADV
ejpam-4498	255	13	the	the	DET
ejpam-4498	255	14	sum	sum	NOUN
ejpam-4498	255	15	of	of	ADP
ejpam-4498	255	16	the	the	DET
ejpam-4498	255	17	sums	sum	NOUN
ejpam-4498	255	18	of	of	ADP
ejpam-4498	255	19	triangular	triangular	NOUN
ejpam-4498	255	20	numbers	number	NOUN
ejpam-4498	255	21	.	.	PUNCT
ejpam-4498	256	1	corresponding	correspond	VERB
ejpam-4498	256	2	to	to	ADP
ejpam-4498	256	3	(	(	PUNCT
ejpam-4498	256	4	30	30	NUM
ejpam-4498	256	5	)	)	PUNCT
ejpam-4498	256	6	,	,	PUNCT
ejpam-4498	256	7	the	the	DET
ejpam-4498	256	8	following	follow	VERB
ejpam-4498	256	9	relationship	relationship	NOUN
ejpam-4498	256	10	holds	hold	VERB
ejpam-4498	256	11	.	.	PUNCT
ejpam-4498	257	1	k−1∑	k−1∑	PROPN
ejpam-4498	257	2	ℓ=1	ℓ=1	VERB
ejpam-4498	257	3	n−ℓck−1	n−ℓck−1	PROPN
ejpam-4498	257	4	=	=	SYM
ejpam-4498	257	5	nck	nck	PROPN
ejpam-4498	257	6	.	.	PUNCT
ejpam-4498	258	1	(	(	PUNCT
ejpam-4498	258	2	36	36	NUM
ejpam-4498	258	3	)	)	PUNCT
ejpam-4498	258	4	yukio	yukio	PROPN
ejpam-4498	258	5	kobayashi	kobayashi	PROPN
ejpam-4498	258	6	/	/	SYM
ejpam-4498	258	7	eur	eur	PROPN
ejpam-4498	258	8	.	.	PUNCT
ejpam-4498	259	1	j.	j.	PROPN
ejpam-4498	259	2	pure	pure	PROPN
ejpam-4498	259	3	appl	appl	PROPN
ejpam-4498	259	4	.	.	PROPN
ejpam-4498	259	5	math	math	PROPN
ejpam-4498	259	6	,	,	PUNCT
ejpam-4498	259	7	15	15	NUM
ejpam-4498	259	8	(	(	PUNCT
ejpam-4498	259	9	3	3	NUM
ejpam-4498	259	10	)	)	PUNCT
ejpam-4498	259	11	(	(	PUNCT
ejpam-4498	259	12	2022	2022	NUM
ejpam-4498	259	13	)	)	PUNCT
ejpam-4498	259	14	,	,	PUNCT
ejpam-4498	259	15	1237	1237	NUM
ejpam-4498	259	16	-	-	SYM
ejpam-4498	259	17	1253	1253	NUM
ejpam-4498	259	18	1250	1250	NUM
ejpam-4498	259	19	table	table	NOUN
ejpam-4498	259	20	2	2	NUM
ejpam-4498	259	21	.	.	PUNCT
ejpam-4498	260	1	addition	addition	NOUN
ejpam-4498	260	2	formulae	formulae	NOUN
ejpam-4498	260	3	for	for	ADP
ejpam-4498	260	4	combinations	combination	NOUN
ejpam-4498	260	5	.	.	PUNCT
ejpam-4498	261	1	2c2	2c2	NUM
ejpam-4498	262	1	=	=	SYM
ejpam-4498	262	2	1	1	NUM
ejpam-4498	262	3	.	.	X
ejpam-4498	263	1	3c2	3c2	NUM
ejpam-4498	263	2	=	=	SYM
ejpam-4498	263	3	1	1	NUM
ejpam-4498	263	4	+	+	NUM
ejpam-4498	263	5	2	2	NUM
ejpam-4498	263	6	.	.	X
ejpam-4498	263	7	4c2	4c2	NUM
ejpam-4498	263	8	=	=	SYM
ejpam-4498	263	9	1	1	NUM
ejpam-4498	264	1	+	+	NUM
ejpam-4498	264	2	2	2	NUM
ejpam-4498	264	3	+	+	SYM
ejpam-4498	264	4	3	3	NUM
ejpam-4498	264	5	.	.	NOUN
ejpam-4498	264	6	5c2	5c2	NUM
ejpam-4498	265	1	=	=	SYM
ejpam-4498	265	2	1	1	NUM
ejpam-4498	265	3	+	+	NUM
ejpam-4498	265	4	2	2	NUM
ejpam-4498	265	5	+	+	NUM
ejpam-4498	265	6	3	3	NUM
ejpam-4498	265	7	+	+	SYM
ejpam-4498	265	8	4	4	NUM
ejpam-4498	265	9	.	.	X
ejpam-4498	265	10	3c3	3c3	NUM
ejpam-4498	265	11	=	=	SYM
ejpam-4498	265	12	1	1	NUM
ejpam-4498	265	13	.	.	NUM
ejpam-4498	265	14	4c3	4c3	NUM
ejpam-4498	266	1	=	=	SYM
ejpam-4498	266	2	1	1	NUM
ejpam-4498	266	3	+	+	NUM
ejpam-4498	266	4	3	3	NUM
ejpam-4498	266	5	.	.	X
ejpam-4498	266	6	5c3	5c3	NUM
ejpam-4498	266	7	=	=	SYM
ejpam-4498	266	8	1	1	NUM
ejpam-4498	266	9	+	+	NUM
ejpam-4498	266	10	3	3	NUM
ejpam-4498	266	11	+	+	SYM
ejpam-4498	266	12	6	6	NUM
ejpam-4498	266	13	.	.	PUNCT
ejpam-4498	266	14	6c3	6c3	NUM
ejpam-4498	266	15	=	=	SYM
ejpam-4498	266	16	1	1	NUM
ejpam-4498	266	17	+	+	NUM
ejpam-4498	266	18	3	3	NUM
ejpam-4498	266	19	+	+	SYM
ejpam-4498	266	20	6	6	NUM
ejpam-4498	266	21	+	+	SYM
ejpam-4498	266	22	10	10	NUM
ejpam-4498	266	23	.	.	PUNCT
ejpam-4498	267	1	4c4	4c4	NUM
ejpam-4498	268	1	=	=	SYM
ejpam-4498	268	2	1	1	NUM
ejpam-4498	268	3	.	.	X
ejpam-4498	268	4	5c4	5c4	NUM
ejpam-4498	269	1	=	=	SYM
ejpam-4498	269	2	1	1	NUM
ejpam-4498	269	3	+	+	NUM
ejpam-4498	269	4	4	4	NUM
ejpam-4498	269	5	.	.	X
ejpam-4498	269	6	6c4	6c4	NUM
ejpam-4498	269	7	=	=	SYM
ejpam-4498	269	8	1	1	NUM
ejpam-4498	269	9	+	+	NUM
ejpam-4498	269	10	4	4	NUM
ejpam-4498	269	11	+	+	SYM
ejpam-4498	269	12	10	10	NUM
ejpam-4498	269	13	.	.	PUNCT
ejpam-4498	269	14	7c4	7c4	NUM
ejpam-4498	270	1	=	=	SYM
ejpam-4498	270	2	1	1	NUM
ejpam-4498	270	3	+	+	NUM
ejpam-4498	270	4	4	4	NUM
ejpam-4498	270	5	+	+	SYM
ejpam-4498	270	6	10	10	NUM
ejpam-4498	270	7	+	+	NUM
ejpam-4498	270	8	20	20	NUM
ejpam-4498	270	9	.	.	PUNCT
ejpam-4498	271	1	5c5	5c5	NUM
ejpam-4498	271	2	=	=	SYM
ejpam-4498	271	3	1	1	NUM
ejpam-4498	271	4	.	.	NOUN
ejpam-4498	272	1	6c5	6c5	NUM
ejpam-4498	272	2	=	=	SYM
ejpam-4498	272	3	1	1	NUM
ejpam-4498	272	4	+	+	NUM
ejpam-4498	272	5	5	5	NUM
ejpam-4498	272	6	.	.	X
ejpam-4498	272	7	7c5	7c5	NUM
ejpam-4498	272	8	=	=	SYM
ejpam-4498	272	9	1	1	NUM
ejpam-4498	272	10	+	+	NUM
ejpam-4498	272	11	5	5	NUM
ejpam-4498	272	12	+	+	SYM
ejpam-4498	272	13	15	15	NUM
ejpam-4498	272	14	.	.	PUNCT
ejpam-4498	272	15	8c5	8c5	NUM
ejpam-4498	273	1	=	=	SYM
ejpam-4498	273	2	1	1	NUM
ejpam-4498	274	1	+	+	NUM
ejpam-4498	274	2	5	5	NUM
ejpam-4498	274	3	+	+	NUM
ejpam-4498	274	4	15	15	NUM
ejpam-4498	274	5	+	+	SYM
ejpam-4498	274	6	35	35	NUM
ejpam-4498	274	7	.	.	PUNCT
ejpam-4498	275	1	see	see	VERB
ejpam-4498	275	2	figures	figure	NOUN
ejpam-4498	275	3	1	1	NUM
ejpam-4498	275	4	and	and	CCONJ
ejpam-4498	275	5	6	6	NUM
ejpam-4498	275	6	.	.	PUNCT
ejpam-4498	275	7	to	to	PART
ejpam-4498	275	8	cultivate	cultivate	VERB
ejpam-4498	275	9	an	an	DET
ejpam-4498	275	10	understanding	understanding	NOUN
ejpam-4498	275	11	of	of	ADP
ejpam-4498	275	12	the	the	DET
ejpam-4498	275	13	addition	addition	NOUN
