id	sid	tid	token	lemma	pos
ejpam-4476	1	1	european	european	PROPN
ejpam-4476	1	2	journal	journal	PROPN
ejpam-4476	1	3	of	of	ADP
ejpam-4476	1	4	pure	pure	ADJ
ejpam-4476	1	5	and	and	CCONJ
ejpam-4476	1	6	applied	apply	VERB
ejpam-4476	1	7	mathematics	mathematic	NOUN
ejpam-4476	1	8	vol	vol	NOUN
ejpam-4476	1	9	.	.	PROPN
ejpam-4476	2	1	15	15	NUM
ejpam-4476	2	2	,	,	PUNCT
ejpam-4476	2	3	no	no	INTJ
ejpam-4476	2	4	.	.	NOUN
ejpam-4476	2	5	3	3	NUM
ejpam-4476	2	6	,	,	PUNCT
ejpam-4476	2	7	2022	2022	NUM
ejpam-4476	2	8	,	,	PUNCT
ejpam-4476	2	9	864	864	NUM
ejpam-4476	2	10	-	-	SYM
ejpam-4476	2	11	877	877	NUM
ejpam-4476	2	12	issn	issn	PROPN
ejpam-4476	2	13	1307	1307	NUM
ejpam-4476	2	14	-	-	SYM
ejpam-4476	2	15	5543	5543	NUM
ejpam-4476	2	16	–	–	PUNCT
ejpam-4476	2	17	ejpam.com	ejpam.com	X
ejpam-4476	2	18	published	publish	VERB
ejpam-4476	2	19	by	by	ADP
ejpam-4476	2	20	new	new	PROPN
ejpam-4476	2	21	york	york	PROPN
ejpam-4476	2	22	business	business	PROPN
ejpam-4476	3	1	global	global	PROPN
ejpam-4476	3	2	a	a	DET
ejpam-4476	3	3	pegagogical	pegagogical	ADJ
ejpam-4476	3	4	study	study	NOUN
ejpam-4476	3	5	of	of	ADP
ejpam-4476	3	6	geometric	geometric	ADJ
ejpam-4476	3	7	relations	relation	NOUN
ejpam-4476	3	8	between	between	ADP
ejpam-4476	3	9	the	the	DET
ejpam-4476	3	10	sums	sum	NOUN
ejpam-4476	3	11	of	of	ADP
ejpam-4476	3	12	cubic	cubic	ADJ
ejpam-4476	3	13	integer	integer	NOUN
ejpam-4476	3	14	numbers	number	NOUN
ejpam-4476	3	15	and	and	CCONJ
ejpam-4476	3	16	the	the	DET
ejpam-4476	3	17	sums	sum	NOUN
ejpam-4476	3	18	of	of	ADP
ejpam-4476	3	19	square	square	ADJ
ejpam-4476	3	20	integer	integer	NOUN
ejpam-4476	3	21	numbers	number	NOUN
ejpam-4476	3	22	with	with	ADP
ejpam-4476	3	23	the	the	DET
ejpam-4476	3	24	sums	sum	NOUN
ejpam-4476	3	25	of	of	ADP
ejpam-4476	3	26	odd	odd	ADJ
ejpam-4476	3	27	numbers	number	NOUN
ejpam-4476	3	28	yukio	yukio	PROPN
ejpam-4476	3	29	kobayashi	kobayashi	PROPN
ejpam-4476	3	30	department	department	PROPN
ejpam-4476	3	31	of	of	ADP
ejpam-4476	3	32	information	information	NOUN
ejpam-4476	3	33	systems	system	NOUN
ejpam-4476	3	34	science	science	NOUN
ejpam-4476	3	35	,	,	PUNCT
ejpam-4476	3	36	faculty	faculty	NOUN
ejpam-4476	3	37	of	of	ADP
ejpam-4476	3	38	science	science	NOUN
ejpam-4476	3	39	and	and	CCONJ
ejpam-4476	3	40	engineering	engineering	NOUN
ejpam-4476	3	41	,	,	PUNCT
ejpam-4476	3	42	soka	soka	PROPN
ejpam-4476	3	43	university	university	NOUN
ejpam-4476	3	44	1	1	NUM
ejpam-4476	3	45	-	-	SYM
ejpam-4476	3	46	236	236	NUM
ejpam-4476	3	47	tangi	tangi	NOUN
ejpam-4476	3	48	-	-	PUNCT
ejpam-4476	3	49	machi	machi	NOUN
ejpam-4476	3	50	,	,	PUNCT
ejpam-4476	3	51	hachioji	hachioji	PROPN
ejpam-4476	3	52	-	-	PUNCT
ejpam-4476	3	53	shi	shi	PROPN
ejpam-4476	3	54	,	,	PUNCT
ejpam-4476	3	55	tokyo	tokyo	PROPN
ejpam-4476	3	56	192	192	NUM
ejpam-4476	3	57	-	-	SYM
ejpam-4476	3	58	8577	8577	NUM
ejpam-4476	3	59	,	,	PUNCT
ejpam-4476	3	60	japan	japan	PROPN
ejpam-4476	3	61	abstract	abstract	PROPN
ejpam-4476	3	62	.	.	PUNCT
ejpam-4476	4	1	the	the	DET
ejpam-4476	4	2	geometric	geometric	ADJ
ejpam-4476	4	3	meaning	meaning	NOUN
ejpam-4476	4	4	of	of	ADP
ejpam-4476	4	5	the	the	DET
ejpam-4476	4	6	sums	sum	NOUN
ejpam-4476	4	7	of	of	ADP
ejpam-4476	4	8	cubic	cubic	ADJ
ejpam-4476	4	9	and	and	CCONJ
ejpam-4476	4	10	square	square	ADJ
ejpam-4476	4	11	numbers	number	NOUN
ejpam-4476	4	12	can	can	AUX
ejpam-4476	4	13	be	be	AUX
ejpam-4476	4	14	explained	explain	VERB
ejpam-4476	4	15	by	by	ADP
ejpam-4476	4	16	expressing	express	VERB
ejpam-4476	4	17	them	they	PRON
ejpam-4476	4	18	as	as	ADP
ejpam-4476	4	19	sums	sum	NOUN
ejpam-4476	4	20	of	of	ADP
ejpam-4476	4	21	consecutive	consecutive	ADJ
ejpam-4476	4	22	odd	odd	ADJ
ejpam-4476	4	23	numbers	number	NOUN
ejpam-4476	4	24	.	.	PUNCT
ejpam-4476	5	1	by	by	ADP
ejpam-4476	5	2	illustrating	illustrate	VERB
ejpam-4476	5	3	not	not	PART
ejpam-4476	5	4	only	only	ADV
ejpam-4476	5	5	the	the	DET
ejpam-4476	5	6	equations	equation	NOUN
ejpam-4476	5	7	but	but	CCONJ
ejpam-4476	5	8	also	also	ADV
ejpam-4476	5	9	simple	simple	ADJ
ejpam-4476	5	10	diagrams	diagram	NOUN
ejpam-4476	5	11	,	,	PUNCT
ejpam-4476	5	12	we	we	PRON
ejpam-4476	5	13	can	can	AUX
ejpam-4476	5	14	understand	understand	VERB
ejpam-4476	5	15	how	how	SCONJ
ejpam-4476	5	16	these	these	DET
ejpam-4476	5	17	sums	sum	NOUN
ejpam-4476	5	18	can	can	AUX
ejpam-4476	5	19	be	be	AUX
ejpam-4476	5	20	obtained	obtain	VERB
ejpam-4476	5	21	.	.	PUNCT
ejpam-4476	6	1	geometrically	geometrically	ADV
ejpam-4476	6	2	viewing	view	VERB
ejpam-4476	6	3	the	the	DET
ejpam-4476	6	4	relationship	relationship	NOUN
ejpam-4476	6	5	between	between	ADP
ejpam-4476	6	6	series	series	NOUN
ejpam-4476	6	7	,	,	PUNCT
ejpam-4476	6	8	rather	rather	ADV
ejpam-4476	6	9	than	than	ADP
ejpam-4476	6	10	merely	merely	ADV
ejpam-4476	6	11	learning	learn	VERB
ejpam-4476	6	12	how	how	SCONJ
ejpam-4476	6	13	to	to	PART
ejpam-4476	6	14	calculate	calculate	VERB
ejpam-4476	6	15	them	they	PRON
ejpam-4476	6	16	,	,	PUNCT
ejpam-4476	6	17	helps	help	VERB
ejpam-4476	6	18	students	student	NOUN
ejpam-4476	6	19	understand	understand	VERB
ejpam-4476	6	20	these	these	DET
ejpam-4476	6	21	series	series	NOUN
ejpam-4476	6	22	systematically	systematically	ADV
ejpam-4476	6	23	.	.	PUNCT
ejpam-4476	7	1	if	if	SCONJ
ejpam-4476	7	2	natural	natural	ADJ
ejpam-4476	7	3	numbers	number	NOUN
ejpam-4476	7	4	are	be	AUX
ejpam-4476	7	5	expressed	express	VERB
ejpam-4476	7	6	as	as	ADP
ejpam-4476	7	7	sums	sum	NOUN
ejpam-4476	7	8	of	of	ADP
ejpam-4476	7	9	only	only	ADV
ejpam-4476	7	10	1	1	NUM
ejpam-4476	7	11	,	,	PUNCT
ejpam-4476	7	12	the	the	DET
ejpam-4476	7	13	sums	sum	NOUN
ejpam-4476	7	14	of	of	ADP
ejpam-4476	7	15	square	square	ADJ
ejpam-4476	7	16	numbers	number	NOUN
ejpam-4476	7	17	and	and	CCONJ
ejpam-4476	7	18	of	of	ADP
ejpam-4476	7	19	cubic	cubic	ADJ
ejpam-4476	7	20	numbers	number	NOUN
ejpam-4476	7	21	can	can	AUX
ejpam-4476	7	22	be	be	AUX
ejpam-4476	7	23	related	relate	VERB
ejpam-4476	7	24	to	to	ADP
ejpam-4476	7	25	the	the	DET
ejpam-4476	7	26	multiplication	multiplication	NOUN
ejpam-4476	7	27	of	of	ADP
ejpam-4476	7	28	a	a	DET
ejpam-4476	7	29	vertical	vertical	ADJ
ejpam-4476	7	30	vector	vector	NOUN
ejpam-4476	7	31	with	with	ADP
ejpam-4476	7	32	a	a	DET
ejpam-4476	7	33	horizontal	horizontal	ADJ
ejpam-4476	7	34	one	one	NUM
ejpam-4476	7	35	whose	whose	DET
ejpam-4476	7	36	components	component	NOUN
ejpam-4476	7	37	are	be	AUX
ejpam-4476	7	38	all	all	PRON
ejpam-4476	7	39	1	1	NUM
ejpam-4476	7	40	’s	’s	NOUN
ejpam-4476	7	41	.	.	PUNCT
ejpam-4476	8	1	this	this	DET
ejpam-4476	8	2	view	view	NOUN
ejpam-4476	8	3	is	be	AUX
ejpam-4476	8	4	educational	educational	ADJ
ejpam-4476	8	5	in	in	ADP
ejpam-4476	8	6	showing	show	VERB
ejpam-4476	8	7	that	that	PRON
ejpam-4476	8	8	matrix	matrix	NOUN
ejpam-4476	8	9	multiplication	multiplication	NOUN
ejpam-4476	8	10	is	be	AUX
ejpam-4476	8	11	related	relate	VERB
ejpam-4476	8	12	to	to	ADP
ejpam-4476	8	13	the	the	DET
ejpam-4476	8	14	calculation	calculation	NOUN
ejpam-4476	8	15	of	of	ADP
ejpam-4476	8	16	series	series	NOUN
ejpam-4476	8	17	.	.	PUNCT
ejpam-4476	9	1	the	the	DET
ejpam-4476	9	2	notion	notion	NOUN
ejpam-4476	9	3	that	that	SCONJ
ejpam-4476	9	4	concepts	concept	NOUN
ejpam-4476	9	5	as	as	ADV
ejpam-4476	9	6	different	different	ADJ
ejpam-4476	9	7	as	as	ADP
ejpam-4476	9	8	series	series	NOUN
ejpam-4476	9	9	and	and	CCONJ
ejpam-4476	9	10	matrices	matrix	NOUN
ejpam-4476	9	11	are	be	AUX
ejpam-4476	9	12	fundamentally	fundamentally	ADV
ejpam-4476	9	13	and	and	CCONJ
ejpam-4476	9	14	closely	closely	ADV
ejpam-4476	9	15	related	relate	VERB
ejpam-4476	9	16	is	be	AUX
ejpam-4476	9	17	an	an	DET
ejpam-4476	9	18	important	important	ADJ
ejpam-4476	9	19	perspective	perspective	NOUN
ejpam-4476	9	20	in	in	ADP
ejpam-4476	9	21	mathematics	mathematic	NOUN
ejpam-4476	9	22	education	education	NOUN
ejpam-4476	9	23	.	.	PUNCT
ejpam-4476	10	1	2020	2020	NUM
ejpam-4476	10	2	mathematics	mathematic	NOUN
ejpam-4476	10	3	subject	subject	NOUN
ejpam-4476	10	4	classifications	classification	NOUN
ejpam-4476	10	5	:	:	PUNCT
ejpam-4476	10	6	97h20	97h20	NUM
ejpam-4476	10	7	key	key	ADJ
ejpam-4476	10	8	words	word	NOUN
ejpam-4476	10	9	and	and	CCONJ
ejpam-4476	10	10	phrases	phrase	NOUN
ejpam-4476	10	11	:	:	PUNCT
ejpam-4476	10	12	square	square	ADJ
ejpam-4476	10	13	numbers	number	NOUN
ejpam-4476	10	14	;	;	PUNCT
ejpam-4476	10	15	cubic	cubic	ADJ
ejpam-4476	10	16	numbers	number	NOUN
ejpam-4476	10	17	;	;	PUNCT
ejpam-4476	10	18	series	series	NOUN
ejpam-4476	10	19	;	;	PUNCT
ejpam-4476	10	20	matrix	matrix	NOUN
ejpam-4476	10	21	multiplication	multiplication	NOUN
ejpam-4476	10	22	;	;	PUNCT
ejpam-4476	10	23	pictorial	pictorial	ADJ
ejpam-4476	10	24	proof	proof	NOUN
ejpam-4476	10	25	;	;	PUNCT
ejpam-4476	10	26	visual	visual	ADJ
ejpam-4476	10	27	thinking	thinking	NOUN
ejpam-4476	10	28	1	1	NUM
ejpam-4476	10	29	.	.	PUNCT
ejpam-4476	11	1	introduction	introduction	NOUN
ejpam-4476	11	2	the	the	DET
ejpam-4476	11	3	calculation	calculation	NOUN
ejpam-4476	11	4	of	of	ADP
ejpam-4476	11	5	n∑	n∑	PROPN
ejpam-4476	11	6	k=1	k=1	PROPN
ejpam-4476	11	7	k3	k3	VERB
ejpam-4476	11	8	and	and	CCONJ
ejpam-4476	11	9	n∑	n∑	NOUN
ejpam-4476	11	10	k=1	k=1	PROPN
ejpam-4476	12	1	k2	k2	PROPN
ejpam-4476	12	2	is	be	AUX
ejpam-4476	12	3	a	a	DET
ejpam-4476	12	4	problem	problem	NOUN
ejpam-4476	12	5	of	of	ADP
ejpam-4476	12	6	counting	count	VERB
ejpam-4476	12	7	geometrical	geometrical	ADJ
ejpam-4476	12	8	arrangements	arrangement	NOUN
ejpam-4476	12	9	of	of	ADP
ejpam-4476	12	10	(	(	PUNCT
ejpam-4476	12	11	1	1	NUM
ejpam-4476	12	12	+	+	NUM
ejpam-4476	12	13	2	2	NUM
ejpam-4476	12	14	+	+	NUM
ejpam-4476	12	15	·	·	PUNCT
ejpam-4476	12	16	·	·	PUNCT
ejpam-4476	12	17	·	·	PUNCT
ejpam-4476	13	1	+	+	NUM
ejpam-4476	13	2	n)2	n)2	NOUN
ejpam-4476	13	3	points	point	NOUN
ejpam-4476	13	4	and	and	CCONJ
ejpam-4476	13	5	(	(	PUNCT
ejpam-4476	13	6	2n	2n	NUM
ejpam-4476	13	7	+	+	CCONJ
ejpam-4476	13	8	1)(1	1)(1	NUM
ejpam-4476	13	9	+	+	CCONJ
ejpam-4476	13	10	2	2	NUM
ejpam-4476	13	11	+	+	NUM
ejpam-4476	13	12	·	·	PUNCT
ejpam-4476	13	13	·	·	PUNCT
ejpam-4476	13	14	·	·	PUNCT
ejpam-4476	13	15	+	+	NUM
ejpam-4476	13	16	n	n	CCONJ
ejpam-4476	13	17	)	)	PUNCT
ejpam-4476	13	18	points	point	NOUN
ejpam-4476	13	19	,	,	PUNCT
ejpam-4476	13	20	respectively	respectively	ADV
ejpam-4476	13	21	[	[	X
ejpam-4476	13	22	3	3	NUM
ejpam-4476	13	23	]	]	PUNCT
ejpam-4476	13	24	.	.	PUNCT
ejpam-4476	14	1	generally	generally	ADV
ejpam-4476	14	2	,	,	PUNCT
ejpam-4476	14	3	n∑	n∑	PROPN
ejpam-4476	14	4	k=0	k=0	PROPN
ejpam-4476	14	5	kℓ	kℓ	NOUN
ejpam-4476	14	6	can	can	AUX
ejpam-4476	14	7	be	be	AUX
ejpam-4476	14	8	represented	represent	VERB
ejpam-4476	14	9	by	by	ADP
ejpam-4476	14	10	n∑	n∑	PROPN
ejpam-4476	14	11	k=0	k=0	PROPN