ejpam-4498	275	14	formulae	formulae	NOUN
ejpam-4498	275	15	,	,	PUNCT
ejpam-4498	275	16	students	student	NOUN
ejpam-4498	275	17	should	should	AUX
ejpam-4498	275	18	check	check	VERB
ejpam-4498	275	19	the	the	DET
ejpam-4498	275	20	total	total	ADJ
ejpam-4498	275	21	number	number	NOUN
ejpam-4498	275	22	of	of	ADP
ejpam-4498	275	23	edges	edge	NOUN
ejpam-4498	275	24	in	in	ADP
ejpam-4498	275	25	the	the	DET
ejpam-4498	275	26	tree	tree	NOUN
ejpam-4498	275	27	diagrams	diagram	NOUN
ejpam-4498	275	28	.	.	PUNCT
ejpam-4498	276	1	the	the	DET
ejpam-4498	276	2	number	number	NOUN
ejpam-4498	276	3	of	of	ADP
ejpam-4498	276	4	edges	edge	NOUN
ejpam-4498	276	5	is	be	AUX
ejpam-4498	276	6	the	the	DET
ejpam-4498	276	7	sum	sum	NOUN
ejpam-4498	276	8	of	of	ADP
ejpam-4498	276	9	triangular	triangular	NOUN
ejpam-4498	276	10	numbers	number	NOUN
ejpam-4498	276	11	.	.	PUNCT
ejpam-4498	277	1	the	the	DET
ejpam-4498	277	2	tree	tree	NOUN
ejpam-4498	277	3	diagrams	diagram	NOUN
ejpam-4498	277	4	for	for	ADP
ejpam-4498	277	5	nck	nck	NOUN
ejpam-4498	277	6	are	be	AUX
ejpam-4498	277	7	inductively	inductively	ADV
ejpam-4498	277	8	extended	extend	VERB
ejpam-4498	277	9	to	to	ADP
ejpam-4498	277	10	those	those	PRON
ejpam-4498	277	11	for	for	ADP
ejpam-4498	277	12	n+1ck	n+1ck	NOUN
ejpam-4498	277	13	.	.	PUNCT
ejpam-4498	278	1	for	for	ADP
ejpam-4498	278	2	example	example	NOUN
ejpam-4498	278	3	,	,	PUNCT
ejpam-4498	278	4	the	the	DET
ejpam-4498	278	5	tree	tree	NOUN
ejpam-4498	278	6	diagrams	diagram	NOUN
ejpam-4498	278	7	for	for	ADP
ejpam-4498	278	8	6c4	6c4	NUM
ejpam-4498	278	9	include	include	VERB
ejpam-4498	278	10	those	those	PRON
ejpam-4498	278	11	for	for	ADP
ejpam-4498	278	12	5c4	5c4	NUM
ejpam-4498	278	13	,	,	PUNCT
ejpam-4498	278	14	as	as	SCONJ
ejpam-4498	278	15	shown	show	VERB
ejpam-4498	278	16	in	in	ADP
ejpam-4498	278	17	figure	figure	NOUN
ejpam-4498	278	18	9	9	NUM
ejpam-4498	278	19	.	.	PUNCT
ejpam-4498	279	1	the	the	DET
ejpam-4498	279	2	sum	sum	NOUN
ejpam-4498	279	3	of	of	ADP
ejpam-4498	279	4	4c3	4c3	NUM
ejpam-4498	279	5	and	and	CCONJ
ejpam-4498	279	6	3c3	3c3	NUM
ejpam-4498	279	7	indicate	indicate	VERB
ejpam-4498	279	8	the	the	DET
ejpam-4498	279	9	method	method	NOUN
ejpam-4498	279	10	of	of	ADP
ejpam-4498	279	11	choosing	choose	VERB
ejpam-4498	279	12	four	four	NUM
ejpam-4498	279	13	cards	card	NOUN
ejpam-4498	279	14	from	from	ADP
ejpam-4498	279	15	two	two	NUM
ejpam-4498	279	16	,	,	PUNCT
ejpam-4498	279	17	three	three	NUM
ejpam-4498	279	18	,	,	PUNCT
ejpam-4498	279	19	four	four	NUM
ejpam-4498	279	20	,	,	PUNCT
ejpam-4498	279	21	five	five	NUM
ejpam-4498	279	22	,	,	PUNCT
ejpam-4498	279	23	and	and	CCONJ
ejpam-4498	279	24	six	six	NUM
ejpam-4498	279	25	cards	card	NOUN
ejpam-4498	279	26	,	,	PUNCT
ejpam-4498	279	27	with	with	ADP
ejpam-4498	279	28	the	the	DET
ejpam-4498	279	29	exception	exception	NOUN
ejpam-4498	279	30	of	of	ADP
ejpam-4498	279	31	one	one	NUM
ejpam-4498	279	32	;	;	PUNCT
ejpam-4498	279	33	thus	thus	ADV
ejpam-4498	279	34	,	,	PUNCT
ejpam-4498	279	35	4c3	4c3	NUM
ejpam-4498	280	1	+	+	CCONJ
ejpam-4498	280	2	3c3	3c3	NUM
ejpam-4498	280	3	=	=	SYM
ejpam-4498	280	4	5c4	5c4	NUM
ejpam-4498	280	5	.	.	NOUN
ejpam-4498	280	6	8	8	X
ejpam-4498	280	7	.	.	X
ejpam-4498	280	8	concluding	conclude	VERB
ejpam-4498	280	9	remarks	remark	NOUN
ejpam-4498	280	10	in	in	ADP
ejpam-4498	280	11	this	this	DET
ejpam-4498	280	12	paper	paper	NOUN
ejpam-4498	280	13	,	,	PUNCT
ejpam-4498	280	14	we	we	PRON
ejpam-4498	280	15	have	have	AUX
ejpam-4498	280	16	proposed	propose	VERB
ejpam-4498	280	17	a	a	DET
ejpam-4498	280	18	pedagogical	pedagogical	ADJ
ejpam-4498	280	19	approach	approach	NOUN
ejpam-4498	280	20	to	to	ADP
ejpam-4498	280	21	the	the	DET
ejpam-4498	280	22	determination	determination	NOUN
ejpam-4498	280	23	of	of	ADP
ejpam-4498	280	24	various	various	ADJ
ejpam-4498	280	25	equations	equation	NOUN
ejpam-4498	280	26	related	relate	VERB
ejpam-4498	280	27	to	to	ADP
ejpam-4498	280	28	the	the	DET
ejpam-4498	280	29	sum	sum	NOUN
ejpam-4498	280	30	of	of	ADP
ejpam-4498	280	31	triangular	triangular	NOUN
ejpam-4498	280	32	numbers	number	NOUN
ejpam-4498	280	33	that	that	PRON
ejpam-4498	280	34	include	include	VERB
ejpam-4498	280	35	the	the	DET
ejpam-4498	280	36	following	follow	VERB
ejpam-4498	280	37	steps	step	NOUN
ejpam-4498	280	38	(	(	PUNCT
ejpam-4498	280	39	in	in	ADP
ejpam-4498	280	40	order	order	NOUN
ejpam-4498	280	41	)	)	PUNCT
ejpam-4498	280	42	;	;	PUNCT
ejpam-4498	280	43	however	however	ADV
ejpam-4498	280	44	,	,	PUNCT
ejpam-4498	280	45	the	the	DET
ejpam-4498	280	46	sum	sum	NOUN
ejpam-4498	280	47	of	of	ADP
ejpam-4498	280	48	triangular	triangular	NOUN
ejpam-4498	280	49	numbers	number	NOUN
ejpam-4498	280	50	can	can	AUX
ejpam-4498	280	51	be	be	AUX
ejpam-4498	280	52	shown	show	VERB
ejpam-4498	280	53	simply	simply	ADV
ejpam-4498	280	54	using	use	VERB
ejpam-4498	280	55	a	a	DET
ejpam-4498	280	56	pictorial	pictorial	ADJ
ejpam-4498	280	57	proof	proof	NOUN
ejpam-4498	280	58	[	[	X
ejpam-4498	280	59	8	8	NUM
ejpam-4498	280	60	]	]	PUNCT
ejpam-4498	280	61	.	.	PUNCT
ejpam-4498	281	1	yukio	yukio	PROPN
ejpam-4498	281	2	kobayashi	kobayashi	PROPN
ejpam-4498	281	3	/	/	SYM
ejpam-4498	281	4	eur	eur	PROPN
ejpam-4498	281	5	.	.	PUNCT
ejpam-4498	282	1	j.	j.	PROPN
ejpam-4498	282	2	pure	pure	PROPN
ejpam-4498	282	3	appl	appl	PROPN