ejpam-4476	14	12	k	k	PROPN
ejpam-4476	14	13	through	through	ADP
ejpam-4476	14	14	recursion	recursion	NOUN
ejpam-4476	14	15	formulae	formulae	NOUN
ejpam-4476	14	16	based	base	VERB
ejpam-4476	14	17	on	on	ADP
ejpam-4476	14	18	the	the	DET
ejpam-4476	14	19	euler	euler	NOUN
ejpam-4476	14	20	-	-	PUNCT
ejpam-4476	14	21	maclaurin	maclaurin	NOUN
ejpam-4476	14	22	sum	sum	NOUN
ejpam-4476	14	23	formulae	formulae	NOUN
ejpam-4476	14	24	which	which	PRON
ejpam-4476	14	25	are	be	AUX
ejpam-4476	14	26	integral	integral	ADJ
ejpam-4476	14	27	representations	representation	NOUN
ejpam-4476	14	28	[	[	X
ejpam-4476	14	29	5	5	NUM
ejpam-4476	14	30	]	]	PUNCT
ejpam-4476	14	31	.	.	PUNCT
ejpam-4476	15	1	this	this	DET
ejpam-4476	15	2	fact	fact	NOUN
ejpam-4476	15	3	means	mean	VERB
ejpam-4476	15	4	that	that	SCONJ
ejpam-4476	15	5	n∑	n∑	PROPN
ejpam-4476	15	6	k=0	k=0	PROPN
ejpam-4476	15	7	kℓ	kℓ	NOUN
ejpam-4476	15	8	can	can	AUX
ejpam-4476	15	9	be	be	AUX
ejpam-4476	15	10	resolved	resolve	VERB
ejpam-4476	15	11	into	into	ADP
ejpam-4476	15	12	the	the	DET
ejpam-4476	15	13	factor	factor	NOUN
ejpam-4476	15	14	n(n	n(n	PROPN
ejpam-4476	15	15	+	+	CCONJ
ejpam-4476	15	16	1	1	NUM
ejpam-4476	15	17	)	)	PUNCT
ejpam-4476	15	18	.	.	PUNCT
ejpam-4476	16	1	the	the	DET
ejpam-4476	16	2	case	case	NOUN
ejpam-4476	16	3	of	of	ADP
ejpam-4476	16	4	ℓ	ℓ	PROPN
ejpam-4476	16	5	=	=	SYM
ejpam-4476	16	6	2	2	NUM
ejpam-4476	16	7	is	be	AUX
ejpam-4476	16	8	the	the	DET
ejpam-4476	16	9	simplest	simple	ADJ
ejpam-4476	16	10	;	;	PUNCT
ejpam-4476	16	11	doi	doi	NOUN
ejpam-4476	16	12	:	:	PUNCT
ejpam-4476	16	13	https://doi.org/10.29020/nybg.ejpam.v15i3.4476	https://doi.org/10.29020/nybg.ejpam.v15i3.4476	VERB
ejpam-4476	16	14	email	email	NOUN
ejpam-4476	16	15	address	address	NOUN
ejpam-4476	16	16	:	:	PUNCT
ejpam-4476	16	17	koba@t.soka.ac.jp	koba@t.soka.ac.jp	PROPN
ejpam-4476	16	18	(	(	PUNCT
ejpam-4476	16	19	y.	y.	PROPN
ejpam-4476	16	20	kobayashi	kobayashi	PROPN
ejpam-4476	16	21	)	)	PUNCT
ejpam-4476	16	22	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4476	16	23	864	864	NUM
ejpam-4476	17	1	©	©	ADP
ejpam-4476	17	2	2022	2022	NUM
ejpam-4476	17	3	ejpam	ejpam	VERB
ejpam-4476	17	4	all	all	DET
ejpam-4476	17	5	rights	right	NOUN
ejpam-4476	17	6	reserved	reserve	VERB
ejpam-4476	17	7	.	.	PUNCT
ejpam-4476	18	1	y.	y.	PROPN
ejpam-4476	18	2	kobayashi	kobayashi	PROPN
ejpam-4476	18	3	/	/	SYM
ejpam-4476	18	4	eur	eur	PROPN
ejpam-4476	18	5	.	.	PUNCT
ejpam-4476	19	1	j.	j.	PROPN
ejpam-4476	19	2	pure	pure	PROPN
ejpam-4476	19	3	appl	appl	PROPN
ejpam-4476	19	4	.	.	PROPN
ejpam-4476	19	5	math	math	PROPN
ejpam-4476	19	6	,	,	PUNCT
ejpam-4476	19	7	15	15	NUM
ejpam-4476	19	8	(	(	PUNCT
ejpam-4476	19	9	3	3	NUM
ejpam-4476	19	10	)	)	PUNCT
ejpam-4476	19	11	(	(	PUNCT
ejpam-4476	19	12	2022	2022	NUM
ejpam-4476	19	13	)	)	PUNCT
ejpam-4476	19	14	,	,	PUNCT
ejpam-4476	19	15	864	864	NUM
ejpam-4476	19	16	-	-	SYM
ejpam-4476	19	17	877	877	NUM
ejpam-4476	19	18	865	865	NUM
ejpam-4476	19	19	thus	thus	ADV
ejpam-4476	19	20	,	,	PUNCT
ejpam-4476	19	21	n∑	n∑	PROPN
ejpam-4476	19	22	k=1	k=1	PROPN
ejpam-4476	19	23	k2	k2	PROPN
ejpam-4476	19	24	can	can	AUX
ejpam-4476	19	25	be	be	AUX
ejpam-4476	19	26	pictorially	pictorially	ADV
ejpam-4476	19	27	proved	prove	VERB
ejpam-4476	19	28	by	by	ADP
ejpam-4476	19	29	a	a	DET
ejpam-4476	19	30	method	method	NOUN
ejpam-4476	19	31	of	of	ADP
ejpam-4476	19	32	calculating	calculate	VERB
ejpam-4476	19	33	area	area	NOUN
ejpam-4476	19	34	called	call	VERB
ejpam-4476	19	35	sectional	sectional	ADJ
ejpam-4476	19	36	mensuration	mensuration	NOUN
ejpam-4476	19	37	[	[	X
ejpam-4476	19	38	4	4	NUM
ejpam-4476	19	39	]	]	PUNCT
ejpam-4476	19	40	.	.	PUNCT
ejpam-4476	20	1	the	the	DET
ejpam-4476	20	2	tree	tree	NOUN
ejpam-4476	20	3	diagrams	diagram	NOUN
ejpam-4476	20	4	for	for	ADP
ejpam-4476	20	5	expressing	express	VERB
ejpam-4476	20	6	n∑	n∑	PROPN
ejpam-4476	20	7	k=1	k=1	PROPN
ejpam-4476	20	8	k3	k3	X
ejpam-4476	20	9	and	and	CCONJ
ejpam-4476	20	10	n∑	n∑	NOUN
ejpam-4476	20	11	k=1	k=1	PROPN
ejpam-4476	20	12	k2	k2	PROPN
ejpam-4476	20	13	reveal	reveal	VERB
ejpam-4476	20	14	the	the	DET
ejpam-4476	20	15	combinatorial	combinatorial	ADJ
ejpam-4476	20	16	meaning	meaning	NOUN
ejpam-4476	20	17	of	of	ADP
ejpam-4476	20	18	the	the	DET
ejpam-4476	20	19	recursion	recursion	NOUN
ejpam-4476	20	20	formulae	formulae	NOUN
ejpam-4476	20	21	[	[	X
ejpam-4476	20	22	5	5	NUM
ejpam-4476	20	23	]	]	PUNCT
ejpam-4476	20	24	.	.	PUNCT
ejpam-4476	21	1	these	these	DET
ejpam-4476	21	2	approaches	approach	NOUN
ejpam-4476	21	3	are	be	AUX
ejpam-4476	21	4	not	not	PART
ejpam-4476	21	5	necessarily	necessarily	ADV
ejpam-4476	21	6	practical	practical	ADJ
ejpam-4476	21	7	methods	method	NOUN
ejpam-4476	21	8	of	of	ADP
ejpam-4476	21	9	calculation	calculation	NOUN
ejpam-4476	21	10	,	,	PUNCT
ejpam-4476	21	11	although	although	SCONJ
ejpam-4476	21	12	,	,	PUNCT
ejpam-4476	21	13	from	from	ADP
ejpam-4476	21	14	a	a	DET
ejpam-4476	21	15	pedagogical	pedagogical	ADJ
ejpam-4476	21	16	standpoint	standpoint	NOUN
ejpam-4476	21	17	,	,	PUNCT
ejpam-4476	21	18	students	student	NOUN
ejpam-4476	21	19	can	can	AUX
ejpam-4476	21	20	learn	learn	VERB
ejpam-4476	21	21	that	that	SCONJ
ejpam-4476	21	22	the	the	DET
ejpam-4476	21	23	algebraic	algebraic	ADJ
ejpam-4476	21	24	formulae	formulae	NOUN
ejpam-4476	21	25	of	of	ADP
ejpam-4476	21	26	the	the	DET
ejpam-4476	21	27	series	series	NOUN
ejpam-4476	22	1	n∑	n∑	PROPN
ejpam-4476	23	1	k=1	k=1	PROPN
ejpam-4476	23	2	k3	k3	VERB
ejpam-4476	23	3	and	and	CCONJ
ejpam-4476	23	4	n∑	n∑	NOUN
ejpam-4476	23	5	k=1	k=1	PROPN
ejpam-4476	24	1	k2	k2	PROPN
ejpam-4476	24	2	are	be	AUX
ejpam-4476	24	3	related	relate	VERB
ejpam-4476	24	4	to	to	ADP
ejpam-4476	24	5	combinations	combination	NOUN
ejpam-4476	24	6	and	and	CCONJ
ejpam-4476	24	7	integrals	integral	NOUN
ejpam-4476	24	8	.	.	PUNCT
ejpam-4476	25	1	one	one	NUM
ejpam-4476	25	2	of	of	ADP
ejpam-4476	25	3	the	the	DET
ejpam-4476	25	4	aims	aim	NOUN
ejpam-4476	25	5	here	here	ADV
ejpam-4476	25	6	is	be	AUX
ejpam-4476	25	7	to	to	PART
ejpam-4476	25	8	show	show	VERB
ejpam-4476	25	9	simple	simple	ADJ
ejpam-4476	25	10	methods	method	NOUN
ejpam-4476	25	11	for	for	ADP
ejpam-4476	25	12	calculating	calculate	VERB
ejpam-4476	25	13	the	the	DET
ejpam-4476	25	14	series	series	NOUN
ejpam-4476	25	15	n∑	n∑	PROPN
ejpam-4476	26	1	k=1	k=1	PROPN
ejpam-4476	26	2	k3	k3	VERB
ejpam-4476	26	3	and	and	CCONJ
ejpam-4476	26	4	n∑	n∑	NOUN
ejpam-4476	26	5	k=1	k=1	PROPN
ejpam-4476	26	6	k2	k2	PROPN
ejpam-4476	26	7	by	by	ADP
ejpam-4476	26	8	using	use	VERB
ejpam-4476	26	9	the	the	DET
ejpam-4476	26	10	features	feature	NOUN
ejpam-4476	26	11	of	of	ADP
ejpam-4476	26	12	each	each	PRON
ejpam-4476	26	13	of	of	ADP
ejpam-4476	26	14	the	the	DET
ejpam-4476	26	15	integer	integer	NOUN
ejpam-4476	26	16	terms	term	NOUN
ejpam-4476	26	17	,	,	PUNCT
ejpam-4476	26	18	k3	k3	ADJ
ejpam-4476	26	19	and	and	CCONJ
ejpam-4476	26	20	k2	k2	NOUN
ejpam-4476	26	21	;	;	PUNCT
ejpam-4476	26	22	that	that	PRON
ejpam-4476	26	23	is	is	ADV
ejpam-4476	26	24	,	,	PUNCT
ejpam-4476	26	25	both	both	CCONJ
ejpam-4476	26	26	square	square	ADJ
ejpam-4476	26	27	and	and	CCONJ
ejpam-4476	26	28	cubic	cubic	ADJ
ejpam-4476	26	29	numbers	number	NOUN
ejpam-4476	26	30	can	can	AUX
ejpam-4476	26	31	be	be	AUX
ejpam-4476	26	32	represented	represent	VERB
ejpam-4476	26	33	as	as	ADP
ejpam-4476	26	34	sums	sum	NOUN
ejpam-4476	26	35	of	of	ADP
ejpam-4476	26	36	consecutive	consecutive	ADJ
ejpam-4476	26	37	odd	odd	ADJ
ejpam-4476	26	38	numbers	number	NOUN
ejpam-4476	26	39	.	.	PUNCT
ejpam-4476	27	1	by	by	ADP
ejpam-4476	27	2	geometrically	geometrically	ADV
ejpam-4476	27	3	viewing	view	VERB
ejpam-4476	27	4	the	the	DET
ejpam-4476	27	5	relationship	relationship	NOUN
ejpam-4476	27	6	between	between	ADP
ejpam-4476	27	7	series	series	NOUN
ejpam-4476	27	8	,	,	PUNCT
ejpam-4476	27	9	rather	rather	ADV
ejpam-4476	27	10	than	than	ADP
ejpam-4476	27	11	merely	merely	ADV
ejpam-4476	27	12	learning	learn	VERB
ejpam-4476	27	13	how	how	SCONJ
ejpam-4476	27	14	to	to	PART
ejpam-4476	27	15	calculate	calculate	VERB
ejpam-4476	27	16	them	they	PRON
ejpam-4476	27	17	,	,	PUNCT
ejpam-4476	27	18	students	student	NOUN
ejpam-4476	27	19	will	will	AUX
ejpam-4476	27	20	be	be	AUX
ejpam-4476	27	21	able	able	ADJ
ejpam-4476	27	22	to	to	PART
ejpam-4476	27	23	systematically	systematically	ADV
ejpam-4476	27	24	understand	understand	VERB
ejpam-4476	27	25	these	these	DET
ejpam-4476	27	26	series	series	NOUN
ejpam-4476	27	27	.	.	PUNCT
ejpam-4476	28	1	another	another	DET
ejpam-4476	28	2	aim	aim	NOUN
ejpam-4476	28	3	is	be	AUX
ejpam-4476	28	4	to	to	PART
ejpam-4476	28	5	show	show	VERB
ejpam-4476	28	6	that	that	SCONJ
ejpam-4476	28	7	the	the	DET
ejpam-4476	28	8	series	series	NOUN
ejpam-4476	28	9	n∑	n∑	PROPN
ejpam-4476	29	1	k=1	k=1	PROPN
ejpam-4476	29	2	k3	k3	VERB
ejpam-4476	29	3	and	and	CCONJ
ejpam-4476	29	4	n∑	n∑	NOUN
ejpam-4476	29	5	k=1	k=1	PROPN
ejpam-4476	30	1	k2	k2	PROPN
ejpam-4476	30	2	can	can	AUX
ejpam-4476	30	3	also	also	ADV
ejpam-4476	30	4	be	be	AUX
ejpam-4476	30	5	understood	understand	VERB
ejpam-4476	30	6	in	in	ADP
ejpam-4476	30	7	terms	term	NOUN
ejpam-4476	30	8	of	of	ADP
ejpam-4476	30	9	the	the	DET
ejpam-4476	30	10	multiplication	multiplication	NOUN
ejpam-4476	30	11	of	of	ADP
ejpam-4476	30	12	a	a	DET
ejpam-4476	30	13	vertical	vertical	ADJ
ejpam-4476	30	14	vector	vector	NOUN
ejpam-4476	30	15	with	with	ADP
ejpam-4476	30	16	a	a	DET
ejpam-4476	30	17	horizontal	horizontal	ADJ
ejpam-4476	30	18	one	one	NUM
ejpam-4476	30	19	whose	whose	DET
ejpam-4476	30	20	components	component	NOUN
ejpam-4476	30	21	are	be	AUX
ejpam-4476	30	22	all	all	PRON
ejpam-4476	30	23	1	1	NUM
ejpam-4476	30	24	’s	’s	NOUN
ejpam-4476	30	25	.	.	PUNCT
ejpam-4476	31	1	from	from	ADP
ejpam-4476	31	2	the	the	DET
ejpam-4476	31	3	standpoint	standpoint	NOUN
ejpam-4476	31	4	of	of	ADP
ejpam-4476	31	5	mathematics	mathematic	NOUN
ejpam-4476	31	6	education	education	NOUN
ejpam-4476	31	7	,	,	PUNCT
ejpam-4476	31	8	it	it	PRON
ejpam-4476	31	9	is	be	AUX
ejpam-4476	31	10	important	important	ADJ
ejpam-4476	31	11	for	for	SCONJ
ejpam-4476	31	12	students	student	NOUN
ejpam-4476	31	13	to	to	PART
ejpam-4476	31	14	understand	understand	VERB
ejpam-4476	31	15	that	that	PRON
ejpam-4476	31	16	matrix	matrix	NOUN
ejpam-4476	31	17	multiplication	multiplication	NOUN
ejpam-4476	31	18	is	be	AUX
ejpam-4476	31	19	related	relate	VERB
ejpam-4476	31	20	to	to	ADP
ejpam-4476	31	21	how	how	SCONJ
ejpam-4476	31	22	series	series	NOUN
ejpam-4476	31	23	are	be	AUX
ejpam-4476	31	24	calculated	calculate	VERB
ejpam-4476	31	25	.	.	PUNCT
ejpam-4476	32	1	the	the	DET
ejpam-4476	32	2	notion	notion	NOUN
ejpam-4476	32	3	that	that	SCONJ
ejpam-4476	32	4	concepts	concept	NOUN
ejpam-4476	32	5	as	as	ADV
ejpam-4476	32	6	different	different	ADJ
ejpam-4476	32	7	as	as	ADP
ejpam-4476	32	8	series	series	NOUN
ejpam-4476	32	9	and	and	CCONJ
ejpam-4476	32	10	matrices	matrix	NOUN
ejpam-4476	32	11	are	be	AUX
ejpam-4476	32	12	fundamentally	fundamentally	ADV
ejpam-4476	32	13	and	and	CCONJ
ejpam-4476	32	14	closely	closely	ADV
ejpam-4476	32	15	related	relate	VERB
ejpam-4476	32	16	is	be	AUX
ejpam-4476	32	17	an	an	DET
ejpam-4476	32	18	important	important	ADJ
ejpam-4476	32	19	perspective	perspective	NOUN
ejpam-4476	32	20	in	in	ADP
ejpam-4476	32	21	mathematics	mathematic	NOUN
ejpam-4476	32	22	education	education	NOUN
ejpam-4476	32	23	.	.	PUNCT
ejpam-4476	33	1	this	this	DET
ejpam-4476	33	2	article	article	NOUN
ejpam-4476	33	3	is	be	AUX
ejpam-4476	33	4	organised	organise	VERB
ejpam-4476	33	5	as	as	SCONJ
ejpam-4476	33	6	follows	follow	VERB
ejpam-4476	33	7	:	:	PUNCT
ejpam-4476	33	8	in	in	ADP
ejpam-4476	33	9	sections	section	NOUN
ejpam-4476	33	10	2	2	NUM
ejpam-4476	33	11	and	and	CCONJ
ejpam-4476	33	12	3	3	NUM
ejpam-4476	33	13	,	,	PUNCT
ejpam-4476	33	14	simple	simple	ADJ
ejpam-4476	33	15	methods	method	NOUN
ejpam-4476	33	16	for	for	ADP
ejpam-4476	33	17	calculating	calculate	VERB
ejpam-4476	34	1	n∑	n∑	PROPN
ejpam-4476	35	1	k=1	k=1	PROPN
ejpam-4476	35	2	k3	k3	VERB
ejpam-4476	35	3	and	and	CCONJ
ejpam-4476	35	4	n∑	n∑	NOUN
ejpam-4476	36	1	k=1	k=1	PROPN
ejpam-4476	36	2	k2	k2	PROPN
ejpam-4476	36	3	are	be	AUX
ejpam-4476	36	4	presented	present	VERB
ejpam-4476	36	5	in	in	ADP
ejpam-4476	36	6	terms	term	NOUN
ejpam-4476	36	7	of	of	ADP
ejpam-4476	36	8	the	the	DET
ejpam-4476	36	9	representations	representation	NOUN
ejpam-4476	36	10	of	of	ADP
ejpam-4476	36	11	k3	k3	NOUN
ejpam-4476	36	12	and	and	CCONJ
ejpam-4476	36	13	k2	k2	ADJ
ejpam-4476	36	14	by	by	ADP
ejpam-4476	36	15	sums	sum	NOUN
ejpam-4476	36	16	of	of	ADP
ejpam-4476	36	17	consecutive	consecutive	ADJ
ejpam-4476	36	18	odd	odd	ADJ
ejpam-4476	36	19	numbers	number	NOUN
ejpam-4476	36	20	.	.	PUNCT
ejpam-4476	37	1	these	these	DET
ejpam-4476	37	2	approaches	approach	NOUN
ejpam-4476	37	3	are	be	AUX
ejpam-4476	37	4	useful	useful	ADJ
ejpam-4476	37	5	for	for	ADP
ejpam-4476	37	6	developing	develop	VERB
ejpam-4476	37	7	the	the	DET
ejpam-4476	37	8	ability	ability	NOUN
ejpam-4476	37	9	to	to	PART
ejpam-4476	37	10	understand	understand	VERB
ejpam-4476	37	11	the	the	DET
ejpam-4476	37	12	characteristics	characteristic	NOUN
ejpam-4476	37	13	of	of	ADP
ejpam-4476	37	14	square	square	ADJ
ejpam-4476	37	15	and	and	CCONJ
ejpam-4476	37	16	cubic	cubic	ADJ
ejpam-4476	37	17	numbers	number	NOUN
ejpam-4476	37	18	rather	rather	ADV
ejpam-4476	37	19	than	than	ADP
ejpam-4476	37	20	mechanically	mechanically	ADV
ejpam-4476	37	21	calculating	calculate	VERB
ejpam-4476	37	22	the	the	DET
ejpam-4476	37	23	sums	sum	NOUN
ejpam-4476	37	24	.	.	PUNCT
ejpam-4476	38	1	in	in	ADP
ejpam-4476	38	2	section	section	NOUN
ejpam-4476	38	3	4	4	NUM
ejpam-4476	38	4	,	,	PUNCT
ejpam-4476	38	5	a	a	DET
ejpam-4476	38	6	common	common	ADJ
ejpam-4476	38	7	feature	feature	NOUN
ejpam-4476	38	8	of	of	ADP
ejpam-4476	38	9	n∑	n∑	NOUN
ejpam-4476	38	10	k=1	k=1	PROPN
ejpam-4476	38	11	k3	k3	VERB
ejpam-4476	38	12	and	and	CCONJ
ejpam-4476	38	13	n∑	n∑	NOUN
ejpam-4476	39	1	k=1	k=1	PROPN
ejpam-4476	39	2	k2	k2	PROPN
ejpam-4476	39	3	is	be	AUX
ejpam-4476	39	4	shown	show	VERB
ejpam-4476	39	5	with	with	ADP
ejpam-4476	39	6	the	the	DET
ejpam-4476	39	7	aid	aid	NOUN
ejpam-4476	39	8	of	of	ADP
ejpam-4476	39	9	matrix	matrix	NOUN
ejpam-4476	39	10	multiplication	multiplication	NOUN
ejpam-4476	39	11	.	.	PUNCT
ejpam-4476	40	1	2	2	X
ejpam-4476	40	2	.	.	X
ejpam-4476	40	3	sum	sum	NOUN
ejpam-4476	40	4	of	of	ADP
ejpam-4476	40	5	cubic	cubic	ADJ
ejpam-4476	40	6	numbers	number	NOUN
ejpam-4476	40	7	2.1	2.1	NUM
ejpam-4476	40	8	.	.	PUNCT
ejpam-4476	41	1	representation	representation	NOUN
ejpam-4476	41	2	based	base	VERB
ejpam-4476	41	3	on	on	ADP
ejpam-4476	41	4	sums	sum	NOUN
ejpam-4476	41	5	of	of	ADP
ejpam-4476	41	6	odd	odd	ADJ
ejpam-4476	41	7	numbers	number	NOUN
ejpam-4476	41	8	to	to	PART
ejpam-4476	41	9	adequately	adequately	ADV
ejpam-4476	41	10	explain	explain	VERB
ejpam-4476	41	11	the	the	DET
ejpam-4476	41	12	reason	reason	NOUN
ejpam-4476	41	13	that	that	PRON
ejpam-4476	41	14	n∑	n∑	ADV
ejpam-4476	41	15	k=1	k=1	PROPN
ejpam-4476	42	1	k3	k3	PROPN
ejpam-4476	42	2	can	can	AUX
ejpam-4476	42	3	be	be	AUX
ejpam-4476	42	4	represented	represent	VERB
ejpam-4476	42	5	as	as	ADP
ejpam-4476	42	6	(	(	PUNCT
ejpam-4476	42	7	n∑	n∑	INTJ
ejpam-4476	42	8	k=1	k=1	PUNCT
ejpam-4476	42	9	k	k	X
ejpam-4476	42	10	)	)	PUNCT
ejpam-4476	42	11	2	2	NUM
ejpam-4476	42	12	,	,	PUNCT
ejpam-4476	42	13	we	we	PRON
ejpam-4476	42	14	directly	directly	ADV
ejpam-4476	42	15	calculate	calculate	VERB
ejpam-4476	42	16	the	the	DET
ejpam-4476	42	17	sum	sum	NOUN
ejpam-4476	42	18	of	of	ADP
ejpam-4476	42	19	consecutive	consecutive	ADJ
ejpam-4476	42	20	odd	odd	ADJ
ejpam-4476	42	21	numbers	number	NOUN
ejpam-4476	42	22	instead	instead	ADV
ejpam-4476	42	23	of	of	ADP
ejpam-4476	42	24	using	use	VERB
ejpam-4476	42	25	the	the	DET
ejpam-4476	42	26	calculation	calculation	NOUN
ejpam-4476	42	27	method	method	NOUN
ejpam-4476	42	28	through	through	ADP
ejpam-4476	42	29	the	the	DET
ejpam-4476	42	30	formula	formula	NOUN
ejpam-4476	42	31	k3	k3	VERB
ejpam-4476	42	32	−	−	PROPN
ejpam-4476	43	1	(	(	PUNCT
ejpam-4476	43	2	k	k	PROPN
ejpam-4476	43	3	−	−	PROPN
ejpam-4476	43	4	1)3	1)3	PROPN
ejpam-4476	43	5	=	=	PUNCT
ejpam-4476	44	1	3k2	3k2	NUM
ejpam-4476	44	2	−	−	NOUN
ejpam-4476	44	3	3k	3k	PRON
ejpam-4476	45	1	+	+	NOUN
ejpam-4476	45	2	1	1	X
ejpam-4476	45	3	.	.	X
ejpam-4476	45	4	this	this	DET
ejpam-4476	45	5	approach	approach	NOUN
ejpam-4476	45	6	can	can	AUX
ejpam-4476	45	7	reveal	reveal	VERB
ejpam-4476	45	8	the	the	DET
ejpam-4476	45	9	y.	y.	PROPN
ejpam-4476	45	10	kobayashi	kobayashi	PROPN
ejpam-4476	45	11	/	/	SYM
ejpam-4476	45	12	eur	eur	PROPN
ejpam-4476	45	13	.	.	PUNCT
ejpam-4476	46	1	j.	j.	PROPN
ejpam-4476	46	2	pure	pure	PROPN
ejpam-4476	46	3	appl	appl	PROPN
ejpam-4476	46	4	.	.	PROPN
ejpam-4476	46	5	math	math	PROPN
ejpam-4476	46	6	,	,	PUNCT
ejpam-4476	46	7	15	15	NUM
ejpam-4476	46	8	(	(	PUNCT
ejpam-4476	46	9	3	3	NUM
ejpam-4476	46	10	)	)	PUNCT
ejpam-4476	46	11	(	(	PUNCT
ejpam-4476	46	12	2022	2022	NUM
ejpam-4476	46	13	)	)	PUNCT
ejpam-4476	46	14	,	,	PUNCT
ejpam-4476	46	15	864	864	NUM
ejpam-4476	46	16	-	-	SYM
ejpam-4476	46	17	877	877	NUM
ejpam-4476	46	18	866	866	NUM
ejpam-4476	46	19	source	source	NOUN
ejpam-4476	46	20	of	of	ADP
ejpam-4476	46	21	n∑	n∑	NOUN
ejpam-4476	46	22	k=1	k=1	PUNCT
ejpam-4476	47	1	k	k	PROPN
ejpam-4476	47	2	in	in	ADP
ejpam-4476	47	3	the	the	DET
ejpam-4476	47	4	processes	process	NOUN
ejpam-4476	47	5	of	of	ADP
ejpam-4476	47	6	rearrangements	rearrangement	NOUN
ejpam-4476	47	7	of	of	ADP
ejpam-4476	47	8	terms	term	NOUN
ejpam-4476	47	9	.	.	PUNCT
ejpam-4476	48	1	as	as	SCONJ
ejpam-4476	48	2	shown	show	VERB
ejpam-4476	48	3	in	in	ADP
ejpam-4476	48	4	table	table	NOUN
ejpam-4476	48	5	1	1	NUM
ejpam-4476	48	6	,	,	PUNCT
ejpam-4476	48	7	a	a	DET
ejpam-4476	48	8	cubic	cubic	ADJ
ejpam-4476	48	9	number	number	NOUN
ejpam-4476	48	10	,	,	PUNCT
ejpam-4476	48	11	k3	k3	VERB
ejpam-4476	48	12	,	,	PUNCT
ejpam-4476	48	13	can	can	AUX
ejpam-4476	48	14	be	be	AUX
ejpam-4476	48	15	represented	represent	VERB
ejpam-4476	48	16	as	as	ADP
ejpam-4476	48	17	a	a	DET
ejpam-4476	48	18	sum	sum	NOUN
ejpam-4476	48	19	of	of	ADP
ejpam-4476	48	20	k	k	PROPN
ejpam-4476	48	21	consecutive	consecutive	ADJ
ejpam-4476	48	22	odd	odd	ADJ
ejpam-4476	48	23	numbers	number	NOUN
ejpam-4476	48	24	.	.	PUNCT
ejpam-4476	49	1	table	table	NOUN
ejpam-4476	49	2	1	1	NUM
ejpam-4476	49	3	.	.	PUNCT
ejpam-4476	49	4	cubic	cubic	ADJ
ejpam-4476	49	5	numbers	number	NOUN
ejpam-4476	49	6	expressed	express	VERB
ejpam-4476	49	7	as	as	ADP
ejpam-4476	49	8	sums	sum	NOUN
ejpam-4476	49	9	of	of	ADP
ejpam-4476	49	10	odd	odd	ADJ
ejpam-4476	49	11	numbers	number	NOUN
ejpam-4476	49	12	.	.	PUNCT
ejpam-4476	50	1	13	13	NUM
ejpam-4476	50	2	=	=	SYM
ejpam-4476	50	3	1	1	NUM
ejpam-4476	50	4	.	.	X
ejpam-4476	50	5	23	23	NUM
ejpam-4476	50	6	=	=	SYM
ejpam-4476	50	7	22	22	NUM
ejpam-4476	50	8	·	·	SYM
ejpam-4476	50	9	2	2	NUM
ejpam-4476	50	10	=	=	SYM
ejpam-4476	50	11	4×	4×	NOUN
ejpam-4476	50	12	2	2	NUM
ejpam-4476	50	13	=	=	SYM
ejpam-4476	50	14	4	4	NUM
ejpam-4476	50	15	+	+	SYM
ejpam-4476	50	16	4	4	NUM
ejpam-4476	50	17	=	=	SYM
ejpam-4476	50	18	(	(	PUNCT
ejpam-4476	50	19	4−	4−	NOUN
ejpam-4476	50	20	1	1	NUM
ejpam-4476	50	21	)	)	PUNCT
ejpam-4476	50	22	+	+	CCONJ
ejpam-4476	50	23	(	(	PUNCT
ejpam-4476	50	24	4	4	NUM
ejpam-4476	50	25	+	+	SYM
ejpam-4476	50	26	1	1	NUM
ejpam-4476	50	27	)	)	PUNCT
ejpam-4476	50	28	=	=	SYM
ejpam-4476	51	1	3	3	NUM
ejpam-4476	51	2	+	+	SYM
ejpam-4476	51	3	5	5	NUM
ejpam-4476	51	4	.	.	X
ejpam-4476	51	5	33	33	NUM
ejpam-4476	51	6	=	=	SYM
ejpam-4476	51	7	32	32	NUM
ejpam-4476	51	8	·	·	SYM
ejpam-4476	51	9	3	3	NUM
ejpam-4476	51	10	=	=	SYM
ejpam-4476	51	11	9×	9×	NUM
ejpam-4476	51	12	3	3	NUM
ejpam-4476	51	13	=	=	SYM
ejpam-4476	51	14	9	9	NUM
ejpam-4476	51	15	+	+	CCONJ
ejpam-4476	51	16	9	9	NUM
ejpam-4476	51	17	+	+	SYM
ejpam-4476	51	18	9	9	NUM
ejpam-4476	51	19	=	=	SYM
ejpam-4476	51	20	(	(	PUNCT
ejpam-4476	51	21	9−	9−	NUM
ejpam-4476	51	22	2	2	NUM
ejpam-4476	51	23	)	)	PUNCT
ejpam-4476	51	24	+	+	CCONJ
ejpam-4476	51	25	9	9	NUM
ejpam-4476	51	26	+	+	CCONJ
ejpam-4476	51	27	(	(	PUNCT
ejpam-4476	51	28	9	9	NUM
ejpam-4476	51	29	+	+	SYM
ejpam-4476	51	30	2	2	NUM
ejpam-4476	51	31	)	)	PUNCT
ejpam-4476	51	32	=	=	SYM
ejpam-4476	51	33	7	7	NUM
ejpam-4476	51	34	+	+	CCONJ
ejpam-4476	51	35	9	9	NUM
ejpam-4476	51	36	+	+	CCONJ
ejpam-4476	51	37	11	11	NUM
ejpam-4476	51	38	.	.	X
ejpam-4476	52	1	43	43	NUM
ejpam-4476	52	2	=	=	SYM
ejpam-4476	52	3	42	42	NUM
ejpam-4476	52	4	·	·	SYM
ejpam-4476	52	5	4	4	NUM
ejpam-4476	52	6	=	=	SYM
ejpam-4476	52	7	16×	16×	NUM
ejpam-4476	52	8	4	4	NUM
ejpam-4476	52	9	=	=	SYM
ejpam-4476	52	10	16	16	NUM
ejpam-4476	53	1	+	+	CCONJ
ejpam-4476	53	2	16	16	NUM
ejpam-4476	54	1	+	+	CCONJ
ejpam-4476	54	2	16	16	NUM
ejpam-4476	54	3	+	+	CCONJ
ejpam-4476	54	4	16	16	NUM
ejpam-4476	54	5	=	=	SYM
ejpam-4476	54	6	(	(	PUNCT
ejpam-4476	54	7	16−	16−	NUM
ejpam-4476	54	8	3	3	NUM
ejpam-4476	54	9	)	)	PUNCT
ejpam-4476	55	1	+	+	CCONJ
ejpam-4476	55	2	(	(	PUNCT
ejpam-4476	55	3	16−	16−	NUM
ejpam-4476	55	4	1	1	NUM
ejpam-4476	55	5	)	)	PUNCT
ejpam-4476	55	6	+	+	CCONJ
ejpam-4476	55	7	(	(	PUNCT
ejpam-4476	55	8	16	16	NUM
ejpam-4476	55	9	+	+	CCONJ
ejpam-4476	55	10	1	1	NUM
ejpam-4476	55	11	)	)	PUNCT
ejpam-4476	55	12	+	+	CCONJ
ejpam-4476	55	13	(	(	PUNCT
ejpam-4476	55	14	16	16	NUM
ejpam-4476	55	15	+	+	CCONJ
ejpam-4476	55	16	3	3	X