ejpam-4498	282	4	.	.	PROPN
ejpam-4498	282	5	math	math	PROPN
ejpam-4498	282	6	,	,	PUNCT
ejpam-4498	282	7	15	15	NUM
ejpam-4498	282	8	(	(	PUNCT
ejpam-4498	282	9	3	3	NUM
ejpam-4498	282	10	)	)	PUNCT
ejpam-4498	282	11	(	(	PUNCT
ejpam-4498	282	12	2022	2022	NUM
ejpam-4498	282	13	)	)	PUNCT
ejpam-4498	282	14	,	,	PUNCT
ejpam-4498	282	15	1237	1237	NUM
ejpam-4498	282	16	-	-	SYM
ejpam-4498	282	17	1253	1253	NUM
ejpam-4498	283	1	1251	1251	NUM
ejpam-4498	283	2	1	1	NUM
ejpam-4498	283	3	2	2	NUM
ejpam-4498	283	4	3	3	NUM
ejpam-4498	283	5	4	4	NUM
ejpam-4498	283	6	5	5	NUM
ejpam-4498	283	7	6	6	NUM
ejpam-4498	283	8	4	4	NUM
ejpam-4498	283	9	5	5	NUM
ejpam-4498	283	10	6	6	NUM
ejpam-4498	283	11	5	5	NUM
ejpam-4498	283	12	6	6	NUM
ejpam-4498	283	13	3	3	NUM
ejpam-4498	283	14	4	4	NUM
ejpam-4498	283	15	5	5	NUM
ejpam-4498	283	16	6	6	NUM
ejpam-4498	283	17	5	5	NUM
ejpam-4498	283	18	6	6	NUM
ejpam-4498	283	19	4	4	NUM
ejpam-4498	283	20	5	5	NUM
ejpam-4498	283	21	6	6	NUM
ejpam-4498	283	22	2	2	NUM
ejpam-4498	283	23	3	3	NUM
ejpam-4498	283	24	4	4	NUM
ejpam-4498	283	25	5	5	NUM
ejpam-4498	283	26	6	6	NUM
ejpam-4498	283	27	5	5	NUM
ejpam-4498	283	28	6	6	NUM
ejpam-4498	283	29	4	4	NUM
ejpam-4498	283	30	5	5	NUM
ejpam-4498	283	31	6	6	NUM
ejpam-4498	283	32	3	3	NUM
ejpam-4498	283	33	4	4	NUM
ejpam-4498	283	34	5	5	NUM
ejpam-4498	283	35	6	6	NUM
ejpam-4498	283	36	6	6	NUM
ejpam-4498	283	37	=	=	SYM
ejpam-4498	283	38	1	1	NUM
ejpam-4498	283	39	+	+	NUM
ejpam-4498	283	40	2	2	NUM
ejpam-4498	283	41	+	+	CCONJ
ejpam-4498	283	42	3	3	NUM
ejpam-4498	283	43	3	3	NUM
ejpam-4498	283	44	=	=	SYM
ejpam-4498	283	45	1	1	NUM
ejpam-4498	283	46	+	+	NUM
ejpam-4498	283	47	2	2	NUM
ejpam-4498	283	48	1	1	NUM
ejpam-4498	283	49	3	3	NUM
ejpam-4498	283	50	=	=	SYM
ejpam-4498	283	51	1	1	NUM
ejpam-4498	283	52	+	+	NUM
ejpam-4498	283	53	2	2	NUM
ejpam-4498	283	54	1	1	NUM
ejpam-4498	283	55	1	1	NUM
ejpam-4498	283	56	5c3	5c3	NUM
ejpam-4498	283	57	=	=	SYM
ejpam-4498	283	58	10	10	NUM
ejpam-4498	283	59	4c3	4c3	NUM
ejpam-4498	283	60	=	=	SYM
ejpam-4498	283	61	4	4	NUM
ejpam-4498	283	62	3c3	3c3	NUM
ejpam-4498	283	63	=	=	SYM
ejpam-4498	283	64	1	1	NUM
ejpam-4498	283	65	no	no	NOUN
ejpam-4498	283	66	.	.	PUNCT
ejpam-4498	284	1	of	of	ADP
ejpam-4498	284	2	edges	edge	VERB
ejpam-4498	284	3	the	the	DET
ejpam-4498	284	4	manner	manner	NOUN
ejpam-4498	284	5	of	of	ADP
ejpam-4498	284	6	choosing	choose	VERB
ejpam-4498	284	7	three	three	NUM
ejpam-4498	284	8	cards	card	NOUN
ejpam-4498	284	9	from	from	ADP
ejpam-4498	284	10	2	2	NUM
ejpam-4498	284	11	,	,	PUNCT
ejpam-4498	284	12	3	3	NUM
ejpam-4498	284	13	,	,	PUNCT
ejpam-4498	284	14	4	4	NUM
ejpam-4498	284	15	,	,	PUNCT
ejpam-4498	284	16	5	5	NUM
ejpam-4498	284	17	and	and	CCONJ
ejpam-4498	284	18	6	6	NUM
ejpam-4498	284	19	,	,	PUNCT
ejpam-4498	284	20	except	except	SCONJ
ejpam-4498	284	21	1	1	NUM
ejpam-4498	284	22	.	.	PUNCT
ejpam-4498	285	1	the	the	DET
ejpam-4498	285	2	manner	manner	NOUN
ejpam-4498	285	3	of	of	ADP
ejpam-4498	285	4	choosing	choose	VERB
ejpam-4498	285	5	three	three	NUM
ejpam-4498	285	6	cards	card	NOUN
ejpam-4498	285	7	from	from	ADP
ejpam-4498	285	8	3	3	NUM
ejpam-4498	285	9	,	,	PUNCT
ejpam-4498	285	10	4	4	NUM
ejpam-4498	285	11	,	,	PUNCT
ejpam-4498	285	12	5	5	NUM
ejpam-4498	285	13	and	and	CCONJ
ejpam-4498	285	14	6	6	NUM
ejpam-4498	285	15	,	,	PUNCT
ejpam-4498	285	16	except	except	SCONJ
ejpam-4498	285	17	1	1	NUM
ejpam-4498	285	18	and	and	CCONJ
ejpam-4498	285	19	2	2	NUM
ejpam-4498	285	20	.	.	PUNCT
ejpam-4498	286	1	the	the	DET
ejpam-4498	286	2	manner	manner	NOUN
ejpam-4498	286	3	of	of	ADP
ejpam-4498	286	4	choosing	choose	VERB
ejpam-4498	286	5	three	three	NUM
ejpam-4498	286	6	cards	card	NOUN
ejpam-4498	286	7	from	from	ADP
ejpam-4498	286	8	4	4	NUM
ejpam-4498	286	9	,	,	PUNCT
ejpam-4498	286	10	5	5	NUM
ejpam-4498	286	11	,	,	PUNCT
ejpam-4498	286	12	6	6	NUM
ejpam-4498	286	13	,	,	PUNCT
ejpam-4498	286	14	except	except	SCONJ
ejpam-4498	286	15	1	1	NUM
ejpam-4498	286	16	,	,	PUNCT
ejpam-4498	286	17	2	2	NUM
ejpam-4498	286	18	and	and	CCONJ
ejpam-4498	286	19	3	3	NUM
ejpam-4498	286	20	.	.	X
ejpam-4498	286	21	figure	figure	NOUN
ejpam-4498	286	22	9	9	NUM
ejpam-4498	286	23	:	:	PUNCT
ejpam-4498	286	24	tree	tree	NOUN
ejpam-4498	286	25	diagrams	diagram	NOUN
ejpam-4498	286	26	for	for	ADP
ejpam-4498	286	27	6c4	6c4	NUM
ejpam-4498	286	28	.	.	PUNCT
ejpam-4498	287	1	the	the	DET
ejpam-4498	287	2	numerals	numeral	NOUN
ejpam-4498	287	3	in	in	ADP
ejpam-4498	287	4	the	the	DET
ejpam-4498	287	5	tree	tree	NOUN
ejpam-4498	287	6	diagrams	diagram	NOUN
ejpam-4498	287	7	indicate	indicate	VERB
ejpam-4498	287	8	the	the	DET
ejpam-4498	287	9	number	number	NOUN
ejpam-4498	287	10	of	of	ADP
ejpam-4498	287	11	cards	card	NOUN
ejpam-4498	287	12	.	.	PUNCT
ejpam-4498	288	1	the	the	DET
ejpam-4498	288	2	edges	edge	NOUN
ejpam-4498	288	3	represent	represent	VERB
ejpam-4498	288	4	how	how	SCONJ
ejpam-4498	288	5	the	the	DET
ejpam-4498	288	6	four	four	NUM
ejpam-4498	288	7	two	two	NUM
ejpam-4498	288	8	-	-	PUNCT
ejpam-4498	288	9	numbered	number	VERB
ejpam-4498	288	10	cards	card	NOUN
ejpam-4498	288	11	are	be	AUX
ejpam-4498	288	12	assembled	assemble	VERB
ejpam-4498	288	13	.	.	PUNCT
ejpam-4498	289	1	the	the	DET
ejpam-4498	289	2	total	total	ADJ
ejpam-4498	289	3	number	number	NOUN
ejpam-4498	289	4	of	of	ADP
ejpam-4498	289	5	edges	edge	NOUN
ejpam-4498	289	6	in	in	ADP
ejpam-4498	289	7	the	the	DET
ejpam-4498	289	8	area	area	NOUN
ejpam-4498	289	9	enclosed	enclose	VERB
ejpam-4498	289	10	by	by	ADP
ejpam-4498	289	11	the	the	DET
ejpam-4498	289	12	dashed	dash	VERB
ejpam-4498	289	13	line	line	NOUN
ejpam-4498	289	14	represents	represent	VERB
ejpam-4498	289	15	5c4s	5c4s	NUM
ejpam-4498	289	16	.	.	PUNCT
ejpam-4498	290	1	list	list	NOUN
ejpam-4498	290	2	of	of	ADP
ejpam-4498	290	3	equations	equation	NOUN
ejpam-4498	290	4	expressing	express	VERB
ejpam-4498	290	5	the	the	DET
ejpam-4498	290	6	sum	sum	NOUN
ejpam-4498	290	7	of	of	ADP
ejpam-4498	290	8	natural	natural	ADJ
ejpam-4498	290	9	numbers	number	NOUN
ejpam-4498	290	10	↓	↓	NOUN
ejpam-4498	290	11	tree	tree	NOUN
ejpam-4498	290	12	diagrams	diagram	NOUN
ejpam-4498	290	13	↓	↓	NOUN
ejpam-4498	290	14	array	array	NOUN
ejpam-4498	290	15	of	of	ADP
ejpam-4498	290	16	filled	fill	VERB
ejpam-4498	290	17	circles	circle	NOUN
ejpam-4498	290	18	↓	↓	NOUN
ejpam-4498	290	19	combinatorics	combinatoric	NOUN
ejpam-4498	290	20	perspective	perspective	VERB
ejpam-4498	290	21	the	the	DET
ejpam-4498	290	22	same	same	ADJ
ejpam-4498	290	23	mathematical	mathematical	ADJ
ejpam-4498	290	24	meaning	meaning	NOUN
ejpam-4498	290	25	can	can	AUX
ejpam-4498	290	26	be	be	AUX
ejpam-4498	290	27	expressed	express	VERB
ejpam-4498	290	28	in	in	ADP
ejpam-4498	290	29	many	many	ADJ
ejpam-4498	290	30	different	different	ADJ
ejpam-4498	290	31	ways	way	NOUN
ejpam-4498	290	32	,	,	PUNCT
ejpam-4498	290	33	as	as	SCONJ
ejpam-4498	290	34	shown	show	VERB
ejpam-4498	290	35	in	in	ADP
ejpam-4498	290	36	equations	equation	NOUN
ejpam-4498	290	37	(	(	PUNCT
ejpam-4498	290	38	1	1	NUM
ejpam-4498	290	39	)	)	PUNCT
ejpam-4498	290	40	,	,	PUNCT
ejpam-4498	290	41	(	(	PUNCT
ejpam-4498	290	42	2	2	NUM
ejpam-4498	290	43	)	)	PUNCT
ejpam-4498	290	44	,	,	PUNCT
ejpam-4498	290	45	(	(	PUNCT
ejpam-4498	290	46	3	3	NUM
ejpam-4498	290	47	)	)	PUNCT
ejpam-4498	290	48	,	,	PUNCT
ejpam-4498	290	49	and	and	CCONJ
ejpam-4498	290	50	(	(	PUNCT
ejpam-4498	290	51	4	4	NUM
ejpam-4498	290	52	)	)	PUNCT
ejpam-4498	290	53	.	.	PUNCT
ejpam-4498	291	1	students	student	NOUN
ejpam-4498	291	2	should	should	AUX
ejpam-4498	291	3	deduce	deduce	VERB
ejpam-4498	291	4	and	and	CCONJ
ejpam-4498	291	5	thus	thus	ADV
ejpam-4498	291	6	perceive	perceive	ADJ
ejpam-4498	291	7	that	that	SCONJ
ejpam-4498	291	8	the	the	DET
ejpam-4498	291	9	different	different	ADJ
ejpam-4498	291	10	expressions	expression	NOUN
ejpam-4498	291	11	have	have	VERB
ejpam-4498	291	12	the	the	DET
ejpam-4498	291	13	same	same	ADJ
ejpam-4498	291	14	meaning	meaning	NOUN
ejpam-4498	291	15	.	.	PUNCT
ejpam-4498	292	1	equations	equation	NOUN
ejpam-4498	292	2	(	(	PUNCT
ejpam-4498	292	3	24	24	NUM
ejpam-4498	292	4	)	)	PUNCT
ejpam-4498	292	5	and	and	CCONJ
ejpam-4498	292	6	(	(	PUNCT
ejpam-4498	292	7	25	25	NUM
ejpam-4498	292	8	)	)	PUNCT
ejpam-4498	292	9	also	also	ADV
ejpam-4498	292	10	serve	serve	VERB
ejpam-4498	292	11	as	as	ADP
ejpam-4498	292	12	good	good	ADJ
ejpam-4498	292	13	examples	example	NOUN
ejpam-4498	292	14	of	of	ADP
ejpam-4498	292	15	this	this	PRON
ejpam-4498	292	16	and	and	CCONJ
ejpam-4498	292	17	are	be	AUX
ejpam-4498	292	18	useful	useful	ADJ
ejpam-4498	292	19	for	for	ADP
ejpam-4498	292	20	calculating	calculate	VERB
ejpam-4498	292	21	the	the	DET
ejpam-4498	292	22	series	series	NOUN
ejpam-4498	292	23	.	.	PUNCT
ejpam-4498	293	1	finding	find	VERB
ejpam-4498	293	2	equations	equation	NOUN
ejpam-4498	293	3	such	such	ADJ
ejpam-4498	293	4	as	as	ADP
ejpam-4498	293	5	(	(	PUNCT
ejpam-4498	293	6	34	34	NUM
ejpam-4498	293	7	)	)	PUNCT
ejpam-4498	293	8	and	and	CCONJ
ejpam-4498	293	9	(	(	PUNCT
ejpam-4498	293	10	35	35	NUM
ejpam-4498	293	11	)	)	PUNCT
ejpam-4498	293	12	can	can	AUX
ejpam-4498	293	13	be	be	AUX
ejpam-4498	293	14	relatively	relatively	ADV
ejpam-4498	293	15	difficult	difficult	ADJ
ejpam-4498	293	16	if	if	SCONJ
ejpam-4498	293	17	students	student	NOUN
ejpam-4498	293	18	consider	consider	VERB
ejpam-4498	293	19	the	the	DET
ejpam-4498	293	20	problem	problem	NOUN
ejpam-4498	293	21	within	within	ADP
ejpam-4498	293	22	the	the	DET
ejpam-4498	293	23	framework	framework	NOUN
ejpam-4498	293	24	of	of	ADP
ejpam-4498	293	25	combination	combination	NOUN
ejpam-4498	293	26	.	.	PUNCT
ejpam-4498	294	1	as	as	SCONJ
ejpam-4498	294	2	we	we	PRON
ejpam-4498	294	3	can	can	AUX
ejpam-4498	294	4	understand	understand	VERB
ejpam-4498	294	5	through	through	ADP
ejpam-4498	294	6	this	this	DET
ejpam-4498	294	7	example	example	NOUN
ejpam-4498	294	8	,	,	PUNCT
ejpam-4498	294	9	the	the	DET
ejpam-4498	294	10	most	most	ADV
ejpam-4498	294	11	fundamental	fundamental	ADJ
ejpam-4498	294	12	number	number	NOUN
ejpam-4498	294	13	of	of	ADP
ejpam-4498	294	14	combinations	combination	NOUN
ejpam-4498	294	15	is	be	AUX
ejpam-4498	294	16	the	the	DET
ejpam-4498	294	17	sum	sum	NOUN
ejpam-4498	294	18	of	of	ADP
ejpam-4498	294	19	triangular	triangular	NOUN
ejpam-4498	294	20	numbers	number	NOUN
ejpam-4498	294	21	.	.	PUNCT
ejpam-4498	295	1	the	the	DET
ejpam-4498	295	2	sum	sum	NOUN
ejpam-4498	295	3	of	of	ADP
ejpam-4498	295	4	triangular	triangular	NOUN
ejpam-4498	295	5	numbers	number	NOUN
ejpam-4498	295	6	expressed	express	VERB
ejpam-4498	295	7	by	by	ADP
ejpam-4498	295	8	equation	equation	NOUN
ejpam-4498	295	9	(	(	PUNCT
ejpam-4498	295	10	1	1	X