ejpam-4476	55	17	)	)	PUNCT
ejpam-4476	55	18	=	=	SYM
ejpam-4476	55	19	13	13	NUM
ejpam-4476	55	20	+	+	NUM
ejpam-4476	55	21	15	15	NUM
ejpam-4476	55	22	+	+	NUM
ejpam-4476	55	23	17	17	NUM
ejpam-4476	55	24	+	+	SYM
ejpam-4476	55	25	19	19	NUM
ejpam-4476	55	26	.	.	PUNCT
ejpam-4476	55	27	·	·	PUNCT
ejpam-4476	55	28	·	·	PUNCT
ejpam-4476	55	29	·	·	PUNCT
ejpam-4476	56	1	k3	k3	X
ejpam-4476	56	2	=	=	PUNCT
ejpam-4476	57	1	[	[	X
ejpam-4476	57	2	k2	k2	X
ejpam-4476	57	3	−	−	PROPN
ejpam-4476	57	4	(	(	PUNCT
ejpam-4476	57	5	k	k	NOUN
ejpam-4476	57	6	−	−	PROPN
ejpam-4476	57	7	1	1	NUM
ejpam-4476	57	8	)	)	PUNCT
ejpam-4476	57	9	]	]	PUNCT
ejpam-4476	58	1	+	+	CCONJ
ejpam-4476	58	2	·	·	PUNCT
ejpam-4476	58	3	·	·	PUNCT
ejpam-4476	58	4	·	·	PUNCT
ejpam-4476	58	5	+	+	PUNCT
ejpam-4476	58	6	[	[	X
ejpam-4476	58	7	k2	k2	X
ejpam-4476	58	8	+	+	CCONJ
ejpam-4476	58	9	(	(	PUNCT
ejpam-4476	58	10	k	k	NOUN
ejpam-4476	58	11	−	−	PROPN
ejpam-4476	58	12	1	1	NUM
ejpam-4476	58	13	)	)	PUNCT
ejpam-4476	58	14	]	]	PUNCT
ejpam-4476	58	15	.	.	PUNCT
ejpam-4476	58	16	·	·	PUNCT
ejpam-4476	58	17	·	·	PUNCT
ejpam-4476	58	18	·	·	PUNCT
ejpam-4476	58	19	n3	n3	NOUN
ejpam-4476	58	20	=	=	PUNCT
ejpam-4476	59	1	[	[	X
ejpam-4476	59	2	n2	n2	NOUN
ejpam-4476	59	3	−	−	PROPN
ejpam-4476	59	4	(	(	PUNCT
ejpam-4476	59	5	n−	n−	NOUN
ejpam-4476	59	6	1	1	NUM
ejpam-4476	59	7	)	)	PUNCT
ejpam-4476	59	8	]	]	PUNCT
ejpam-4476	60	1	+	+	CCONJ
ejpam-4476	60	2	·	·	PUNCT
ejpam-4476	60	3	·	·	PUNCT
ejpam-4476	60	4	·	·	PUNCT
ejpam-4476	60	5	+	+	PUNCT
ejpam-4476	60	6	[	[	X
ejpam-4476	60	7	n2	n2	ADJ
ejpam-4476	60	8	+	+	CCONJ
ejpam-4476	60	9	(	(	PUNCT
ejpam-4476	60	10	n−	n−	NOUN
ejpam-4476	60	11	1	1	NUM
ejpam-4476	60	12	)	)	PUNCT
ejpam-4476	60	13	]	]	PUNCT
ejpam-4476	60	14	.	.	PUNCT
ejpam-4476	61	1	thus	thus	ADV
ejpam-4476	61	2	,	,	PUNCT
ejpam-4476	61	3	13	13	NUM
ejpam-4476	61	4	+	+	SYM
ejpam-4476	61	5	23	23	NUM
ejpam-4476	61	6	+	+	SYM
ejpam-4476	61	7	33	33	NUM
ejpam-4476	61	8	+	+	CCONJ
ejpam-4476	61	9	·	·	PUNCT
ejpam-4476	61	10	·	·	PUNCT
ejpam-4476	61	11	·	·	PUNCT
ejpam-4476	61	12	+	+	CCONJ
ejpam-4476	61	13	k3	k3	ADJ
ejpam-4476	61	14	+	+	X
ejpam-4476	61	15	·	·	PUNCT
ejpam-4476	61	16	·	·	PUNCT
ejpam-4476	61	17	·	·	PUNCT
ejpam-4476	62	1	+	+	NUM
ejpam-4476	62	2	n3	n3	NOUN
ejpam-4476	62	3	=	=	SYM
ejpam-4476	62	4	1︸︷︷︸	1︸︷︷︸	NUM
ejpam-4476	62	5	1	1	NUM
ejpam-4476	62	6	term	term	NOUN
ejpam-4476	62	7	+	+	NOUN
ejpam-4476	62	8	(	(	PUNCT
ejpam-4476	62	9	3	3	NUM
ejpam-4476	62	10	+	+	NUM
ejpam-4476	62	11	5)︸	5)︸	NOUN
ejpam-4476	62	12	︷︷	︷︷	NOUN
ejpam-4476	62	13	︸	︸	ADP
ejpam-4476	62	14	2	2	NUM
ejpam-4476	62	15	terms	term	NOUN
ejpam-4476	62	16	+	+	PROPN
ejpam-4476	62	17	(	(	PUNCT
ejpam-4476	62	18	7	7	NUM
ejpam-4476	62	19	+	+	CCONJ
ejpam-4476	62	20	9	9	NUM
ejpam-4476	62	21	+	+	SYM
ejpam-4476	62	22	11)︸	11)︸	NUM
ejpam-4476	62	23	︷︷	︷︷	NOUN
ejpam-4476	62	24	︸	︸	ADP
ejpam-4476	62	25	3	3	NUM
ejpam-4476	62	26	terms	term	NOUN
ejpam-4476	62	27	+	+	PROPN
ejpam-4476	62	28	(	(	PUNCT
ejpam-4476	62	29	13	13	NUM
ejpam-4476	62	30	+	+	NUM
ejpam-4476	62	31	15	15	NUM
ejpam-4476	62	32	+	+	NUM
ejpam-4476	62	33	17	17	NUM
ejpam-4476	62	34	+	+	NUM
ejpam-4476	62	35	19)︸	19)︸	NUM
ejpam-4476	62	36	︷︷	︷︷	PROPN
ejpam-4476	62	37	︸	︸	ADP
ejpam-4476	62	38	4	4	NUM
ejpam-4476	62	39	terms	term	NOUN
ejpam-4476	62	40	+	+	X
ejpam-4476	62	41	·	·	PUNCT
ejpam-4476	62	42	·	·	PUNCT
ejpam-4476	62	43	·	·	PUNCT
ejpam-4476	62	44	+	+	PUNCT
ejpam-4476	63	1	[	[	X
ejpam-4476	63	2	k2	k2	ADJ
ejpam-4476	63	3	−	−	PROPN
ejpam-4476	63	4	(	(	PUNCT
ejpam-4476	63	5	k	k	NOUN
ejpam-4476	63	6	−	−	PROPN
ejpam-4476	63	7	1	1	NUM
ejpam-4476	63	8	)	)	PUNCT
ejpam-4476	63	9	]	]	PUNCT
ejpam-4476	64	1	+	+	CCONJ
ejpam-4476	64	2	·	·	PUNCT
ejpam-4476	64	3	·	·	PUNCT
ejpam-4476	64	4	·	·	PUNCT
ejpam-4476	64	5	+	+	PUNCT
ejpam-4476	64	6	[	[	X
ejpam-4476	64	7	k2	k2	X
ejpam-4476	64	8	+	+	CCONJ
ejpam-4476	64	9	(	(	PUNCT
ejpam-4476	64	10	k	k	PROPN
ejpam-4476	64	11	−	−	PROPN
ejpam-4476	64	12	1)]︸	1)]︸	PROPN
ejpam-4476	64	13	︷︷	︷︷	PROPN
ejpam-4476	64	14	︸	︸	X
ejpam-4476	64	15	k	k	PROPN
ejpam-4476	64	16	terms	term	NOUN
ejpam-4476	64	17	+	+	X
ejpam-4476	64	18	·	·	PUNCT
ejpam-4476	64	19	·	·	PUNCT
ejpam-4476	64	20	·	·	PUNCT
ejpam-4476	64	21	+	+	PUNCT
ejpam-4476	65	1	[	[	X
ejpam-4476	65	2	n2	n2	ADJ
ejpam-4476	65	3	−	−	PROPN
ejpam-4476	65	4	(	(	PUNCT
ejpam-4476	65	5	n−	n−	NOUN
ejpam-4476	65	6	1	1	NUM
ejpam-4476	65	7	)	)	PUNCT
ejpam-4476	65	8	]	]	PUNCT
ejpam-4476	66	1	+	+	CCONJ
ejpam-4476	66	2	·	·	PUNCT
ejpam-4476	66	3	·	·	PUNCT
ejpam-4476	66	4	·	·	PUNCT
ejpam-4476	66	5	+	+	PUNCT
ejpam-4476	66	6	[	[	X
ejpam-4476	66	7	n2	n2	ADJ
ejpam-4476	66	8	+	+	CCONJ
ejpam-4476	66	9	(	(	PUNCT
ejpam-4476	66	10	n−	n−	NOUN
ejpam-4476	66	11	1)]︸	1)]︸	NUM
ejpam-4476	66	12	︷︷	︷︷	PROPN
ejpam-4476	66	13	︸	︸	ADP
ejpam-4476	66	14	n	n	PROPN
ejpam-4476	66	15	terms	term	NOUN
ejpam-4476	66	16	.	.	PUNCT
ejpam-4476	67	1	(	(	PUNCT
ejpam-4476	67	2	1	1	X
ejpam-4476	67	3	)	)	PUNCT
ejpam-4476	67	4	equation	equation	NOUN
ejpam-4476	67	5	(	(	PUNCT
ejpam-4476	67	6	1	1	X
ejpam-4476	67	7	)	)	PUNCT
ejpam-4476	67	8	means	mean	VERB
ejpam-4476	67	9	that	that	SCONJ
ejpam-4476	67	10	n∑	n∑	PROPN
ejpam-4476	67	11	k=1	k=1	PROPN
ejpam-4476	67	12	k3	k3	X
ejpam-4476	67	13	is	be	AUX
ejpam-4476	67	14	the	the	DET
ejpam-4476	67	15	sum	sum	NOUN
ejpam-4476	67	16	of	of	ADP
ejpam-4476	67	17	consecutive	consecutive	ADJ
ejpam-4476	67	18	odd	odd	ADJ
ejpam-4476	67	19	numbers	number	NOUN
ejpam-4476	67	20	from	from	ADP
ejpam-4476	67	21	1	1	NUM
ejpam-4476	67	22	to	to	ADP
ejpam-4476	67	23	[	[	NOUN
ejpam-4476	67	24	n2	n2	ADJ
ejpam-4476	67	25	+	+	CCONJ
ejpam-4476	67	26	(	(	PUNCT
ejpam-4476	67	27	n−	n−	NOUN
ejpam-4476	67	28	1	1	NUM
ejpam-4476	67	29	)	)	PUNCT
ejpam-4476	67	30	]	]	PUNCT
ejpam-4476	67	31	,	,	PUNCT
ejpam-4476	67	32	in	in	ADP
ejpam-4476	67	33	which	which	PRON
ejpam-4476	67	34	the	the	DET
ejpam-4476	67	35	number	number	NOUN
ejpam-4476	67	36	of	of	ADP
ejpam-4476	67	37	terms	term	NOUN
ejpam-4476	67	38	is	be	AUX
ejpam-4476	67	39	(	(	PUNCT
ejpam-4476	67	40	1	1	NUM
ejpam-4476	67	41	+	+	NUM
ejpam-4476	67	42	2	2	NUM
ejpam-4476	67	43	+	+	NUM
ejpam-4476	67	44	3	3	NUM
ejpam-4476	67	45	+	+	NUM
ejpam-4476	67	46	4	4	NUM
ejpam-4476	67	47	+	+	NUM
ejpam-4476	67	48	·	·	PUNCT
ejpam-4476	67	49	·	·	PUNCT
ejpam-4476	67	50	·	·	PUNCT
ejpam-4476	67	51	+	+	NOUN
ejpam-4476	67	52	n	n	CCONJ
ejpam-4476	67	53	)	)	PUNCT
ejpam-4476	67	54	;	;	PUNCT
ejpam-4476	67	55	that	that	PRON
ejpam-4476	67	56	is	is	ADV
ejpam-4476	67	57	,	,	PUNCT
ejpam-4476	67	58	the	the	DET
ejpam-4476	67	59	initial	initial	ADJ
ejpam-4476	67	60	term	term	NOUN
ejpam-4476	67	61	of	of	ADP
ejpam-4476	67	62	the	the	DET
ejpam-4476	67	63	arithmetic	arithmetic	ADJ
ejpam-4476	67	64	progression	progression	NOUN
ejpam-4476	67	65	is	be	AUX
ejpam-4476	67	66	1	1	NUM
ejpam-4476	67	67	and	and	CCONJ
ejpam-4476	67	68	the	the	DET
ejpam-4476	67	69	common	common	ADJ
ejpam-4476	67	70	difference	difference	NOUN
ejpam-4476	67	71	between	between	ADP
ejpam-4476	67	72	successive	successive	ADJ
ejpam-4476	67	73	terms	term	NOUN
ejpam-4476	67	74	is	be	AUX
ejpam-4476	67	75	2	2	NUM
ejpam-4476	67	76	.	.	PUNCT
ejpam-4476	67	77	by	by	ADP
ejpam-4476	67	78	a	a	DET
ejpam-4476	67	79	simple	simple	ADJ
ejpam-4476	67	80	calculation	calculation	NOUN
ejpam-4476	67	81	of	of	ADP
ejpam-4476	67	82	1	1	NUM
ejpam-4476	67	83	2	2	NUM
ejpam-4476	67	84	×	×	NOUN
ejpam-4476	67	85	(	(	PUNCT
ejpam-4476	67	86	initial	initial	ADJ
ejpam-4476	67	87	term+final	term+final	ADJ
ejpam-4476	67	88	term)×	term)×	NOUN
ejpam-4476	67	89	(	(	PUNCT
ejpam-4476	67	90	number	number	NOUN
ejpam-4476	67	91	of	of	ADP
ejpam-4476	67	92	terms	term	NOUN
ejpam-4476	67	93	)	)	PUNCT
ejpam-4476	67	94	,	,	PUNCT
ejpam-4476	67	95	we	we	PRON
ejpam-4476	67	96	obtain	obtain	VERB
ejpam-4476	68	1	n∑	n∑	NOUN
ejpam-4476	68	2	k=1	k=1	PROPN
ejpam-4476	69	1	k3	k3	PROPN
ejpam-4476	69	2	=	=	NOUN
ejpam-4476	69	3	1	1	NUM
ejpam-4476	69	4	2	2	NUM
ejpam-4476	69	5	×	×	NOUN
ejpam-4476	69	6	[	[	PUNCT
ejpam-4476	69	7	1	1	NUM
ejpam-4476	69	8	+	+	CCONJ
ejpam-4476	69	9	(	(	PUNCT
ejpam-4476	69	10	n2	n2	ADJ
ejpam-4476	69	11	+	+	CCONJ
ejpam-4476	69	12	n−	n−	NOUN
ejpam-4476	69	13	1	1	NUM
ejpam-4476	69	14	)	)	PUNCT
ejpam-4476	69	15	]	]	PUNCT
ejpam-4476	70	1	×	×	NOUN
ejpam-4476	70	2	(	(	PUNCT
ejpam-4476	70	3	1	1	NUM
ejpam-4476	70	4	+	+	NUM
ejpam-4476	70	5	2	2	NUM
ejpam-4476	70	6	+	+	NUM
ejpam-4476	70	7	3	3	NUM
ejpam-4476	70	8	+	+	SYM
ejpam-4476	70	9	4	4	NUM
ejpam-4476	70	10	+	+	CCONJ
ejpam-4476	70	11	·	·	PUNCT
ejpam-4476	70	12	·	·	PUNCT
ejpam-4476	70	13	·	·	PUNCT
ejpam-4476	70	14	+	+	NUM
ejpam-4476	70	15	n	n	CCONJ
ejpam-4476	70	16	)	)	PUNCT
ejpam-4476	70	17	=	=	PUNCT
ejpam-4476	70	18	[	[	PUNCT
ejpam-4476	70	19	n(n+	n(n+	NOUN
ejpam-4476	70	20	1	1	NUM
ejpam-4476	70	21	)	)	PUNCT
ejpam-4476	70	22	2	2	NUM
ejpam-4476	70	23	]	]	SYM
ejpam-4476	70	24	2	2	NUM
ejpam-4476	70	25	.	.	PUNCT
ejpam-4476	71	1	(	(	PUNCT
ejpam-4476	71	2	2	2	X
ejpam-4476	71	3	)	)	PUNCT
ejpam-4476	71	4	practically	practically	ADV
ejpam-4476	71	5	,	,	PUNCT
ejpam-4476	71	6	this	this	DET
ejpam-4476	71	7	calculation	calculation	NOUN
ejpam-4476	71	8	is	be	AUX
ejpam-4476	71	9	not	not	PART
ejpam-4476	71	10	necessary	necessary	ADJ
ejpam-4476	71	11	,	,	PUNCT
ejpam-4476	71	12	as	as	SCONJ
ejpam-4476	71	13	shown	show	VERB
ejpam-4476	71	14	in	in	ADP
ejpam-4476	71	15	the	the	DET
ejpam-4476	71	16	following	following	ADJ
ejpam-4476	71	17	way	way	NOUN
ejpam-4476	71	18	.	.	PUNCT
ejpam-4476	72	1	the	the	DET
ejpam-4476	72	2	second	second	ADJ
ejpam-4476	72	3	power	power	NOUN
ejpam-4476	72	4	of	of	ADP
ejpam-4476	72	5	a	a	DET
ejpam-4476	72	6	natural	natural	ADJ
ejpam-4476	72	7	number	number	NOUN
ejpam-4476	72	8	,	,	PUNCT
ejpam-4476	72	9	k	k	NOUN
ejpam-4476	72	10	,	,	PUNCT
ejpam-4476	72	11	is	be	AUX
ejpam-4476	72	12	a	a	DET
ejpam-4476	72	13	sum	sum	NOUN
ejpam-4476	72	14	of	of	ADP
ejpam-4476	72	15	consecutive	consecutive	ADJ
ejpam-4476	72	16	odd	odd	ADJ
ejpam-4476	72	17	numbers	number	NOUN
ejpam-4476	72	18	in	in	ADP
ejpam-4476	72	19	which	which	PRON
ejpam-4476	72	20	the	the	DET
ejpam-4476	72	21	number	number	NOUN
ejpam-4476	72	22	of	of	ADP
ejpam-4476	72	23	terms	term	NOUN
ejpam-4476	72	24	is	be	AUX
ejpam-4476	72	25	k	k	NOUN
ejpam-4476	72	26	,	,	PUNCT
ejpam-4476	72	27	such	such	ADJ
ejpam-4476	72	28	as	as	ADP
ejpam-4476	72	29	42	42	NUM
ejpam-4476	72	30	=	=	SYM
ejpam-4476	72	31	1	1	NUM
ejpam-4476	72	32	+	+	NUM
ejpam-4476	72	33	3	3	NUM
ejpam-4476	72	34	+	+	SYM
ejpam-4476	72	35	5	5	NUM
ejpam-4476	72	36	+	+	SYM
ejpam-4476	72	37	7	7	NUM
ejpam-4476	72	38	.	.	PUNCT
ejpam-4476	73	1	the	the	DET
ejpam-4476	73	2	sum	sum	NOUN
ejpam-4476	73	3	n∑	n∑	PROPN
ejpam-4476	74	1	k=1	k=1	PROPN