ejpam-4498	295	11	)	)	PUNCT
ejpam-4498	295	12	is	be	AUX
ejpam-4498	295	13	the	the	DET
ejpam-4498	295	14	basis	basis	NOUN
ejpam-4498	295	15	,	,	PUNCT
ejpam-4498	295	16	and	and	CCONJ
ejpam-4498	295	17	thus	thus	ADV
ejpam-4498	295	18	equations	equation	NOUN
ejpam-4498	295	19	(	(	PUNCT
ejpam-4498	295	20	2	2	NUM
ejpam-4498	295	21	)	)	PUNCT
ejpam-4498	295	22	,	,	PUNCT
ejpam-4498	295	23	(	(	PUNCT
ejpam-4498	295	24	3	3	NUM
ejpam-4498	295	25	)	)	PUNCT
ejpam-4498	295	26	,	,	PUNCT
ejpam-4498	295	27	and	and	CCONJ
ejpam-4498	295	28	(	(	PUNCT
ejpam-4498	295	29	4	4	X
ejpam-4498	295	30	)	)	PUNCT
ejpam-4498	295	31	are	be	AUX
ejpam-4498	295	32	different	different	ADJ
ejpam-4498	295	33	references	reference	NOUN
ejpam-4498	295	34	1252	1252	NUM
ejpam-4498	295	35	expressions	expression	NOUN
ejpam-4498	295	36	of	of	ADP
ejpam-4498	295	37	(	(	PUNCT
ejpam-4498	295	38	1	1	NUM
ejpam-4498	295	39	)	)	PUNCT
ejpam-4498	295	40	.	.	PUNCT
ejpam-4498	296	1	the	the	DET
ejpam-4498	296	2	utility	utility	NOUN
ejpam-4498	296	3	of	of	ADP
ejpam-4498	296	4	the	the	DET
ejpam-4498	296	5	sum	sum	NOUN
ejpam-4498	296	6	of	of	ADP
ejpam-4498	296	7	triangular	triangular	NOUN
ejpam-4498	296	8	numbers	number	NOUN
ejpam-4498	296	9	is	be	AUX
ejpam-4498	296	10	significant	significant	ADJ
ejpam-4498	296	11	.	.	PUNCT
ejpam-4498	297	1	although	although	SCONJ
ejpam-4498	297	2	indirectly	indirectly	ADV
ejpam-4498	297	3	related	relate	VERB
ejpam-4498	297	4	to	to	ADP
ejpam-4498	297	5	this	this	DET
ejpam-4498	297	6	study	study	NOUN
ejpam-4498	297	7	,	,	PUNCT
ejpam-4498	297	8	understanding	understand	VERB
ejpam-4498	297	9	the	the	DET
ejpam-4498	297	10	principles	principle	NOUN
ejpam-4498	297	11	of	of	ADP
ejpam-4498	297	12	statistical	statistical	ADJ
ejpam-4498	297	13	mechanics	mechanic	NOUN
ejpam-4498	297	14	is	be	AUX
ejpam-4498	297	15	difficult	difficult	ADJ
ejpam-4498	297	16	if	if	SCONJ
ejpam-4498	297	17	we	we	PRON
ejpam-4498	297	18	are	be	AUX
ejpam-4498	297	19	unfamiliar	unfamiliar	ADJ
ejpam-4498	297	20	with	with	ADP
ejpam-4498	297	21	an	an	DET
ejpam-4498	297	22	equation	equation	NOUN
ejpam-4498	297	23	of	of	ADP
ejpam-4498	297	24	the	the	DET
ejpam-4498	297	25	form	form	NOUN
ejpam-4498	297	26	[	[	X
ejpam-4498	297	27	2	2	NUM
ejpam-4498	297	28	]	]	PUNCT
ejpam-4498	297	29	η(e0	η(e0	ADJ
ejpam-4498	297	30	−	−	ADP
ejpam-4498	297	31	er	er	INTJ
ejpam-4498	297	32	)	)	PUNCT
ejpam-4498	297	33	η(e0	η(e0	NOUN
ejpam-4498	297	34	−	−	PROPN
ejpam-4498	297	35	er′	er′	PROPN
ejpam-4498	297	36	)	)	PUNCT
ejpam-4498	297	37	=	=	PROPN
ejpam-4498	297	38	eln	eln	PROPN
ejpam-4498	297	39	η(e0−er)−ln	η(e0−er)−ln	PROPN
ejpam-4498	297	40	η(e0−er′	η(e0−er′	PROPN
ejpam-4498	297	41	)	)	PUNCT
ejpam-4498	297	42	,	,	PUNCT
ejpam-4498	297	43	(	(	PUNCT
ejpam-4498	297	44	37	37	NUM
ejpam-4498	297	45	)	)	PUNCT
ejpam-4498	297	46	in	in	ADP
ejpam-4498	297	47	which	which	PRON
ejpam-4498	297	48	the	the	DET
ejpam-4498	297	49	leftand	leftand	NOUN
ejpam-4498	297	50	right	right	ADJ
ejpam-4498	297	51	-	-	PUNCT
ejpam-4498	297	52	hand	hand	NOUN
ejpam-4498	297	53	sides	side	NOUN
ejpam-4498	297	54	appear	appear	VERB
ejpam-4498	297	55	completely	completely	ADV
ejpam-4498	297	56	different	different	ADJ
ejpam-4498	297	57	.	.	PUNCT
ejpam-4498	298	1	the	the	DET
ejpam-4498	298	2	ability	ability	NOUN
ejpam-4498	298	3	to	to	PART
ejpam-4498	298	4	calculate	calculate	VERB
ejpam-4498	298	5	using	use	VERB
ejpam-4498	298	6	memorised	memorised	ADJ
ejpam-4498	298	7	formulas	formula	NOUN
ejpam-4498	298	8	is	be	AUX
ejpam-4498	298	9	important	important	ADJ
ejpam-4498	298	10	;	;	PUNCT
ejpam-4498	298	11	however	however	ADV
ejpam-4498	298	12	,	,	PUNCT
ejpam-4498	298	13	practice	practice	NOUN
ejpam-4498	298	14	in	in	ADP
ejpam-4498	298	15	expressing	express	VERB
ejpam-4498	298	16	mathematical	mathematical	ADJ
ejpam-4498	298	17	meanings	meaning	NOUN
ejpam-4498	298	18	with	with	ADP
ejpam-4498	298	19	equations	equation	NOUN
ejpam-4498	298	20	helps	help	VERB
ejpam-4498	298	21	students	student	NOUN
ejpam-4498	298	22	become	become	VERB
ejpam-4498	298	23	familiar	familiar	ADJ
ejpam-4498	298	24	with	with	ADP
ejpam-4498	298	25	a	a	DET
ejpam-4498	298	26	variety	variety	NOUN
ejpam-4498	298	27	of	of	ADP
ejpam-4498	298	28	expressions	expression	NOUN
ejpam-4498	298	29	using	use	VERB
ejpam-4498	298	30	heuristic	heuristic	ADJ
ejpam-4498	298	31	methods	method	NOUN
ejpam-4498	298	32	.	.	PUNCT
ejpam-4498	299	1	by	by	ADP
ejpam-4498	299	2	combining	combine	VERB
ejpam-4498	299	3	different	different	ADJ
ejpam-4498	299	4	topics	topic	NOUN
ejpam-4498	299	5	that	that	PRON
ejpam-4498	299	6	are	be	AUX
ejpam-4498	299	7	normally	normally	ADV
ejpam-4498	299	8	learned	learn	VERB
ejpam-4498	299	9	separately	separately	ADV
ejpam-4498	299	10	such	such	ADJ
ejpam-4498	299	11	as	as	ADP
ejpam-4498	299	12	combinatorics	combinatoric	NOUN
ejpam-4498	299	13	and	and	CCONJ
ejpam-4498	299	14	algebra	algebra	NOUN
ejpam-4498	299	15	,	,	PUNCT
ejpam-4498	299	16	students	student	NOUN
ejpam-4498	299	17	can	can	AUX
ejpam-4498	299	18	understand	understand	VERB
ejpam-4498	299	19	that	that	SCONJ
ejpam-4498	299	20	different	different	ADJ
ejpam-4498	299	21	notions	notion	NOUN
ejpam-4498	299	22	are	be	AUX
ejpam-4498	299	23	connected	connect	VERB
ejpam-4498	299	24	at	at	ADP
ejpam-4498	299	25	the	the	DET
ejpam-4498	299	26	root	root	NOUN
ejpam-4498	299	27	of	of	ADP
ejpam-4498	299	28	mathematics	mathematic	NOUN
ejpam-4498	299	29	,	,	PUNCT
ejpam-4498	299	30	and	and	CCONJ
ejpam-4498	299	31	they	they	PRON
ejpam-4498	299	32	can	can	AUX
ejpam-4498	299	33	acquire	acquire	VERB
ejpam-4498	299	34	the	the	DET
ejpam-4498	299	35	idea	idea	NOUN
ejpam-4498	299	36	of	of	ADP
ejpam-4498	299	37	reading	read	VERB
ejpam-4498	299	38	the	the	DET
ejpam-4498	299	39	mathematical	mathematical	ADJ
ejpam-4498	299	40	meanings	meaning	NOUN
ejpam-4498	299	41	of	of	ADP
ejpam-4498	299	42	equations	equation	NOUN
ejpam-4498	299	43	and	and	CCONJ
ejpam-4498	299	44	expressing	express	VERB
ejpam-4498	299	45	mathematical	mathematical	ADJ
ejpam-4498	299	46	meanings	meaning	NOUN
ejpam-4498	299	47	by	by	ADP
ejpam-4498	299	48	equations	equation	NOUN
ejpam-4498	299	49	.	.	PUNCT
ejpam-4498	300	1	slater	slater	NOUN
ejpam-4498	300	2	and	and	CCONJ
ejpam-4498	300	3	frank	frank	ADJ
ejpam-4498	301	1	[	[	X
ejpam-4498	301	2	12	12	NUM
ejpam-4498	301	3	]	]	PUNCT
ejpam-4498	301	4	posited	posit	VERB
ejpam-4498	301	5	that	that	SCONJ
ejpam-4498	301	6	the	the	DET
ejpam-4498	301	7	greatest	great	ADJ
ejpam-4498	301	8	difficulty	difficulty	NOUN
ejpam-4498	301	9	that	that	PRON
ejpam-4498	301	10	students	student	NOUN
ejpam-4498	301	11	have	have	VERB
ejpam-4498	301	12	in	in	ADP
ejpam-4498	301	13	mastering	master	VERB
ejpam-4498	301	14	theoretical	theoretical	ADJ
ejpam-4498	301	15	physics	physics	NOUN
ejpam-4498	301	16	involves	involve	VERB
ejpam-4498	301	17	learning	learn	VERB
ejpam-4498	301	18	how	how	SCONJ
ejpam-4498	301	19	to	to	PART
ejpam-4498	301	20	apply	apply	VERB
ejpam-4498	301	21	mathematics	mathematic	NOUN
ejpam-4498	301	22	to	to	ADP
ejpam-4498	301	23	a	a	DET
ejpam-4498	301	24	physical	physical	ADJ
ejpam-4498	301	25	situation	situation	NOUN
ejpam-4498	301	26	and	and	CCONJ
ejpam-4498	301	27	mathematically	mathematically	ADV
ejpam-4498	301	28	formulate	formulate	VERB
ejpam-4498	301	29	a	a	DET
ejpam-4498	301	30	problem	problem	NOUN
ejpam-4498	301	31	,	,	PUNCT
ejpam-4498	301	32	rather	rather	ADV
ejpam-4498	301	33	than	than	ADP
ejpam-4498	301	34	solving	solve	VERB
ejpam-4498	301	35	the	the	DET
ejpam-4498	301	36	problem	problem	NOUN
ejpam-4498	301	37	when	when	SCONJ
ejpam-4498	301	38	formulated	formulate	VERB
ejpam-4498	301	39	.	.	PUNCT
ejpam-4498	302	1	heuristic	heuristic	ADJ
ejpam-4498	302	2	methods	method	NOUN
ejpam-4498	302	3	are	be	AUX
ejpam-4498	302	4	effective	effective	ADJ
ejpam-4498	302	5	in	in	ADP
ejpam-4498	302	6	implementing	implement	VERB
ejpam-4498	302	7	learning	learning	NOUN
ejpam-4498	302	8	,	,	PUNCT
ejpam-4498	302	9	as	as	SCONJ
ejpam-4498	302	10	noted	note	VERB
ejpam-4498	302	11	by	by	ADP
ejpam-4498	302	12	slater	slater	PROPN
ejpam-4498	302	13	and	and	CCONJ
ejpam-4498	302	14	frank	frank	PROPN
ejpam-4498	302	15	.	.	PUNCT
ejpam-4498	303	1	acknowledgements	acknowledgement	VERB
ejpam-4498	303	2	the	the	DET
ejpam-4498	303	3	author	author	NOUN
ejpam-4498	303	4	thanks	thank	NOUN
ejpam-4498	303	5	editage	editage	NOUN
ejpam-4498	303	6	(	(	PUNCT
ejpam-4498	303	7	www.editage.jp	www.editage.jp	ADV
ejpam-4498	303	8	)	)	PUNCT
ejpam-4498	303	9	for	for	ADP
ejpam-4498	303	10	english	english	ADJ
ejpam-4498	303	11	language	language	NOUN
ejpam-4498	303	12	editing	editing	NOUN
ejpam-4498	303	13	.	.	PUNCT
ejpam-4498	304	1	references	reference	NOUN
ejpam-4498	304	2	[	[	X
ejpam-4498	304	3	1	1	NUM
ejpam-4498	304	4	]	]	PUNCT
ejpam-4498	304	5	c.	c.	PROPN
ejpam-4498	304	6	alsina	alsina	PROPN
ejpam-4498	304	7	and	and	CCONJ
ejpam-4498	304	8	r.	r.	PROPN
ejpam-4498	304	9	b.	b.	PROPN
ejpam-4498	304	10	nelsen	nelsen	PROPN
ejpam-4498	304	11	.	.	PUNCT
ejpam-4498	305	1	an	an	DET
ejpam-4498	305	2	invitation	invitation	NOUN
ejpam-4498	305	3	to	to	AUX
ejpam-4498	305	4	proofs	proof	NOUN
ejpam-4498	305	5	without	without	ADP
ejpam-4498	305	6	words	word	NOUN
ejpam-4498	305	7	.	.	PUNCT
ejpam-4498	306	1	european	european	ADJ
ejpam-4498	306	2	journal	journal	PROPN
ejpam-4498	306	3	of	of	ADP
ejpam-4498	306	4	pure	pure	ADJ
ejpam-4498	306	5	and	and	CCONJ
ejpam-4498	306	6	applied	applied	ADJ
ejpam-4498	306	7	mathematics	mathematic	NOUN
ejpam-4498	306	8	.	.	PUNCT
ejpam-4498	306	9	,	,	PUNCT
ejpam-4498	306	10	3:118–127	3:118–127	NUM
ejpam-4498	306	11	,	,	PUNCT
ejpam-4498	306	12	2010	2010	NUM
ejpam-4498	306	13	.	.	PUNCT
ejpam-4498	307	1	[	[	X
ejpam-4498	307	2	2	2	NUM
ejpam-4498	307	3	]	]	PUNCT
ejpam-4498	307	4	r.	r.	PROPN
ejpam-4498	307	5	p.	p.	PROPN
ejpam-4498	307	6	feynman	feynman	PROPN
ejpam-4498	307	7	.	.	PUNCT
ejpam-4498	308	1	statistical	statistical	ADJ
ejpam-4498	308	2	mechanics	mechanic	NOUN
ejpam-4498	308	3	:	:	PUNCT
ejpam-4498	308	4	a	a	DET
ejpam-4498	308	5	set	set	NOUN
ejpam-4498	308	6	of	of	ADP
ejpam-4498	308	7	lectures	lecture	NOUN
ejpam-4498	308	8	.	.	PUNCT
ejpam-4498	309	1	benjamin	benjamin	PROPN
ejpam-4498	309	2	,	,	PUNCT
ejpam-4498	309	3	1972	1972	NUM
ejpam-4498	309	4	.	.	PUNCT
ejpam-4498	310	1	[	[	X
ejpam-4498	310	2	3	3	X
ejpam-4498	310	3	]	]	X
ejpam-4498	310	4	y.	y.	PROPN
ejpam-4498	310	5	kobayashi	kobayashi	PROPN
ejpam-4498	310	6	.	.	PUNCT
ejpam-4498	311	1	integral	integral	ADJ
ejpam-4498	311	2	representation	representation	NOUN
ejpam-4498	311	3	of	of	ADP
ejpam-4498	311	4	the	the	DET
ejpam-4498	311	5	pictorial	pictorial	ADJ