ejpam-4476	74	2	k3	k3	PROPN
ejpam-4476	74	3	is	be	AUX
ejpam-4476	74	4	also	also	ADV
ejpam-4476	74	5	an	an	DET
ejpam-4476	74	6	arithmetic	arithmetic	ADJ
ejpam-4476	74	7	series	series	NOUN
ejpam-4476	74	8	of	of	ADP
ejpam-4476	74	9	consecutive	consecutive	ADJ
ejpam-4476	74	10	odd	odd	ADJ
ejpam-4476	74	11	numbers	number	NOUN
ejpam-4476	74	12	,	,	PUNCT
ejpam-4476	74	13	one	one	NUM
ejpam-4476	74	14	in	in	ADP
ejpam-4476	74	15	which	which	PRON
ejpam-4476	74	16	the	the	DET
ejpam-4476	74	17	number	number	NOUN
ejpam-4476	74	18	of	of	ADP
ejpam-4476	74	19	terms	term	NOUN
ejpam-4476	74	20	is	be	AUX
ejpam-4476	74	21	n(n+	n(n+	PROPN
ejpam-4476	74	22	1)/2	1)/2	NUM
ejpam-4476	74	23	.	.	PUNCT
ejpam-4476	75	1	thus	thus	ADV
ejpam-4476	75	2	,	,	PUNCT
ejpam-4476	75	3	we	we	PRON
ejpam-4476	75	4	can	can	AUX
ejpam-4476	75	5	obtain	obtain	VERB
ejpam-4476	75	6	equation	equation	NOUN
ejpam-4476	75	7	(	(	PUNCT
ejpam-4476	75	8	2	2	NUM
ejpam-4476	75	9	)	)	PUNCT
ejpam-4476	75	10	without	without	ADP
ejpam-4476	75	11	tedious	tedious	ADJ
ejpam-4476	75	12	calculations	calculation	NOUN
ejpam-4476	75	13	.	.	PUNCT
ejpam-4476	76	1	the	the	DET
ejpam-4476	76	2	sum	sum	NOUN
ejpam-4476	76	3	1	1	NUM
ejpam-4476	76	4	+	+	NOUN
ejpam-4476	76	5	2	2	NUM
ejpam-4476	76	6	+	+	NOUN
ejpam-4476	76	7	3	3	NUM
ejpam-4476	76	8	+	+	SYM
ejpam-4476	76	9	4	4	NUM
ejpam-4476	76	10	+	+	NUM
ejpam-4476	76	11	·	·	PUNCT
ejpam-4476	76	12	·	·	PUNCT
ejpam-4476	76	13	·	·	PUNCT
ejpam-4476	76	14	+	+	NOUN
ejpam-4476	76	15	n	n	ADP
ejpam-4476	76	16	indicates	indicate	VERB
ejpam-4476	76	17	y.	y.	PROPN
ejpam-4476	76	18	kobayashi	kobayashi	PROPN
ejpam-4476	76	19	/	/	SYM
ejpam-4476	76	20	eur	eur	PROPN
ejpam-4476	76	21	.	.	PUNCT
ejpam-4476	77	1	j.	j.	PROPN
ejpam-4476	77	2	pure	pure	PROPN
ejpam-4476	77	3	appl	appl	PROPN
ejpam-4476	77	4	.	.	PROPN
ejpam-4476	77	5	math	math	PROPN
ejpam-4476	77	6	,	,	PUNCT
ejpam-4476	77	7	15	15	NUM
ejpam-4476	77	8	(	(	PUNCT
ejpam-4476	77	9	3	3	NUM
ejpam-4476	77	10	)	)	PUNCT
ejpam-4476	77	11	(	(	PUNCT
ejpam-4476	77	12	2022	2022	NUM
ejpam-4476	77	13	)	)	PUNCT
ejpam-4476	77	14	,	,	PUNCT
ejpam-4476	77	15	864	864	NUM
ejpam-4476	77	16	-	-	SYM
ejpam-4476	77	17	877	877	NUM
ejpam-4476	77	18	867	867	NUM
ejpam-4476	77	19	two	two	NUM
ejpam-4476	77	20	meanings	meaning	NOUN
ejpam-4476	77	21	:	:	PUNCT
ejpam-4476	77	22	first	first	ADV
ejpam-4476	77	23	,	,	PUNCT
ejpam-4476	77	24	the	the	DET
ejpam-4476	77	25	arithmetic	arithmetic	ADJ
ejpam-4476	77	26	mean	mean	NOUN
ejpam-4476	77	27	of	of	ADP
ejpam-4476	77	28	the	the	DET
ejpam-4476	77	29	initial	initial	ADJ
ejpam-4476	77	30	and	and	CCONJ
ejpam-4476	77	31	final	final	ADJ
ejpam-4476	77	32	terms	term	NOUN
ejpam-4476	77	33	of	of	ADP
ejpam-4476	77	34	1	1	NUM
ejpam-4476	77	35	+	+	NOUN
ejpam-4476	77	36	3	3	NUM
ejpam-4476	77	37	+	+	ADJ
ejpam-4476	77	38	5	5	NUM
ejpam-4476	77	39	+	+	NUM
ejpam-4476	77	40	·	·	PUNCT
ejpam-4476	77	41	·	·	PUNCT
ejpam-4476	77	42	·	·	PUNCT
ejpam-4476	78	1	+	+	PUNCT
ejpam-4476	78	2	[	[	X
ejpam-4476	78	3	n2+(n−1	n2+(n−1	NUM
ejpam-4476	78	4	)	)	PUNCT
ejpam-4476	78	5	]	]	PUNCT
ejpam-4476	78	6	in	in	ADP
ejpam-4476	78	7	equation	equation	NOUN
ejpam-4476	78	8	(	(	PUNCT
ejpam-4476	78	9	1	1	NUM
ejpam-4476	78	10	)	)	PUNCT
ejpam-4476	78	11	;	;	PUNCT
ejpam-4476	78	12	second	second	X
ejpam-4476	78	13	,	,	PUNCT
ejpam-4476	78	14	the	the	DET
ejpam-4476	78	15	number	number	NOUN
ejpam-4476	78	16	of	of	ADP
ejpam-4476	78	17	terms	term	NOUN
ejpam-4476	78	18	of	of	ADP
ejpam-4476	78	19	1	1	NUM
ejpam-4476	78	20	+	+	NOUN
ejpam-4476	78	21	3	3	NUM
ejpam-4476	78	22	+	+	ADJ
ejpam-4476	78	23	5	5	NUM
ejpam-4476	78	24	+	+	NUM
ejpam-4476	78	25	·	·	PUNCT
ejpam-4476	78	26	·	·	PUNCT
ejpam-4476	78	27	·	·	PUNCT
ejpam-4476	79	1	+	+	PROPN
ejpam-4476	79	2	[	[	X
ejpam-4476	79	3	k2+(k−1	k2+(k−1	PROPN
ejpam-4476	79	4	)	)	PUNCT
ejpam-4476	79	5	]	]	PUNCT
ejpam-4476	79	6	,	,	PUNCT
ejpam-4476	79	7	with	with	ADP
ejpam-4476	79	8	k	k	PROPN
ejpam-4476	79	9	from	from	ADP
ejpam-4476	79	10	1	1	NUM
ejpam-4476	79	11	to	to	ADP
ejpam-4476	79	12	n	n	CCONJ
ejpam-4476	79	13	,	,	PUNCT
ejpam-4476	79	14	in	in	ADP
ejpam-4476	79	15	equation	equation	NOUN
ejpam-4476	79	16	(	(	PUNCT
ejpam-4476	79	17	1	1	NUM
ejpam-4476	79	18	)	)	PUNCT
ejpam-4476	79	19	.	.	PUNCT
ejpam-4476	80	1	2.2	2.2	NUM
ejpam-4476	80	2	.	.	PUNCT
ejpam-4476	80	3	visual	visual	ADJ
ejpam-4476	80	4	thinking	think	VERB
ejpam-4476	80	5	students	student	NOUN
ejpam-4476	80	6	can	can	AUX
ejpam-4476	80	7	cultivate	cultivate	VERB
ejpam-4476	80	8	a	a	DET
ejpam-4476	80	9	sense	sense	NOUN
ejpam-4476	80	10	of	of	ADP
ejpam-4476	80	11	mathematics	mathematic	NOUN
ejpam-4476	80	12	not	not	PART
ejpam-4476	80	13	only	only	ADV
ejpam-4476	80	14	by	by	ADP
ejpam-4476	80	15	performing	perform	VERB
ejpam-4476	80	16	calculations	calculation	NOUN
ejpam-4476	80	17	but	but	CCONJ
ejpam-4476	80	18	also	also	ADV
ejpam-4476	80	19	by	by	ADP
ejpam-4476	80	20	visualising	visualise	VERB
ejpam-4476	80	21	how	how	SCONJ
ejpam-4476	80	22	the	the	DET
ejpam-4476	80	23	calculations	calculation	NOUN
ejpam-4476	80	24	work	work	VERB
ejpam-4476	80	25	.	.	PUNCT
ejpam-4476	81	1	the	the	DET
ejpam-4476	81	2	formula	formula	NOUN
ejpam-4476	81	3	n∑	n∑	INTJ
ejpam-4476	82	1	k=1	k=1	PROPN
ejpam-4476	82	2	k3	k3	PROPN
ejpam-4476	82	3	=	=	SYM
ejpam-4476	82	4	(	(	PUNCT
ejpam-4476	82	5	n∑	n∑	INTJ
ejpam-4476	82	6	k=1	k=1	PUNCT
ejpam-4476	82	7	k	k	X
ejpam-4476	82	8	)	)	PUNCT
ejpam-4476	82	9	2	2	NUM
ejpam-4476	82	10	can	can	AUX
ejpam-4476	82	11	be	be	AUX
ejpam-4476	82	12	proved	prove	VERB
ejpam-4476	82	13	pictorially	pictorially	ADV
ejpam-4476	82	14	by	by	ADP
ejpam-4476	82	15	expressing	express	VERB
ejpam-4476	82	16	square	square	ADJ
ejpam-4476	82	17	numbers	number	NOUN
ejpam-4476	82	18	as	as	ADP
ejpam-4476	82	19	sums	sum	NOUN
ejpam-4476	82	20	of	of	ADP
ejpam-4476	82	21	consecutive	consecutive	ADJ
ejpam-4476	82	22	odd	odd	ADJ
ejpam-4476	82	23	numbers	number	NOUN
ejpam-4476	82	24	,	,	PUNCT
ejpam-4476	82	25	with	with	ADP
ejpam-4476	82	26	slightly	slightly	ADV
ejpam-4476	82	27	different	different	ADJ
ejpam-4476	82	28	diagrams	diagram	NOUN
ejpam-4476	82	29	from	from	ADP
ejpam-4476	82	30	those	those	PRON
ejpam-4476	82	31	of	of	ADP
ejpam-4476	82	32	well	well	ADV
ejpam-4476	82	33	-	-	PUNCT
ejpam-4476	82	34	known	know	VERB
ejpam-4476	82	35	pictorial	pictorial	ADJ
ejpam-4476	82	36	proofs	proof	NOUN
ejpam-4476	82	37	[	[	X
ejpam-4476	82	38	1	1	NUM
ejpam-4476	82	39	,	,	PUNCT
ejpam-4476	82	40	2	2	NUM
ejpam-4476	82	41	,	,	PUNCT
ejpam-4476	82	42	9	9	NUM
ejpam-4476	82	43	,	,	PUNCT
ejpam-4476	82	44	10	10	NUM
ejpam-4476	82	45	]	]	PUNCT
ejpam-4476	82	46	.	.	PUNCT
ejpam-4476	83	1	two	two	NUM
ejpam-4476	83	2	different	different	ADJ
ejpam-4476	83	3	diagrams	diagram	NOUN
ejpam-4476	83	4	,	,	PUNCT
ejpam-4476	83	5	as	as	SCONJ
ejpam-4476	83	6	shown	show	VERB
ejpam-4476	83	7	in	in	ADP
ejpam-4476	83	8	figures	figure	NOUN
ejpam-4476	83	9	1	1	NUM
ejpam-4476	83	10	and	and	CCONJ
ejpam-4476	83	11	2	2	NUM
ejpam-4476	83	12	,	,	PUNCT
ejpam-4476	83	13	can	can	AUX
ejpam-4476	83	14	be	be	AUX
ejpam-4476	83	15	drawn	draw	VERB
ejpam-4476	83	16	,	,	PUNCT
ejpam-4476	83	17	depending	depend	VERB
ejpam-4476	83	18	on	on	ADP
ejpam-4476	83	19	the	the	DET
ejpam-4476	83	20	different	different	ADJ
ejpam-4476	83	21	ways	way	NOUN
ejpam-4476	83	22	of	of	ADP
ejpam-4476	83	23	counting	count	VERB
ejpam-4476	83	24	the	the	DET
ejpam-4476	83	25	number	number	NOUN
ejpam-4476	83	26	of	of	ADP
ejpam-4476	83	27	squares	square	NOUN
ejpam-4476	83	28	.	.	PUNCT
ejpam-4476	84	1	y.	y.	PROPN
ejpam-4476	84	2	kobayashi	kobayashi	PROPN
ejpam-4476	84	3	/	/	SYM
ejpam-4476	84	4	eur	eur	PROPN
ejpam-4476	84	5	.	.	PUNCT
ejpam-4476	85	1	j.	j.	PROPN
ejpam-4476	85	2	pure	pure	PROPN
ejpam-4476	85	3	appl	appl	PROPN
ejpam-4476	85	4	.	.	PROPN
ejpam-4476	85	5	math	math	PROPN
ejpam-4476	85	6	,	,	PUNCT
ejpam-4476	85	7	15	15	NUM
ejpam-4476	85	8	(	(	PUNCT
ejpam-4476	85	9	3	3	NUM
ejpam-4476	85	10	)	)	PUNCT
ejpam-4476	85	11	(	(	PUNCT
ejpam-4476	85	12	2022	2022	NUM
ejpam-4476	85	13	)	)	PUNCT
ejpam-4476	85	14	,	,	PUNCT
ejpam-4476	85	15	864	864	NUM
ejpam-4476	85	16	-	-	SYM
ejpam-4476	85	17	877	877	NUM
ejpam-4476	85	18	868	868	NUM
ejpam-4476	85	19	10	10	NUM
ejpam-4476	85	20	10	10	NUM
ejpam-4476	85	21	1	1	NUM
ejpam-4476	85	22	3	3	NUM
ejpam-4476	85	23	5	5	NUM
ejpam-4476	85	24	7	7	NUM
ejpam-4476	85	25	9	9	NUM
ejpam-4476	85	26	11	11	NUM
ejpam-4476	85	27	13	13	NUM
ejpam-4476	85	28	15	15	NUM
ejpam-4476	85	29	17	17	NUM
ejpam-4476	85	30	19	19	NUM
ejpam-4476	85	31	102	102	NUM
ejpam-4476	85	32	=	=	SYM
ejpam-4476	85	33	1	1	NUM
ejpam-4476	85	34	+	+	NUM
ejpam-4476	85	35	3	3	NUM
ejpam-4476	86	1	+	+	SYM
ejpam-4476	86	2	5	5	NUM
ejpam-4476	86	3	+	+	CCONJ
ejpam-4476	86	4	7	7	NUM
ejpam-4476	86	5	+	+	CCONJ
ejpam-4476	86	6	9	9	NUM
ejpam-4476	86	7	+	+	NUM
ejpam-4476	86	8	11	11	NUM
ejpam-4476	86	9	+	+	SYM
ejpam-4476	86	10	13	13	NUM
ejpam-4476	86	11	+	+	NUM
ejpam-4476	86	12	15	15	NUM
ejpam-4476	86	13	+	+	NUM
ejpam-4476	86	14	17	17	NUM
ejpam-4476	86	15	+	+	SYM
ejpam-4476	86	16	19︸	19︸	NUM
ejpam-4476	86	17	︷︷	︷︷	PROPN
ejpam-4476	86	18	︸	︸	ADP
ejpam-4476	86	19	10	10	NUM
ejpam-4476	86	20	terms	term	NOUN
ejpam-4476	86	21	.	.	PUNCT
ejpam-4476	87	1	1	1	NUM
ejpam-4476	87	2	3	3	NUM
ejpam-4476	87	3	5	5	NUM
ejpam-4476	87	4	7	7	NUM
ejpam-4476	87	5	9	9	NUM
ejpam-4476	87	6	11	11	NUM
ejpam-4476	87	7	12	12	NUM
ejpam-4476	87	8	×	×	NOUN
ejpam-4476	87	9	1	1	NUM
ejpam-4476	87	10	︸	︸	X
ejpam-4476	87	11	︷︷	︷︷	VERB
ejpam-4476	87	12	︸	︸	ADP
ejpam-4476	87	13	8	8	NUM
ejpam-4476	87	14	(=	(=	NOUN
ejpam-4476	87	15	4	4	NUM
ejpam-4476	87	16	+	+	SYM
ejpam-4476	87	17	4	4	NUM
ejpam-4476	87	18	)	)	PUNCT
ejpam-4476	87	19	22	22	NUM
ejpam-4476	87	20	×	×	NOUN
ejpam-4476	87	21	2	2	NUM
ejpam-4476	87	22	︸	︸	X
ejpam-4476	87	23	︷︷	︷︷	VERB
ejpam-4476	87	24	︸	︸	ADP
ejpam-4476	87	25	27	27	NUM
ejpam-4476	87	26	(=	(=	NOUN
ejpam-4476	87	27	9	9	NUM
ejpam-4476	87	28	+	+	CCONJ
ejpam-4476	87	29	9	9	NUM
ejpam-4476	87	30	+	+	CCONJ
ejpam-4476	87	31	9	9	NUM
ejpam-4476	87	32	)	)	PUNCT
ejpam-4476	87	33	32	32	NUM
ejpam-4476	87	34	×	×	NOUN
ejpam-4476	87	35	3	3	NUM
ejpam-4476	87	36	13	13	NUM
ejpam-4476	87	37	15	15	NUM
ejpam-4476	87	38	17	17	NUM
ejpam-4476	87	39	19︸	19︸	NUM
ejpam-4476	87	40	︷︷	︷︷	PROPN
ejpam-4476	87	41	︸	︸	ADP
ejpam-4476	87	42	64	64	NUM
ejpam-4476	87	43	(=	(=	NOUN
ejpam-4476	87	44	16	16	NUM
ejpam-4476	88	1	+	+	CCONJ
ejpam-4476	88	2	16	16	NUM
ejpam-4476	88	3	+	+	CCONJ
ejpam-4476	88	4	16	16	NUM
ejpam-4476	88	5	+	+	CCONJ
ejpam-4476	88	6	16	16	NUM
ejpam-4476	88	7	)	)	PUNCT
ejpam-4476	88	8	42	42	NUM
ejpam-4476	88	9	×	×	NOUN
ejpam-4476	88	10	4	4	NUM
ejpam-4476	88	11	figure	figure	NOUN
ejpam-4476	88	12	1	1	NUM
ejpam-4476	88	13	:	:	PUNCT
ejpam-4476	88	14	visual	visual	ADJ
ejpam-4476	88	15	thinking	thinking	NOUN
ejpam-4476	88	16	for	for	ADP
ejpam-4476	88	17	the	the	DET
ejpam-4476	88	18	sum	sum	NOUN
ejpam-4476	88	19	of	of	ADP
ejpam-4476	88	20	n	n	DET
ejpam-4476	88	21	cubic	cubic	ADJ
ejpam-4476	88	22	numbers	number	NOUN
ejpam-4476	88	23	.	.	PUNCT
ejpam-4476	89	1	y.	y.	PROPN
ejpam-4476	89	2	kobayashi	kobayashi	PROPN
ejpam-4476	89	3	/	/	SYM
ejpam-4476	89	4	eur	eur	PROPN
ejpam-4476	89	5	.	.	PUNCT
ejpam-4476	90	1	j.	j.	PROPN
ejpam-4476	90	2	pure	pure	PROPN
ejpam-4476	90	3	appl	appl	PROPN
ejpam-4476	90	4	.	.	PROPN
ejpam-4476	90	5	math	math	PROPN
ejpam-4476	90	6	,	,	PUNCT
ejpam-4476	90	7	15	15	NUM
ejpam-4476	90	8	(	(	PUNCT
ejpam-4476	90	9	3	3	NUM
ejpam-4476	90	10	)	)	PUNCT
ejpam-4476	90	11	(	(	PUNCT
ejpam-4476	90	12	2022	2022	NUM
ejpam-4476	90	13	)	)	PUNCT
ejpam-4476	90	14	,	,	PUNCT
ejpam-4476	90	15	864	864	NUM
ejpam-4476	90	16	-	-	SYM
ejpam-4476	90	17	877	877	NUM
ejpam-4476	90	18	869	869	NUM
ejpam-4476	90	19	1	1	NUM
ejpam-4476	90	20	3	3	NUM
ejpam-4476	90	21	+	+	CCONJ
ejpam-4476	90	22	5	5	NUM
ejpam-4476	90	23	7	7	NUM
ejpam-4476	90	24	+	+	CCONJ
ejpam-4476	90	25	9	9	NUM
ejpam-4476	90	26	+	+	CCONJ
ejpam-4476	90	27	11	11	NUM
ejpam-4476	90	28	13	13	NUM
ejpam-4476	90	29	+	+	NUM
ejpam-4476	90	30	15	15	NUM
ejpam-4476	90	31	+	+	NUM
ejpam-4476	90	32	17	17	NUM
ejpam-4476	90	33	+	+	SYM
ejpam-4476	90	34	19	19	NUM
ejpam-4476	90	35	12	12	NUM
ejpam-4476	90	36	22	22	NUM
ejpam-4476	90	37	×	×	NOUN
ejpam-4476	90	38	2	2	NUM
ejpam-4476	90	39	32	32	NUM
ejpam-4476	90	40	×	×	NOUN
ejpam-4476	90	41	3	3	NUM
ejpam-4476	90	42	42	42	NUM
ejpam-4476	90	43	×	×	NOUN
ejpam-4476	90	44	4	4	NUM
ejpam-4476	90	45	figure	figure	NOUN
ejpam-4476	90	46	2	2	NUM
ejpam-4476	90	47	:	:	PUNCT
ejpam-4476	90	48	a	a	DET
ejpam-4476	90	49	sum	sum	NOUN
ejpam-4476	90	50	of	of	ADP
ejpam-4476	90	51	odd	odd	ADJ
ejpam-4476	90	52	numbers	number	NOUN
ejpam-4476	90	53	can	can	AUX
ejpam-4476	90	54	be	be	AUX
ejpam-4476	90	55	expressed	express	VERB
ejpam-4476	90	56	as	as	ADP
ejpam-4476	90	57	a	a	DET
ejpam-4476	90	58	sum	sum	NOUN
ejpam-4476	90	59	of	of	ADP
ejpam-4476	90	60	cubic	cubic	ADJ
ejpam-4476	90	61	numbers	number	NOUN
ejpam-4476	90	62	.	.	PUNCT
ejpam-4476	91	1	diagrams	diagram	NOUN
ejpam-4476	91	2	above	above	ADP
ejpam-4476	91	3	and	and	CCONJ
ejpam-4476	91	4	below	below	ADP
ejpam-4476	91	5	illustrate	illustrate	VERB
ejpam-4476	91	6	the	the	DET
ejpam-4476	91	7	different	different	ADJ
ejpam-4476	91	8	ways	way	NOUN
ejpam-4476	91	9	of	of	ADP
ejpam-4476	91	10	counting	count	VERB
ejpam-4476	91	11	the	the	DET
ejpam-4476	91	12	squares	square	NOUN
ejpam-4476	91	13	.	.	PUNCT
ejpam-4476	92	1	a	a	DET
ejpam-4476	92	2	visual	visual	ADJ
ejpam-4476	92	3	clue	clue	NOUN
ejpam-4476	92	4	to	to	ADP
ejpam-4476	92	5	the	the	DET
ejpam-4476	92	6	fact	fact	NOUN
ejpam-4476	92	7	that	that	SCONJ
ejpam-4476	93	1	n∑	n∑	NOUN
ejpam-4476	93	2	k=1	k=1	PROPN
ejpam-4476	94	1	k3	k3	PROPN
ejpam-4476	94	2	=	=	SYM
ejpam-4476	94	3	(	(	PUNCT
ejpam-4476	94	4	n∑	n∑	INTJ
ejpam-4476	94	5	k=1	k=1	PUNCT
ejpam-4476	94	6	k	k	X
ejpam-4476	94	7	)	)	PUNCT
ejpam-4476	94	8	2	2	NUM
ejpam-4476	94	9	can	can	AUX
ejpam-4476	94	10	also	also	ADV
ejpam-4476	94	11	be	be	AUX
ejpam-4476	94	12	presented	present	VERB
ejpam-4476	94	13	through	through	ADP
ejpam-4476	94	14	a	a	DET
ejpam-4476	94	15	similarity	similarity	NOUN
ejpam-4476	94	16	ratio	ratio	NOUN
ejpam-4476	94	17	,	,	PUNCT
ejpam-4476	94	18	as	as	SCONJ
ejpam-4476	94	19	shown	show	VERB
ejpam-4476	94	20	in	in	ADP
ejpam-4476	94	21	figure	figure	NOUN
ejpam-4476	94	22	3	3	NUM
ejpam-4476	94	23	.	.	PUNCT
ejpam-4476	95	1	a	a	DET
ejpam-4476	95	2	visual	visual	ADJ
ejpam-4476	95	3	proof	proof	NOUN
ejpam-4476	95	4	that	that	PRON
ejpam-4476	95	5	is	be	AUX
ejpam-4476	95	6	based	base	VERB
ejpam-4476	95	7	on	on	ADP
ejpam-4476	95	8	the	the	DET
ejpam-4476	95	9	fact	fact	NOUN
ejpam-4476	95	10	that	that	SCONJ
ejpam-4476	95	11	(	(	PUNCT
ejpam-4476	95	12	area	area	NOUN
ejpam-4476	95	13	ratio	ratio	NOUN
ejpam-4476	95	14	)	)	PUNCT
ejpam-4476	95	15	=	=	PUNCT
ejpam-4476	96	1	(	(	PUNCT
ejpam-4476	96	2	similarity	similarity	NOUN
ejpam-4476	96	3	ratio)2	ratio)2	NOUN
ejpam-4476	96	4	for	for	ADP
ejpam-4476	96	5	similar	similar	ADJ
ejpam-4476	96	6	triangles	triangle	NOUN
ejpam-4476	96	7	is	be	AUX
ejpam-4476	96	8	presented	present	VERB
ejpam-4476	96	9	without	without	ADP
ejpam-4476	96	10	calculating	calculate	VERB
ejpam-4476	96	11	half	half	DET
ejpam-4476	96	12	the	the	DET
ejpam-4476	96	13	base	base	NOUN
ejpam-4476	96	14	and	and	CCONJ
ejpam-4476	96	15	height	height	NOUN
ejpam-4476	96	16	.	.	PUNCT
ejpam-4476	97	1	these	these	DET
ejpam-4476	97	2	diagrams	diagram	NOUN
ejpam-4476	97	3	illustrate	illustrate	VERB
ejpam-4476	97	4	that	that	SCONJ
ejpam-4476	97	5	12×1	12×1	PROPN
ejpam-4476	97	6	+	+	PROPN
ejpam-4476	97	7	22×2	22×2	PROPN
ejpam-4476	97	8	+	+	ADJ
ejpam-4476	97	9	32×3	32×3	NOUN
ejpam-4476	97	10	=	=	SYM
ejpam-4476	97	11	(	(	PUNCT
ejpam-4476	97	12	1	1	NUM
ejpam-4476	97	13	+	+	NOUN
ejpam-4476	97	14	2	2	NUM
ejpam-4476	97	15	+	+	NOUN
ejpam-4476	97	16	3)2	3)2	NUM
ejpam-4476	97	17	.	.	PUNCT
ejpam-4476	98	1	right	right	ADJ
ejpam-4476	98	2	-	-	PUNCT
ejpam-4476	98	3	angled	angle	VERB
ejpam-4476	98	4	isosceles	isoscele	NOUN
ejpam-4476	98	5	triangles	triangle	NOUN
ejpam-4476	98	6	can	can	AUX
ejpam-4476	98	7	also	also	ADV
ejpam-4476	98	8	be	be	AUX
ejpam-4476	98	9	used	use	VERB
ejpam-4476	98	10	instead	instead	ADV
ejpam-4476	98	11	of	of	ADP
ejpam-4476	98	12	equilateral	equilateral	ADJ
ejpam-4476	98	13	triangles	triangle	NOUN
ejpam-4476	98	14	.	.	PUNCT
ejpam-4476	99	1	y.	y.	PROPN
ejpam-4476	99	2	kobayashi	kobayashi	PROPN
ejpam-4476	99	3	/	/	SYM
ejpam-4476	99	4	eur	eur	PROPN
ejpam-4476	99	5	.	.	PUNCT
ejpam-4476	100	1	j.	j.	PROPN
ejpam-4476	100	2	pure	pure	PROPN
ejpam-4476	100	3	appl	appl	PROPN
ejpam-4476	100	4	.	.	PROPN
ejpam-4476	100	5	math	math	PROPN
ejpam-4476	100	6	,	,	PUNCT
ejpam-4476	100	7	15	15	NUM
ejpam-4476	100	8	(	(	PUNCT
ejpam-4476	100	9	3	3	NUM
ejpam-4476	100	10	)	)	PUNCT
ejpam-4476	100	11	(	(	PUNCT
ejpam-4476	100	12	2022	2022	NUM
ejpam-4476	100	13	)	)	PUNCT
ejpam-4476	100	14	,	,	PUNCT
ejpam-4476	100	15	864	864	NUM
ejpam-4476	100	16	-	-	SYM
ejpam-4476	100	17	877	877	NUM
ejpam-4476	100	18	870	870	NUM
ejpam-4476	100	19	3	3	NUM
ejpam-4476	100	20	3	3	NUM
ejpam-4476	100	21	3	3	NUM
ejpam-4476	100	22	3	3	NUM
ejpam-4476	100	23	3	3	NUM
ejpam-4476	100	24	3	3	NUM
ejpam-4476	100	25	2	2	NUM
ejpam-4476	100	26	2	2	NUM
ejpam-4476	100	27	2	2	NUM
ejpam-4476	100	28	1	1	NUM
ejpam-4476	100	29	1	1	NUM
ejpam-4476	100	30	2	2	NUM
ejpam-4476	100	31	1	1	NUM
ejpam-4476	100	32	32	32	NUM
ejpam-4476	100	33	22	22	NUM
ejpam-4476	100	34	12	12	NUM
ejpam-4476	100	35	1	1	NUM
ejpam-4476	100	36	+	+	NUM
ejpam-4476	100	37	2	2	NUM
ejpam-4476	100	38	+	+	CCONJ
ejpam-4476	100	39	3	3	NUM
ejpam-4476	100	40	1	1	NUM
ejpam-4476	100	41	+	+	NUM
ejpam-4476	100	42	2	2	NUM
ejpam-4476	100	43	+	+	CCONJ
ejpam-4476	100	44	3	3	NUM
ejpam-4476	100	45	1	1	NUM
ejpam-4476	100	46	+	+	NUM
ejpam-4476	100	47	2	2	NUM
ejpam-4476	100	48	+	+	SYM
ejpam-4476	100	49	3(1	3(1	NUM
ejpam-4476	100	50	+	+	CCONJ
ejpam-4476	100	51	2	2	NUM
ejpam-4476	100	52	+	+	CCONJ
ejpam-4476	100	53	3)2	3)2	NUM
ejpam-4476	100	54	figure	figure	NOUN
ejpam-4476	100	55	3	3	NUM
ejpam-4476	100	56	:	:	PUNCT
ejpam-4476	100	57	visual	visual	ADJ
ejpam-4476	100	58	clue	clue	NOUN
ejpam-4476	100	59	to	to	ADP
ejpam-4476	100	60	n∑	n∑	PROPN
ejpam-4476	100	61	k=1	k=1	PROPN
ejpam-4476	101	1	k3	k3	PROPN
ejpam-4476	101	2	=	=	SYM
ejpam-4476	101	3	(	(	PUNCT
ejpam-4476	101	4	n∑	n∑	INTJ
ejpam-4476	101	5	k=1	k=1	PUNCT
ejpam-4476	101	6	k	k	X
ejpam-4476	101	7	)	)	PUNCT
ejpam-4476	101	8	2	2	NUM
ejpam-4476	101	9	based	base	VERB
ejpam-4476	101	10	on	on	ADP
ejpam-4476	101	11	the	the	DET
ejpam-4476	101	12	similarity	similarity	NOUN
ejpam-4476	101	13	ratio	ratio	NOUN
ejpam-4476	101	14	.	.	PUNCT
ejpam-4476	102	1	3	3	X
ejpam-4476	102	2	.	.	X
ejpam-4476	102	3	sum	sum	NOUN
ejpam-4476	102	4	of	of	ADP
ejpam-4476	102	5	square	square	ADJ
ejpam-4476	102	6	numbers	number	NOUN
ejpam-4476	102	7	3.1	3.1	NUM
ejpam-4476	102	8	.	.	PUNCT
ejpam-4476	103	1	representation	representation	NOUN
ejpam-4476	103	2	based	base	VERB
ejpam-4476	103	3	on	on	ADP
ejpam-4476	103	4	sums	sum	NOUN
ejpam-4476	103	5	of	of	ADP
ejpam-4476	103	6	odd	odd	ADJ
ejpam-4476	103	7	numbers	number	NOUN
ejpam-4476	103	8	as	as	SCONJ
ejpam-4476	103	9	shown	show	VERB
ejpam-4476	103	10	in	in	ADP
ejpam-4476	103	11	table	table	NOUN
ejpam-4476	103	12	2	2	NUM
ejpam-4476	103	13	,	,	PUNCT
ejpam-4476	103	14	a	a	DET
ejpam-4476	103	15	square	square	ADJ
ejpam-4476	103	16	number	number	NOUN
ejpam-4476	103	17	,	,	PUNCT
ejpam-4476	103	18	k2	k2	NOUN
ejpam-4476	103	19	,	,	PUNCT
ejpam-4476	103	20	can	can	AUX
ejpam-4476	103	21	be	be	AUX
ejpam-4476	103	22	represented	represent	VERB
ejpam-4476	103	23	as	as	ADP
ejpam-4476	103	24	the	the	DET
ejpam-4476	103	25	sum	sum	NOUN
ejpam-4476	103	26	of	of	ADP
ejpam-4476	103	27	the	the	DET
ejpam-4476	103	28	consecutive	consecutive	ADJ
ejpam-4476	103	29	odd	odd	ADJ
ejpam-4476	103	30	numbers	number	NOUN
ejpam-4476	103	31	1	1	NUM
ejpam-4476	103	32	,	,	PUNCT
ejpam-4476	103	33	3	3	NUM
ejpam-4476	103	34	,	,	PUNCT
ejpam-4476	103	35	5	5	NUM
ejpam-4476	103	36	,	,	PUNCT
ejpam-4476	103	37	.	.	PUNCT
ejpam-4476	103	38	.	.	PUNCT
ejpam-4476	104	1	.	.	PUNCT
ejpam-4476	105	1	,	,	PUNCT
ejpam-4476	105	2	2k	2k	NOUN
ejpam-4476	105	3	−	−	NOUN
ejpam-4476	105	4	1	1	NUM
ejpam-4476	105	5	,	,	PUNCT
ejpam-4476	105	6	in	in	ADP
ejpam-4476	105	7	which	which	PRON
ejpam-4476	105	8	the	the	DET
ejpam-4476	105	9	number	number	NOUN
ejpam-4476	105	10	of	of	ADP
ejpam-4476	105	11	terms	term	NOUN
ejpam-4476	105	12	is	be	AUX
ejpam-4476	105	13	k.	k.	PROPN
ejpam-4476	105	14	thus	thus	ADV
ejpam-4476	105	15	,	,	PUNCT
ejpam-4476	105	16	12	12	NUM
ejpam-4476	106	1	+	+	SYM
ejpam-4476	106	2	22	22	NUM
ejpam-4476	106	3	+	+	NUM
ejpam-4476	106	4	32	32	NUM
ejpam-4476	106	5	+	+	CCONJ
ejpam-4476	106	6	·	·	PUNCT
ejpam-4476	106	7	·	·	PUNCT
ejpam-4476	106	8	·	·	PUNCT
ejpam-4476	106	9	+	+	NUM
ejpam-4476	106	10	k2	k2	X
ejpam-4476	106	11	+	+	X
ejpam-4476	106	12	·	·	PUNCT
ejpam-4476	106	13	·	·	PUNCT
ejpam-4476	106	14	·	·	PUNCT
ejpam-4476	106	15	+	+	NUM
ejpam-4476	106	16	n2	n2	NOUN
ejpam-4476	106	17	=	=	SYM
ejpam-4476	106	18	(	(	PUNCT
ejpam-4476	106	19	1	1	NUM
ejpam-4476	106	20	+	+	NUM
ejpam-4476	106	21	1	1	NUM
ejpam-4476	106	22	+	+	NUM
ejpam-4476	106	23	1	1	NUM
ejpam-4476	106	24	+	+	NUM
ejpam-4476	106	25	1	1	NUM
ejpam-4476	106	26	+	+	CCONJ
ejpam-4476	106	27	·	·	PUNCT
ejpam-4476	106	28	·	·	PUNCT
ejpam-4476	106	29	·	·	PUNCT