ejpam-4498	311	6	proof	proof	NOUN
ejpam-4498	311	7	of	of	ADP
ejpam-4498	311	8	n∑	n∑	PROPN
ejpam-4498	311	9	k=1	k=1	PROPN
ejpam-4498	311	10	k2	k2	PROPN
ejpam-4498	311	11	=	=	NOUN
ejpam-4498	311	12	1	1	NUM
ejpam-4498	311	13	6	6	NUM
ejpam-4498	311	14	n(n+1)(2n+	n(n+1)(2n+	ADP
ejpam-4498	311	15	1	1	NUM
ejpam-4498	311	16	)	)	PUNCT
ejpam-4498	311	17	.	.	PUNCT
ejpam-4498	312	1	int	int	NOUN
ejpam-4498	312	2	.	.	PUNCT
ejpam-4498	313	1	j.	j.	PROPN
ejpam-4498	313	2	math	math	PROPN
ejpam-4498	313	3	.	.	PUNCT
ejpam-4498	314	1	sci	sci	PROPN
ejpam-4498	314	2	.	.	PUNCT
ejpam-4498	314	3	tech	tech	PROPN
ejpam-4498	314	4	.	.	PUNCT
ejpam-4498	314	5	,	,	PUNCT
ejpam-4498	314	6	42:235–239	42:235–239	NUM
ejpam-4498	314	7	,	,	PUNCT
ejpam-4498	314	8	2011	2011	NUM
ejpam-4498	314	9	.	.	PUNCT
ejpam-4498	315	1	[	[	X
ejpam-4498	315	2	4	4	X
ejpam-4498	315	3	]	]	X
ejpam-4498	315	4	y.	y.	PROPN
ejpam-4498	315	5	kobayashi	kobayashi	PROPN
ejpam-4498	315	6	.	.	PUNCT
ejpam-4498	316	1	geometrical	geometrical	ADJ
ejpam-4498	316	2	meaning	meaning	NOUN
ejpam-4498	316	3	of	of	ADP
ejpam-4498	316	4	arithmetic	arithmetic	ADJ
ejpam-4498	316	5	series	series	NOUN
ejpam-4498	316	6	n∑	n∑	PROPN
ejpam-4498	316	7	k=1	k=1	PROPN
ejpam-4498	317	1	k	k	X
ejpam-4498	317	2	,	,	PUNCT
ejpam-4498	317	3	n∑	n∑	NOUN
ejpam-4498	317	4	k=1	k=1	PROPN
ejpam-4498	317	5	k2	k2	PROPN
ejpam-4498	317	6	,	,	PUNCT
ejpam-4498	317	7	and	and	CCONJ
ejpam-4498	317	8	n∑	n∑	NOUN
ejpam-4498	317	9	k=1	k=1	PROPN
ejpam-4498	318	1	k3	k3	VERB
ejpam-4498	318	2	in	in	ADP
ejpam-4498	318	3	terms	term	NOUN
ejpam-4498	318	4	of	of	ADP
ejpam-4498	318	5	the	the	DET
ejpam-4498	318	6	elementary	elementary	ADJ
ejpam-4498	318	7	combinatorics	combinatoric	NOUN
ejpam-4498	318	8	.	.	PUNCT
ejpam-4498	319	1	int	int	NOUN
ejpam-4498	319	2	.	.	PUNCT
ejpam-4498	320	1	j.	j.	PROPN
ejpam-4498	320	2	math	math	PROPN
ejpam-4498	320	3	.	.	PUNCT
ejpam-4498	321	1	sci	sci	PROPN
ejpam-4498	321	2	.	.	PUNCT
ejpam-4498	321	3	tech	tech	PROPN
ejpam-4498	321	4	.	.	PROPN
ejpam-4498	321	5	,	,	PUNCT
ejpam-4498	321	6	42:657–664	42:657–664	PROPN
ejpam-4498	321	7	,	,	PUNCT
ejpam-4498	321	8	2011	2011	NUM
ejpam-4498	321	9	.	.	PUNCT
ejpam-4498	322	1	[	[	X
ejpam-4498	322	2	corrigendum	corrigendum	NOUN
ejpam-4498	322	3	42	42	NUM
ejpam-4498	322	4	(	(	PUNCT
ejpam-4498	322	5	2011	2011	NUM
ejpam-4498	322	6	)	)	PUNCT
ejpam-4498	322	7	,	,	PUNCT
ejpam-4498	322	8	1123	1123	NUM
ejpam-4498	322	9	]	]	PUNCT
ejpam-4498	322	10	.	.	PUNCT
ejpam-4498	323	1	[	[	X
ejpam-4498	323	2	5	5	X
ejpam-4498	323	3	]	]	X
ejpam-4498	323	4	y.	y.	PROPN
ejpam-4498	323	5	kobayashi	kobayashi	PROPN
ejpam-4498	323	6	.	.	PUNCT
ejpam-4498	324	1	recursion	recursion	NOUN
ejpam-4498	324	2	formulae	formulae	NOUN
ejpam-4498	324	3	of	of	ADP
ejpam-4498	324	4	1ℓ+2ℓ+	1ℓ+2ℓ+	NUM
ejpam-4498	324	5	·	·	PUNCT
ejpam-4498	324	6	·	·	PUNCT
ejpam-4498	325	1	·	·	PUNCT
ejpam-4498	325	2	+	+	NOUN
ejpam-4498	325	3	nℓ	nℓ	NOUN
ejpam-4498	325	4	and	and	CCONJ
ejpam-4498	325	5	their	their	PRON
ejpam-4498	325	6	combinatoric	combinatoric	ADJ
ejpam-4498	325	7	meaning	meaning	NOUN
ejpam-4498	325	8	in	in	ADP
ejpam-4498	325	9	terms	term	NOUN
ejpam-4498	325	10	of	of	ADP
ejpam-4498	325	11	the	the	DET
ejpam-4498	325	12	tree	tree	NOUN
ejpam-4498	325	13	diagrams	diagram	NOUN
ejpam-4498	325	14	.	.	PUNCT
ejpam-4498	326	1	int	int	NOUN
ejpam-4498	326	2	.	.	PUNCT
ejpam-4498	327	1	j.	j.	PROPN
ejpam-4498	327	2	math	math	PROPN
ejpam-4498	327	3	.	.	PUNCT
ejpam-4498	328	1	sci	sci	PROPN
ejpam-4498	328	2	.	.	PUNCT
ejpam-4498	328	3	tech	tech	PROPN
ejpam-4498	328	4	.	.	PUNCT
ejpam-4498	328	5	,	,	PUNCT
ejpam-4498	328	6	44:132–142	44:132–142	PROPN
ejpam-4498	328	7	,	,	PUNCT
ejpam-4498	328	8	2013	2013	NUM
ejpam-4498	328	9	.	.	PUNCT
ejpam-4498	329	1	[	[	X
ejpam-4498	329	2	6	6	NUM
ejpam-4498	329	3	]	]	X
ejpam-4498	329	4	y.	y.	PROPN
ejpam-4498	329	5	kobayashi	kobayashi	PROPN
ejpam-4498	329	6	.	.	PUNCT
ejpam-4498	330	1	derivation	derivation	NOUN
ejpam-4498	330	2	of	of	ADP
ejpam-4498	330	3	some	some	DET
ejpam-4498	330	4	formulae	formulae	NOUN
ejpam-4498	330	5	in	in	ADP
ejpam-4498	330	6	combinatrics	combinatric	NOUN
ejpam-4498	330	7	by	by	ADP
ejpam-4498	330	8	heuristic	heuristic	ADJ
ejpam-4498	330	9	method	method	NOUN
ejpam-4498	330	10	.	.	PUNCT
ejpam-4498	331	1	int	int	NOUN
ejpam-4498	331	2	.	.	PUNCT
ejpam-4498	332	1	j.	j.	PROPN
ejpam-4498	332	2	math	math	PROPN
ejpam-4498	332	3	.	.	PUNCT
ejpam-4498	333	1	sci	sci	PROPN
ejpam-4498	333	2	.	.	PUNCT
ejpam-4498	333	3	tech	tech	PROPN
ejpam-4498	333	4	.	.	PROPN
ejpam-4498	333	5	,	,	PUNCT
ejpam-4498	333	6	46:469–476	46:469–476	PROPN
ejpam-4498	333	7	,	,	PUNCT
ejpam-4498	333	8	2015	2015	NUM
ejpam-4498	333	9	.	.	PUNCT
ejpam-4498	334	1	[	[	X
ejpam-4498	334	2	erratum	erratum	NOUN
ejpam-4498	334	3	46	46	NUM
ejpam-4498	334	4	(	(	PUNCT
ejpam-4498	334	5	2015	2015	NUM
ejpam-4498	334	6	)	)	PUNCT
ejpam-4498	334	7	,	,	PUNCT
ejpam-4498	334	8	795	795	NUM
ejpam-4498	334	9	]	]	PUNCT
ejpam-4498	334	10	.	.	PUNCT
ejpam-4498	335	1	references	reference	NOUN
ejpam-4498	335	2	1253	1253	NUM
ejpam-4498	335	3	[	[	X
ejpam-4498	335	4	7	7	X
ejpam-4498	335	5	]	]	X
ejpam-4498	335	6	y.	y.	PROPN
ejpam-4498	335	7	kobayashi	kobayashi	PROPN
ejpam-4498	335	8	.	.	PUNCT
ejpam-4498	336	1	proof	proof	NOUN
ejpam-4498	336	2	without	without	ADP