ejpam-4476	106	30	+	+	NUM
ejpam-4476	106	31	1)︸	1)︸	NUM
ejpam-4476	106	32	︷︷	︷︷	PROPN
ejpam-4476	106	33	︸	︸	ADP
ejpam-4476	106	34	n	n	PROPN
ejpam-4476	106	35	terms	term	NOUN
ejpam-4476	106	36	+	+	PROPN
ejpam-4476	106	37	(	(	PUNCT
ejpam-4476	106	38	3	3	NUM
ejpam-4476	106	39	+	+	SYM
ejpam-4476	106	40	3	3	NUM
ejpam-4476	106	41	+	+	NUM
ejpam-4476	106	42	3	3	NUM
ejpam-4476	106	43	+	+	NUM
ejpam-4476	106	44	·	·	PUNCT
ejpam-4476	106	45	·	·	PUNCT
ejpam-4476	106	46	·	·	PUNCT
ejpam-4476	107	1	+	+	NUM
ejpam-4476	107	2	3)︸	3)︸	NUM
ejpam-4476	107	3	︷︷	︷︷	NOUN
ejpam-4476	107	4	︸	︸	X
ejpam-4476	107	5	(	(	PUNCT
ejpam-4476	107	6	n−1	n−1	PROPN
ejpam-4476	107	7	)	)	PUNCT
ejpam-4476	107	8	terms	term	NOUN
ejpam-4476	108	1	+	+	PROPN
ejpam-4476	108	2	(	(	PUNCT
ejpam-4476	108	3	5	5	NUM
ejpam-4476	108	4	+	+	SYM
ejpam-4476	108	5	5	5	NUM
ejpam-4476	108	6	+	+	CCONJ
ejpam-4476	108	7	·	·	PUNCT
ejpam-4476	108	8	·	·	PUNCT
ejpam-4476	108	9	·	·	PUNCT
ejpam-4476	108	10	+	+	NUM
ejpam-4476	108	11	5)︸	5)︸	NUM
ejpam-4476	108	12	︷︷	︷︷	PROPN
ejpam-4476	108	13	︸	︸	X
ejpam-4476	108	14	(	(	PUNCT
ejpam-4476	108	15	n−2	n−2	PROPN
ejpam-4476	108	16	)	)	PUNCT
ejpam-4476	108	17	terms	term	NOUN
ejpam-4476	108	18	+	+	X
ejpam-4476	108	19	·	·	PUNCT
ejpam-4476	108	20	·	·	PUNCT
ejpam-4476	108	21	·	·	PUNCT
ejpam-4476	109	1	+	+	PUNCT
ejpam-4476	109	2	[	[	X
ejpam-4476	109	3	(	(	PUNCT
ejpam-4476	109	4	2k	2k	NOUN
ejpam-4476	109	5	−	−	NOUN
ejpam-4476	109	6	1	1	NUM
ejpam-4476	109	7	)	)	PUNCT
ejpam-4476	109	8	+	+	CCONJ
ejpam-4476	109	9	·	·	PUNCT
ejpam-4476	109	10	·	·	PUNCT
ejpam-4476	109	11	·	·	PUNCT
ejpam-4476	109	12	+	+	NUM
ejpam-4476	109	13	(	(	PUNCT
ejpam-4476	109	14	2k	2k	NUM
ejpam-4476	109	15	−	−	PROPN
ejpam-4476	109	16	1)]︸	1)]︸	NUM
ejpam-4476	109	17	︷︷	︷︷	PROPN
ejpam-4476	109	18	︸	︸	X
ejpam-4476	110	1	[	[	X
ejpam-4476	110	2	n−(k+1	n−(k+1	NOUN
ejpam-4476	110	3	)	)	PUNCT
ejpam-4476	110	4	]	]	PUNCT
ejpam-4476	110	5	terns	tern	VERB
ejpam-4476	110	6	+	+	X
ejpam-4476	110	7	·	·	PUNCT
ejpam-4476	110	8	·	·	PUNCT
ejpam-4476	110	9	·	·	PUNCT
ejpam-4476	111	1	+	+	PUNCT
ejpam-4476	111	2	(	(	PUNCT
ejpam-4476	111	3	2n−	2n−	NUM
ejpam-4476	111	4	1)︸	1)︸	NUM
ejpam-4476	111	5	︷︷	︷︷	NOUN
ejpam-4476	111	6	︸	︸	ADP
ejpam-4476	111	7	1	1	NUM
ejpam-4476	111	8	term	term	NOUN
ejpam-4476	111	9	.	.	PUNCT
ejpam-4476	112	1	(	(	PUNCT
ejpam-4476	112	2	3	3	X
ejpam-4476	112	3	)	)	PUNCT
ejpam-4476	112	4	by	by	ADP
ejpam-4476	112	5	expressing	express	VERB
ejpam-4476	112	6	equation	equation	NOUN
ejpam-4476	112	7	(	(	PUNCT
ejpam-4476	112	8	3	3	NUM
ejpam-4476	112	9	)	)	PUNCT
ejpam-4476	112	10	with	with	ADP
ejpam-4476	112	11	summation	summation	NOUN
ejpam-4476	112	12	signs	sign	NOUN
ejpam-4476	112	13	,	,	PUNCT
ejpam-4476	112	14	the	the	DET
ejpam-4476	112	15	calculation	calculation	NOUN
ejpam-4476	112	16	can	can	AUX
ejpam-4476	112	17	be	be	AUX
ejpam-4476	112	18	simplified	simplify	VERB
ejpam-4476	112	19	to	to	PART
ejpam-4476	112	20	obtain	obtain	VERB
ejpam-4476	113	1	n∑	n∑	X
ejpam-4476	113	2	k=1	k=1	PROPN
ejpam-4476	113	3	k2	k2	PROPN
ejpam-4476	113	4	=	=	SYM
ejpam-4476	113	5	n∑	n∑	NOUN
ejpam-4476	113	6	k=1	k=1	PROPN
ejpam-4476	114	1	(	(	PUNCT
ejpam-4476	114	2	2k	2k	NOUN
ejpam-4476	114	3	−	−	PROPN
ejpam-4476	115	1	1)[n−	1)[n−	NUM
ejpam-4476	115	2	(	(	PUNCT
ejpam-4476	115	3	k	k	NOUN
ejpam-4476	115	4	−	−	PROPN
ejpam-4476	115	5	1	1	NUM
ejpam-4476	115	6	)	)	PUNCT
ejpam-4476	115	7	]	]	PUNCT
ejpam-4476	116	1	=	=	PUNCT
ejpam-4476	116	2	(	(	PUNCT
ejpam-4476	116	3	2n+	2n+	NUM
ejpam-4476	116	4	3	3	NUM
ejpam-4476	116	5	)	)	PUNCT
ejpam-4476	116	6	n∑	n∑	NOUN
ejpam-4476	116	7	k=1	k=1	PUNCT
ejpam-4476	117	1	k	k	X
ejpam-4476	118	1	−	−	PROPN
ejpam-4476	118	2	2	2	NUM
ejpam-4476	118	3	n∑	n∑	NOUN
ejpam-4476	118	4	k=1	k=1	PROPN
ejpam-4476	118	5	k2	k2	PROPN
ejpam-4476	118	6	−	−	PROPN
ejpam-4476	119	1	n∑	n∑	PROPN
ejpam-4476	120	1	k=1	k=1	X
ejpam-4476	120	2	(	(	PUNCT
ejpam-4476	120	3	n+	n+	NOUN
ejpam-4476	120	4	1	1	NUM
ejpam-4476	120	5	)	)	PUNCT
ejpam-4476	120	6	.	.	PUNCT
ejpam-4476	121	1	(	(	PUNCT
ejpam-4476	121	2	4	4	X
ejpam-4476	121	3	)	)	PUNCT
ejpam-4476	121	4	with	with	ADP
ejpam-4476	121	5	slight	slight	ADJ
ejpam-4476	121	6	rearrangements	rearrangement	NOUN
ejpam-4476	121	7	,	,	PUNCT
ejpam-4476	121	8	equation	equation	NOUN
ejpam-4476	121	9	(	(	PUNCT
ejpam-4476	121	10	4	4	X
ejpam-4476	121	11	)	)	PUNCT
ejpam-4476	121	12	can	can	AUX
ejpam-4476	121	13	be	be	AUX
ejpam-4476	121	14	rewritten	rewrite	VERB
ejpam-4476	121	15	as	as	ADP
ejpam-4476	121	16	3	3	NUM
ejpam-4476	121	17	n∑	n∑	NOUN
ejpam-4476	121	18	k=1	k=1	PROPN
ejpam-4476	121	19	k2	k2	PROPN
ejpam-4476	121	20	=	=	SYM
ejpam-4476	121	21	(	(	PUNCT
ejpam-4476	121	22	2n+	2n+	NUM
ejpam-4476	121	23	3	3	NUM
ejpam-4476	121	24	)	)	PUNCT
ejpam-4476	121	25	n∑	n∑	NOUN
ejpam-4476	121	26	k=1	k=1	PUNCT
ejpam-4476	122	1	k	k	X
ejpam-4476	122	2	−	−	PROPN
ejpam-4476	122	3	n(n+	n(n+	PROPN
ejpam-4476	122	4	1	1	NUM
ejpam-4476	122	5	)	)	PUNCT
ejpam-4476	122	6	,	,	PUNCT
ejpam-4476	122	7	(	(	PUNCT
ejpam-4476	122	8	5	5	X
ejpam-4476	122	9	)	)	PUNCT
ejpam-4476	122	10	y.	y.	PROPN
ejpam-4476	122	11	kobayashi	kobayashi	PROPN
ejpam-4476	122	12	/	/	SYM
ejpam-4476	122	13	eur	eur	PROPN
ejpam-4476	122	14	.	.	PUNCT
ejpam-4476	123	1	j.	j.	PROPN
ejpam-4476	123	2	pure	pure	PROPN
ejpam-4476	123	3	appl	appl	PROPN
ejpam-4476	123	4	.	.	PROPN
ejpam-4476	123	5	math	math	PROPN
ejpam-4476	123	6	,	,	PUNCT
ejpam-4476	123	7	15	15	NUM
ejpam-4476	123	8	(	(	PUNCT
ejpam-4476	123	9	3	3	NUM
ejpam-4476	123	10	)	)	PUNCT
ejpam-4476	123	11	(	(	PUNCT
ejpam-4476	123	12	2022	2022	NUM
ejpam-4476	123	13	)	)	PUNCT
ejpam-4476	123	14	,	,	PUNCT
ejpam-4476	123	15	864	864	NUM
ejpam-4476	123	16	-	-	SYM
ejpam-4476	123	17	877	877	NUM
ejpam-4476	123	18	871	871	NUM
ejpam-4476	123	19	and	and	CCONJ
ejpam-4476	123	20	thus	thus	ADV
ejpam-4476	123	21	we	we	PRON
ejpam-4476	123	22	obtain	obtain	VERB
ejpam-4476	123	23	n∑	n∑	X
ejpam-4476	123	24	k=1	k=1	PROPN
ejpam-4476	124	1	k2	k2	PROPN
ejpam-4476	125	1	=	=	SYM
ejpam-4476	126	1	2n+	2n+	NUM
ejpam-4476	127	1	3	3	NUM
ejpam-4476	127	2	3	3	NUM
ejpam-4476	127	3	n∑	n∑	NOUN
ejpam-4476	127	4	k=1	k=1	PUNCT
ejpam-4476	128	1	k	k	X
ejpam-4476	129	1	−	−	PROPN
ejpam-4476	129	2	2	2	NUM
ejpam-4476	129	3	3	3	NUM
ejpam-4476	129	4	n∑	n∑	NOUN
ejpam-4476	129	5	k=1	k=1	PUNCT
ejpam-4476	130	1	k	k	X
ejpam-4476	130	2	=	=	SYM
ejpam-4476	130	3	2n+	2n+	NUM
ejpam-4476	130	4	1	1	NUM
ejpam-4476	130	5	3	3	NUM
ejpam-4476	130	6	n∑	n∑	NOUN
ejpam-4476	130	7	k=1	k=1	PUNCT
ejpam-4476	131	1	k	k	X
ejpam-4476	131	2	=	=	SYM
ejpam-4476	131	3	1	1	NUM
ejpam-4476	131	4	6	6	NUM
ejpam-4476	131	5	n(n+	n(n+	NOUN
ejpam-4476	131	6	1)(2n+	1)(2n+	NUM
ejpam-4476	131	7	1	1	NUM
ejpam-4476	131	8	)	)	PUNCT
ejpam-4476	131	9	(	(	PUNCT
ejpam-4476	131	10	6	6	X
ejpam-4476	131	11	)	)	PUNCT
ejpam-4476	131	12	using	use	VERB
ejpam-4476	131	13	n∑	n∑	PROPN
ejpam-4476	131	14	k=1	k=1	PUNCT
ejpam-4476	132	1	k	k	X
ejpam-4476	132	2	=	=	PUNCT
ejpam-4476	132	3	n(n+	n(n+	PROPN
ejpam-4476	132	4	1	1	NUM
ejpam-4476	132	5	)	)	SYM
ejpam-4476	132	6	2	2	NUM
ejpam-4476	132	7	.	.	PUNCT
ejpam-4476	132	8	table	table	NOUN
ejpam-4476	132	9	2	2	NUM
ejpam-4476	132	10	.	.	PUNCT
ejpam-4476	132	11	square	square	ADJ
ejpam-4476	132	12	numbers	number	NOUN
ejpam-4476	132	13	expressed	express	VERB
ejpam-4476	132	14	as	as	ADP
ejpam-4476	132	15	sums	sum	NOUN
ejpam-4476	132	16	of	of	ADP
ejpam-4476	132	17	odd	odd	ADJ
ejpam-4476	132	18	numbers	number	NOUN
ejpam-4476	132	19	.	.	PUNCT
ejpam-4476	133	1	12	12	NUM
ejpam-4476	133	2	=	=	SYM
ejpam-4476	133	3	1	1	NUM
ejpam-4476	133	4	22	22	NUM
ejpam-4476	133	5	=	=	SYM
ejpam-4476	133	6	1	1	NUM
ejpam-4476	133	7	+	+	NUM
ejpam-4476	133	8	3	3	NUM
ejpam-4476	133	9	32	32	NUM
ejpam-4476	133	10	=	=	SYM
ejpam-4476	133	11	1	1	NUM
ejpam-4476	133	12	+	+	NUM
ejpam-4476	133	13	3	3	NUM
ejpam-4476	133	14	+	+	SYM
ejpam-4476	133	15	5	5	NUM
ejpam-4476	133	16	·	·	PUNCT
ejpam-4476	133	17	·	·	PUNCT
ejpam-4476	133	18	·	·	PUNCT
ejpam-4476	133	19	k2	k2	X
ejpam-4476	133	20	=	=	NOUN
ejpam-4476	133	21	1	1	NUM
ejpam-4476	133	22	+	+	NUM
ejpam-4476	133	23	3	3	NUM
ejpam-4476	133	24	+	+	SYM
ejpam-4476	133	25	5	5	NUM
ejpam-4476	133	26	+	+	CCONJ
ejpam-4476	133	27	7	7	NUM
ejpam-4476	133	28	+	+	CCONJ
ejpam-4476	133	29	·	·	PUNCT
ejpam-4476	133	30	·	·	PUNCT
ejpam-4476	133	31	·	·	PUNCT
ejpam-4476	133	32	+	+	CCONJ
ejpam-4476	133	33	(	(	PUNCT
ejpam-4476	133	34	2k	2k	NOUN
ejpam-4476	133	35	−	−	NOUN
ejpam-4476	133	36	1	1	NUM
ejpam-4476	133	37	)	)	PUNCT
ejpam-4476	133	38	·	·	PUNCT
ejpam-4476	133	39	·	·	PUNCT
ejpam-4476	133	40	·	·	PUNCT
ejpam-4476	133	41	n2	n2	NOUN
ejpam-4476	133	42	=	=	NOUN
ejpam-4476	133	43	1	1	NUM
ejpam-4476	133	44	+	+	NUM
ejpam-4476	133	45	3	3	NUM
ejpam-4476	133	46	+	+	SYM
ejpam-4476	133	47	5	5	NUM
ejpam-4476	133	48	+	+	CCONJ
ejpam-4476	133	49	7	7	NUM
ejpam-4476	133	50	+	+	CCONJ
ejpam-4476	133	51	·	·	PUNCT
ejpam-4476	133	52	·	·	PUNCT
ejpam-4476	133	53	·	·	PUNCT
ejpam-4476	133	54	+	+	CCONJ
ejpam-4476	133	55	(	(	PUNCT
ejpam-4476	133	56	2k	2k	NOUN
ejpam-4476	133	57	−	−	NOUN
ejpam-4476	133	58	1	1	NUM
ejpam-4476	133	59	)	)	PUNCT
ejpam-4476	133	60	+	+	CCONJ
ejpam-4476	133	61	·	·	PUNCT
ejpam-4476	133	62	·	·	PUNCT
ejpam-4476	133	63	·	·	PUNCT
ejpam-4476	133	64	+	+	PUNCT
ejpam-4476	133	65	(	(	PUNCT
ejpam-4476	133	66	2n−	2n−	PROPN
ejpam-4476	133	67	1	1	NUM
ejpam-4476	133	68	)	)	PUNCT
ejpam-4476	133	69	a	a	DET
ejpam-4476	133	70	redundant	redundant	ADJ
ejpam-4476	133	71	calculation	calculation	NOUN
ejpam-4476	133	72	without	without	ADP
ejpam-4476	133	73	summation	summation	NOUN
ejpam-4476	133	74	signs	sign	NOUN
ejpam-4476	133	75	clearly	clearly	ADV
ejpam-4476	133	76	shows	show	VERB
ejpam-4476	133	77	the	the	DET
ejpam-4476	133	78	process	process	NOUN
ejpam-4476	133	79	of	of	ADP
ejpam-4476	133	80	calculation	calculation	NOUN
ejpam-4476	133	81	,	,	PUNCT
ejpam-4476	133	82	which	which	PRON
ejpam-4476	133	83	is	be	AUX
ejpam-4476	133	84	preferable	preferable	ADJ
ejpam-4476	133	85	from	from	ADP
ejpam-4476	133	86	a	a	DET
ejpam-4476	133	87	pedagogical	pedagogical	ADJ
ejpam-4476	133	88	standpoint	standpoint	NOUN
ejpam-4476	133	89	.	.	PUNCT
ejpam-4476	134	1	equation	equation	NOUN
ejpam-4476	134	2	(	(	PUNCT
ejpam-4476	134	3	3	3	X
ejpam-4476	134	4	)	)	PUNCT
ejpam-4476	134	5	can	can	AUX
ejpam-4476	134	6	be	be	AUX
ejpam-4476	134	7	written	write	VERB
ejpam-4476	134	8	as	as	ADP
ejpam-4476	134	9	12	12	NUM
ejpam-4476	134	10	+	+	CCONJ
ejpam-4476	134	11	22	22	NUM
ejpam-4476	134	12	+	+	NUM
ejpam-4476	134	13	32	32	NUM
ejpam-4476	134	14	+	+	CCONJ
ejpam-4476	134	15	·	·	PUNCT
ejpam-4476	134	16	·	·	PUNCT
ejpam-4476	134	17	·	·	PUNCT
ejpam-4476	135	1	+	+	NUM
ejpam-4476	135	2	k2	k2	X
ejpam-4476	135	3	+	+	X