ejpam-4498	336	3	words	word	NOUN
ejpam-4498	336	4	:	:	PUNCT
ejpam-4498	336	5	relationship	relationship	NOUN
ejpam-4498	336	6	between	between	ADP
ejpam-4498	336	7	combination	combination	NOUN
ejpam-4498	336	8	nc2	nc2	NOUN
ejpam-4498	336	9	and	and	CCONJ
ejpam-4498	336	10	arithmetic	arithmetic	ADJ
ejpam-4498	336	11	series	series	NOUN
ejpam-4498	336	12	.	.	PUNCT
ejpam-4498	337	1	far	far	PROPN
ejpam-4498	337	2	east	east	PROPN
ejpam-4498	337	3	j.	j.	PROPN
ejpam-4498	337	4	math	math	PROPN
ejpam-4498	337	5	.	.	PUNCT
ejpam-4498	338	1	educ	educ	PROPN
ejpam-4498	338	2	.	.	PUNCT
ejpam-4498	338	3	,	,	PUNCT
ejpam-4498	338	4	19:57–58	19:57–58	NUM
ejpam-4498	338	5	,	,	PUNCT
ejpam-4498	338	6	2019	2019	NUM
ejpam-4498	338	7	.	.	PUNCT
ejpam-4498	339	1	[	[	X
ejpam-4498	339	2	8	8	NUM
ejpam-4498	339	3	]	]	X
ejpam-4498	339	4	y.	y.	PROPN
ejpam-4498	339	5	kobayashi	kobayashi	PROPN
ejpam-4498	339	6	.	.	PUNCT
ejpam-4498	340	1	proof	proof	NOUN
ejpam-4498	340	2	without	without	ADP
ejpam-4498	340	3	words	word	NOUN
ejpam-4498	340	4	:	:	PUNCT
ejpam-4498	340	5	relationship	relationship	NOUN
ejpam-4498	340	6	between	between	ADP
ejpam-4498	340	7	the	the	DET
ejpam-4498	340	8	sum	sum	NOUN
ejpam-4498	340	9	of	of	ADP
ejpam-4498	340	10	triangular	triangular	NOUN
ejpam-4498	340	11	numbers	number	NOUN
ejpam-4498	340	12	,	,	PUNCT
ejpam-4498	340	13	sum	sum	NOUN
ejpam-4498	340	14	of	of	ADP
ejpam-4498	340	15	natural	natural	ADJ
ejpam-4498	340	16	numbers	number	NOUN
ejpam-4498	340	17	,	,	PUNCT
ejpam-4498	340	18	and	and	CCONJ
ejpam-4498	340	19	sum	sum	NOUN
ejpam-4498	340	20	of	of	ADP
ejpam-4498	340	21	square	square	ADJ
ejpam-4498	340	22	numbers	number	NOUN
ejpam-4498	340	23	.	.	PUNCT
ejpam-4498	341	1	far	far	ADV
ejpam-4498	341	2	east	east	PROPN
ejpam-4498	341	3	j.	j.	PROPN
ejpam-4498	341	4	math	math	PROPN
ejpam-4498	341	5	.	.	PUNCT
ejpam-4498	342	1	educ	educ	PROPN
ejpam-4498	342	2	.	.	PUNCT
ejpam-4498	342	3	,	,	PUNCT
ejpam-4498	342	4	21:83–91	21:83–91	NUM
ejpam-4498	342	5	,	,	PUNCT
ejpam-4498	342	6	2021	2021	NUM
ejpam-4498	342	7	.	.	PUNCT
ejpam-4498	343	1	[	[	X
ejpam-4498	343	2	9	9	NUM
ejpam-4498	343	3	]	]	SYM
ejpam-4498	343	4	l.	l.	PROPN
ejpam-4498	343	5	c.	c.	PROPN
ejpam-4498	343	6	larson	larson	PROPN
ejpam-4498	343	7	.	.	PUNCT
ejpam-4498	344	1	a	a	DET
ejpam-4498	344	2	discrete	discrete	ADJ
ejpam-4498	344	3	look	look	NOUN
ejpam-4498	344	4	at	at	ADP
ejpam-4498	344	5	1	1	NUM
ejpam-4498	344	6	+	+	SYM
ejpam-4498	344	7	2	2	NUM
ejpam-4498	344	8	+	+	CCONJ
ejpam-4498	344	9	·	·	PUNCT
ejpam-4498	344	10	·	·	PUNCT
ejpam-4498	344	11	·	·	PUNCT
ejpam-4498	344	12	+	+	NUM
ejpam-4498	344	13	n.	n.	PROPN
ejpam-4498	344	14	college	college	NOUN
ejpam-4498	344	15	math	math	NOUN
ejpam-4498	344	16	.	.	PUNCT
ejpam-4498	345	1	j.	j.	PROPN
ejpam-4498	345	2	,	,	PUNCT
ejpam-4498	345	3	16:369–382	16:369–382	PROPN
ejpam-4498	345	4	,	,	PUNCT
ejpam-4498	345	5	1985	1985	NUM
ejpam-4498	345	6	.	.	PUNCT
ejpam-4498	346	1	[	[	X
ejpam-4498	346	2	10	10	NUM
ejpam-4498	346	3	]	]	X
ejpam-4498	346	4	zhi	zhi	PROPN
ejpam-4498	346	5	-	-	PUNCT
ejpam-4498	346	6	guo	guo	PROPN
ejpam-4498	346	7	liu	liu	PROPN
ejpam-4498	346	8	.	.	PUNCT
ejpam-4498	347	1	an	an	DET
ejpam-4498	347	2	identity	identity	NOUN
ejpam-4498	347	3	of	of	ADP
ejpam-4498	347	4	ramanujan	ramanujan	NOUN
ejpam-4498	347	5	and	and	CCONJ
ejpam-4498	347	6	the	the	DET
ejpam-4498	347	7	representation	representation	NOUN
ejpam-4498	347	8	of	of	ADP
ejpam-4498	347	9	integers	integer	NOUN
ejpam-4498	347	10	as	as	ADP
ejpam-4498	347	11	sums	sum	NOUN
ejpam-4498	347	12	of	of	ADP
ejpam-4498	347	13	triangular	triangular	NOUN
ejpam-4498	347	14	numbers	number	NOUN
ejpam-4498	347	15	.	.	PUNCT
ejpam-4498	348	1	the	the	DET
ejpam-4498	348	2	ramanujan	ramanujan	PROPN
ejpam-4498	348	3	journal	journal	PROPN
ejpam-4498	348	4	.	.	PUNCT
ejpam-4498	348	5	,	,	PUNCT
ejpam-4498	348	6	7:407–434	7:407–434	NUM
ejpam-4498	348	7	,	,	PUNCT
ejpam-4498	348	8	2003	2003	NUM
ejpam-4498	348	9	.	.	PUNCT
ejpam-4498	349	1	[	[	X
ejpam-4498	349	2	11	11	NUM
ejpam-4498	349	3	]	]	PUNCT
ejpam-4498	349	4	r.	r.	PROPN
ejpam-4498	349	5	b.	b.	PROPN
ejpam-4498	349	6	nelsen	nelsen	PROPN
ejpam-4498	349	7	,	,	PUNCT
ejpam-4498	349	8	editor	editor	NOUN
ejpam-4498	349	9	.	.	PUNCT
ejpam-4498	350	1	proofs	proof	NOUN
ejpam-4498	350	2	without	without	ADP
ejpam-4498	350	3	words	word	NOUN
ejpam-4498	350	4	.	.	PUNCT
ejpam-4498	351	1	the	the	DET
ejpam-4498	351	2	mathematical	mathematical	ADJ
ejpam-4498	351	3	association	association	PROPN
ejpam-4498	351	4	of	of	ADP
ejpam-4498	351	5	america	america	PROPN
ejpam-4498	351	6	,	,	PUNCT
ejpam-4498	351	7	2000	2000	NUM
ejpam-4498	351	8	.	.	PUNCT
ejpam-4498	352	1	[	[	X
ejpam-4498	352	2	12	12	NUM
ejpam-4498	352	3	]	]	PUNCT
ejpam-4498	352	4	j.	j.	PROPN
ejpam-4498	352	5	c.	c.	PROPN
ejpam-4498	352	6	slater	slater	PROPN
ejpam-4498	352	7	and	and	CCONJ
ejpam-4498	352	8	n.	n.	PROPN
ejpam-4498	352	9	h.	h.	PROPN
ejpam-4498	352	10	frank	frank	PROPN
ejpam-4498	352	11	.	.	PUNCT
ejpam-4498	353	1	introduction	introduction	NOUN
ejpam-4498	353	2	to	to	ADP
ejpam-4498	353	3	theoretical	theoretical	ADJ
ejpam-4498	353	4	physics	physics	NOUN
ejpam-4498	353	5	.	.	PUNCT
ejpam-4498	354	1	mcgrawhill	mcgrawhill	PROPN
ejpam-4498	354	2	,	,	PUNCT
ejpam-4498	354	3	1933	1933	NUM
ejpam-4498	354	4	.	.	PUNCT