ejpam-4476	135	4	·	·	PUNCT
ejpam-4476	135	5	·	·	PUNCT
ejpam-4476	135	6	·	·	PUNCT
ejpam-4476	135	7	+	+	NUM
ejpam-4476	135	8	n2	n2	ADJ
ejpam-4476	135	9	=	=	SYM
ejpam-4476	135	10	1×	1×	NUM
ejpam-4476	135	11	n+	n+	NUM
ejpam-4476	135	12	3×	3×	NUM
ejpam-4476	135	13	(	(	PUNCT
ejpam-4476	135	14	n−	n−	NOUN
ejpam-4476	135	15	1	1	NUM
ejpam-4476	135	16	)	)	PUNCT
ejpam-4476	135	17	+	+	CCONJ
ejpam-4476	135	18	5×	5×	NOUN
ejpam-4476	135	19	(	(	PUNCT
ejpam-4476	135	20	n−	n−	NOUN
ejpam-4476	135	21	2	2	NUM
ejpam-4476	135	22	)	)	PUNCT
ejpam-4476	135	23	+	+	CCONJ
ejpam-4476	135	24	·	·	PUNCT
ejpam-4476	135	25	·	·	PUNCT
ejpam-4476	135	26	·	·	PUNCT
ejpam-4476	136	1	+	+	X
ejpam-4476	136	2	(	(	PUNCT
ejpam-4476	136	3	2k	2k	NOUN
ejpam-4476	136	4	−	−	NOUN
ejpam-4476	136	5	1)×	1)×	NUM
ejpam-4476	136	6	[	[	X
ejpam-4476	136	7	n−	n−	PROPN
ejpam-4476	136	8	(	(	PUNCT
ejpam-4476	136	9	k	k	NOUN
ejpam-4476	136	10	−	−	PROPN
ejpam-4476	136	11	1	1	NUM
ejpam-4476	136	12	)	)	PUNCT
ejpam-4476	136	13	]	]	PUNCT
ejpam-4476	137	1	+	+	CCONJ
ejpam-4476	137	2	·	·	PUNCT
ejpam-4476	137	3	·	·	PUNCT
ejpam-4476	137	4	·	·	PUNCT
ejpam-4476	137	5	+	+	PUNCT
ejpam-4476	137	6	(	(	PUNCT
ejpam-4476	137	7	2n−	2n−	NUM
ejpam-4476	137	8	1)×	1)×	NUM
ejpam-4476	137	9	[	[	X
ejpam-4476	137	10	n−	n−	NOUN
ejpam-4476	137	11	(	(	PUNCT
ejpam-4476	137	12	n−	n−	NOUN
ejpam-4476	137	13	1	1	NUM
ejpam-4476	137	14	)	)	PUNCT
ejpam-4476	137	15	]	]	PUNCT
ejpam-4476	137	16	.	.	PUNCT
ejpam-4476	138	1	(	(	PUNCT
ejpam-4476	138	2	7	7	X
ejpam-4476	138	3	)	)	PUNCT
ejpam-4476	138	4	the	the	DET
ejpam-4476	138	5	right	right	ADJ
ejpam-4476	138	6	side	side	NOUN
ejpam-4476	138	7	of	of	ADP
ejpam-4476	138	8	equation	equation	NOUN
ejpam-4476	138	9	(	(	PUNCT
ejpam-4476	138	10	7	7	X
ejpam-4476	138	11	)	)	PUNCT
ejpam-4476	138	12	can	can	AUX
ejpam-4476	138	13	be	be	AUX
ejpam-4476	138	14	expressed	express	VERB
ejpam-4476	138	15	as	as	ADP
ejpam-4476	138	16	[	[	X
ejpam-4476	138	17	1	1	NUM
ejpam-4476	138	18	+	+	NUM
ejpam-4476	138	19	3	3	NUM
ejpam-4476	138	20	+	+	SYM
ejpam-4476	138	21	5	5	NUM
ejpam-4476	138	22	+	+	CCONJ
ejpam-4476	138	23	·	·	PUNCT
ejpam-4476	138	24	·	·	PUNCT
ejpam-4476	138	25	·	·	PUNCT
ejpam-4476	139	1	+	+	CCONJ
ejpam-4476	139	2	(	(	PUNCT
ejpam-4476	139	3	2k	2k	NOUN
ejpam-4476	139	4	−	−	NOUN
ejpam-4476	139	5	1	1	NUM
ejpam-4476	139	6	)	)	PUNCT
ejpam-4476	139	7	+	+	CCONJ
ejpam-4476	139	8	·	·	PUNCT
ejpam-4476	139	9	·	·	PUNCT
ejpam-4476	139	10	·	·	PUNCT
ejpam-4476	139	11	+	+	PUNCT
ejpam-4476	139	12	(	(	PUNCT
ejpam-4476	139	13	2n−	2n−	NUM
ejpam-4476	139	14	1)]n	1)]n	NUM
ejpam-4476	139	15	−[3×	−[3×	X
ejpam-4476	139	16	1	1	NUM
ejpam-4476	139	17	+	+	CCONJ
ejpam-4476	139	18	5×	5×	NUM
ejpam-4476	139	19	2	2	NUM
ejpam-4476	139	20	+	+	CCONJ
ejpam-4476	139	21	·	·	PUNCT
ejpam-4476	139	22	·	·	PUNCT
ejpam-4476	139	23	·	·	PUNCT
ejpam-4476	139	24	+	+	NUM
ejpam-4476	139	25	(	(	PUNCT
ejpam-4476	139	26	2k	2k	NOUN
ejpam-4476	139	27	−	−	PROPN
ejpam-4476	139	28	1)×	1)×	NUM
ejpam-4476	139	29	(	(	PUNCT
ejpam-4476	139	30	k	k	NOUN
ejpam-4476	139	31	−	−	PROPN
ejpam-4476	139	32	1	1	NUM
ejpam-4476	139	33	)	)	PUNCT
ejpam-4476	139	34	+	+	CCONJ
ejpam-4476	139	35	·	·	PUNCT
ejpam-4476	139	36	·	·	PUNCT
ejpam-4476	139	37	·	·	PUNCT
ejpam-4476	140	1	+	+	PUNCT
ejpam-4476	140	2	(	(	PUNCT
ejpam-4476	140	3	2n−	2n−	NUM
ejpam-4476	140	4	1)×	1)×	NUM
ejpam-4476	140	5	(	(	PUNCT
ejpam-4476	140	6	n−	n−	NOUN
ejpam-4476	140	7	1	1	NUM
ejpam-4476	140	8	)	)	PUNCT
ejpam-4476	140	9	]	]	PUNCT
ejpam-4476	140	10	.	.	PUNCT
ejpam-4476	141	1	(	(	PUNCT
ejpam-4476	141	2	8)	8)	NUM
ejpam-4476	141	3	we	we	PRON
ejpam-4476	141	4	can	can	AUX
ejpam-4476	141	5	simply	simply	ADV
ejpam-4476	141	6	obtain	obtain	VERB
ejpam-4476	141	7	[	[	PUNCT
ejpam-4476	141	8	1	1	NUM
ejpam-4476	141	9	+	+	NUM
ejpam-4476	141	10	3	3	NUM
ejpam-4476	141	11	+	+	SYM
ejpam-4476	141	12	5	5	NUM
ejpam-4476	141	13	+	+	CCONJ
ejpam-4476	141	14	·	·	PUNCT
ejpam-4476	141	15	·	·	PUNCT
ejpam-4476	141	16	·	·	PUNCT
ejpam-4476	142	1	+	+	CCONJ
ejpam-4476	142	2	(	(	PUNCT
ejpam-4476	142	3	2k	2k	NOUN
ejpam-4476	142	4	−	−	NOUN
ejpam-4476	142	5	1	1	NUM
ejpam-4476	142	6	)	)	PUNCT
ejpam-4476	142	7	+	+	CCONJ
ejpam-4476	142	8	·	·	PUNCT
ejpam-4476	142	9	·	·	PUNCT
ejpam-4476	142	10	·	·	PUNCT
ejpam-4476	142	11	+	+	PUNCT
ejpam-4476	142	12	(	(	PUNCT
ejpam-4476	142	13	2n−	2n−	NUM
ejpam-4476	142	14	1)]n	1)]n	NUM
ejpam-4476	142	15	=	=	SYM
ejpam-4476	142	16	n3	n3	NOUN
ejpam-4476	142	17	(	(	PUNCT
ejpam-4476	142	18	9	9	NUM
ejpam-4476	142	19	)	)	PUNCT
ejpam-4476	142	20	from	from	ADP
ejpam-4476	142	21	n2	n2	ADJ
ejpam-4476	142	22	=	=	PROPN
ejpam-4476	142	23	1	1	NUM
ejpam-4476	142	24	+	+	NOUN
ejpam-4476	142	25	3	3	NUM
ejpam-4476	142	26	+	+	SYM
ejpam-4476	142	27	5	5	NUM
ejpam-4476	142	28	+	+	ADJ
ejpam-4476	142	29	7	7	NUM
ejpam-4476	142	30	+	+	NUM
ejpam-4476	142	31	·	·	PUNCT
ejpam-4476	142	32	·	·	PUNCT
ejpam-4476	142	33	·	·	PUNCT
ejpam-4476	143	1	+	+	PROPN
ejpam-4476	143	2	(	(	PUNCT
ejpam-4476	143	3	2k−	2k−	PROPN
ejpam-4476	143	4	1)+	1)+	NUM
ejpam-4476	143	5	·	·	PUNCT
ejpam-4476	143	6	·	·	PUNCT
ejpam-4476	143	7	·	·	PUNCT
ejpam-4476	143	8	+	+	PROPN
ejpam-4476	143	9	(	(	PUNCT
ejpam-4476	143	10	2n−	2n−	PROPN
ejpam-4476	143	11	1	1	NUM
ejpam-4476	143	12	)	)	PUNCT
ejpam-4476	143	13	,	,	PUNCT
ejpam-4476	143	14	although	although	SCONJ
ejpam-4476	143	15	the	the	DET
ejpam-4476	143	16	calculation	calculation	NOUN
ejpam-4476	143	17	is	be	AUX
ejpam-4476	143	18	a	a	DET
ejpam-4476	143	19	little	little	ADV
ejpam-4476	143	20	complicated	complicated	ADJ
ejpam-4476	143	21	:	:	PUNCT
ejpam-4476	143	22	3×	3×	NUM
ejpam-4476	143	23	1	1	NUM
ejpam-4476	143	24	+	+	CCONJ
ejpam-4476	143	25	5×	5×	NUM
ejpam-4476	143	26	2	2	NUM
ejpam-4476	143	27	+	+	CCONJ
ejpam-4476	143	28	·	·	PUNCT
ejpam-4476	143	29	·	·	PUNCT
ejpam-4476	143	30	·	·	PUNCT
ejpam-4476	143	31	+	+	NUM
ejpam-4476	143	32	(	(	PUNCT
ejpam-4476	143	33	2k	2k	NOUN
ejpam-4476	143	34	−	−	NOUN
ejpam-4476	143	35	1)×	1)×	NUM
ejpam-4476	143	36	k	k	NOUN
ejpam-4476	143	37	+	+	CCONJ
ejpam-4476	143	38	·	·	PUNCT
ejpam-4476	143	39	·	·	PUNCT
ejpam-4476	143	40	·	·	PUNCT
ejpam-4476	143	41	+	+	PUNCT
ejpam-4476	143	42	(	(	PUNCT
ejpam-4476	143	43	2n−	2n−	NUM
ejpam-4476	143	44	1)×	1)×	NUM
ejpam-4476	143	45	(	(	PUNCT
ejpam-4476	143	46	n−	n−	NOUN
ejpam-4476	143	47	1	1	NUM
ejpam-4476	143	48	)	)	PUNCT
ejpam-4476	143	49	=	=	NOUN
ejpam-4476	143	50	(	(	PUNCT
ejpam-4476	143	51	2	2	NUM
ejpam-4476	143	52	+	+	SYM
ejpam-4476	143	53	1)×	1)×	NUM
ejpam-4476	143	54	1	1	NUM
ejpam-4476	143	55	+	+	CCONJ
ejpam-4476	143	56	(	(	PUNCT
ejpam-4476	143	57	3	3	NUM
ejpam-4476	143	58	+	+	SYM
ejpam-4476	143	59	2)×	2)×	NUM
ejpam-4476	143	60	2	2	NUM
ejpam-4476	143	61	+	+	NUM
ejpam-4476	143	62	·	·	PUNCT
ejpam-4476	143	63	·	·	PUNCT
ejpam-4476	143	64	·	·	PUNCT
ejpam-4476	144	1	+	+	PUNCT
ejpam-4476	145	1	[	[	X
ejpam-4476	145	2	k	k	X
ejpam-4476	145	3	+	+	X
ejpam-4476	145	4	(	(	PUNCT
ejpam-4476	145	5	k	k	PROPN
ejpam-4476	145	6	−	−	PROPN
ejpam-4476	145	7	1)]×	1)]×	NUM
ejpam-4476	145	8	(	(	PUNCT
ejpam-4476	145	9	k	k	NOUN
ejpam-4476	145	10	−	−	PROPN
ejpam-4476	145	11	1	1	NUM
ejpam-4476	145	12	)	)	PUNCT
ejpam-4476	145	13	+	+	CCONJ
ejpam-4476	145	14	·	·	PUNCT
ejpam-4476	145	15	·	·	PUNCT
ejpam-4476	145	16	·	·	PUNCT
ejpam-4476	145	17	+	+	PUNCT
ejpam-4476	145	18	[	[	X
ejpam-4476	145	19	n+	n+	X
ejpam-4476	145	20	(	(	PUNCT
ejpam-4476	145	21	n−	n−	NOUN
ejpam-4476	145	22	1)]×	1)]×	NUM
ejpam-4476	145	23	(	(	PUNCT
ejpam-4476	145	24	n−	n−	NOUN
ejpam-4476	145	25	1	1	NUM
ejpam-4476	145	26	)	)	PUNCT
ejpam-4476	145	27	=	=	PUNCT
ejpam-4476	145	28	2×	2×	NOUN
ejpam-4476	145	29	1	1	NUM
ejpam-4476	145	30	+	+	CCONJ
ejpam-4476	145	31	12	12	NUM
ejpam-4476	145	32	+	+	NUM
ejpam-4476	145	33	3×	3×	NUM
ejpam-4476	145	34	2	2	NUM
ejpam-4476	145	35	+	+	NUM
ejpam-4476	145	36	22	22	NUM
ejpam-4476	145	37	+	+	CCONJ
ejpam-4476	145	38	·	·	PUNCT
ejpam-4476	145	39	·	·	PUNCT
ejpam-4476	145	40	·	·	PUNCT
ejpam-4476	145	41	+	+	NUM
ejpam-4476	146	1	k(k	k(k	ADJ
ejpam-4476	146	2	−	−	NOUN
ejpam-4476	146	3	1	1	NUM
ejpam-4476	146	4	)	)	PUNCT
ejpam-4476	146	5	+	+	CCONJ
ejpam-4476	146	6	(	(	PUNCT
ejpam-4476	146	7	k	k	X
ejpam-4476	146	8	−	−	PROPN
ejpam-4476	147	1	1)2	1)2	NUM
ejpam-4476	148	1	+	+	X
ejpam-4476	148	2	·	·	PUNCT
ejpam-4476	148	3	·	·	PUNCT
ejpam-4476	148	4	·	·	PUNCT
ejpam-4476	148	5	+	+	NUM
ejpam-4476	148	6	n(n−	n(n−	NOUN
ejpam-4476	148	7	1	1	NUM
ejpam-4476	148	8	)	)	PUNCT
ejpam-4476	148	9	+	+	CCONJ
ejpam-4476	148	10	(	(	PUNCT
ejpam-4476	148	11	n−	n−	NOUN
ejpam-4476	148	12	1)2	1)2	NUM
ejpam-4476	148	13	y.	y.	PROPN
ejpam-4476	148	14	kobayashi	kobayashi	PROPN
ejpam-4476	148	15	/	/	SYM
ejpam-4476	148	16	eur	eur	PROPN
ejpam-4476	148	17	.	.	PUNCT
ejpam-4476	149	1	j.	j.	PROPN
ejpam-4476	149	2	pure	pure	PROPN
ejpam-4476	149	3	appl	appl	PROPN
ejpam-4476	149	4	.	.	PROPN
ejpam-4476	149	5	math	math	PROPN
ejpam-4476	149	6	,	,	PUNCT
ejpam-4476	149	7	15	15	NUM
ejpam-4476	149	8	(	(	PUNCT
ejpam-4476	149	9	3	3	NUM
ejpam-4476	149	10	)	)	PUNCT
ejpam-4476	149	11	(	(	PUNCT
ejpam-4476	149	12	2022	2022	NUM
ejpam-4476	149	13	)	)	PUNCT
ejpam-4476	149	14	,	,	PUNCT
ejpam-4476	149	15	864	864	NUM
ejpam-4476	149	16	-	-	SYM
ejpam-4476	149	17	877	877	NUM
ejpam-4476	149	18	872	872	NUM
ejpam-4476	149	19	=	=	SYM
ejpam-4476	149	20	(	(	PUNCT
ejpam-4476	149	21	1	1	NUM
ejpam-4476	149	22	+	+	SYM
ejpam-4476	149	23	1)×	1)×	NUM
ejpam-4476	149	24	1	1	NUM
ejpam-4476	149	25	+	+	NUM
ejpam-4476	149	26	12	12	NUM
ejpam-4476	149	27	+	+	CCONJ
ejpam-4476	149	28	(	(	PUNCT
ejpam-4476	149	29	2	2	NUM
ejpam-4476	149	30	+	+	SYM
ejpam-4476	149	31	1)×	1)×	NUM
ejpam-4476	149	32	2	2	NUM
ejpam-4476	149	33	+	+	CCONJ
ejpam-4476	149	34	22	22	NUM
ejpam-4476	149	35	+	+	CCONJ
ejpam-4476	149	36	·	·	PUNCT
ejpam-4476	149	37	·	·	PUNCT
ejpam-4476	149	38	·	·	PUNCT
ejpam-4476	150	1	+	+	PUNCT
ejpam-4476	150	2	[	[	X
ejpam-4476	150	3	(	(	PUNCT
ejpam-4476	150	4	k	k	NOUN
ejpam-4476	150	5	−	−	PROPN
ejpam-4476	150	6	1	1	NUM
ejpam-4476	150	7	)	)	PUNCT
ejpam-4476	150	8	+	+	PUNCT
ejpam-4476	151	1	1](k	1](k	NUM
ejpam-4476	151	2	−	−	NOUN
ejpam-4476	151	3	1	1	NUM
ejpam-4476	151	4	)	)	PUNCT
ejpam-4476	151	5	+	+	CCONJ
ejpam-4476	151	6	(	(	PUNCT
ejpam-4476	151	7	k	k	X
ejpam-4476	151	8	−	−	PROPN
ejpam-4476	151	9	1)2	1)2	NUM
ejpam-4476	151	10	+	+	X
ejpam-4476	151	11	·	·	PUNCT
ejpam-4476	151	12	·	·	PUNCT
ejpam-4476	151	13	·	·	PUNCT
ejpam-4476	151	14	+	+	PUNCT
ejpam-4476	152	1	[	[	X
ejpam-4476	152	2	(	(	PUNCT
ejpam-4476	152	3	n−	n−	NOUN
ejpam-4476	152	4	1	1	NUM
ejpam-4476	152	5	)	)	PUNCT
ejpam-4476	153	1	+	+	CCONJ
ejpam-4476	153	2	1](n−	1](n−	NUM
ejpam-4476	153	3	1	1	NUM
ejpam-4476	153	4	)	)	PUNCT
ejpam-4476	154	1	+	+	CCONJ
ejpam-4476	154	2	(	(	PUNCT
ejpam-4476	154	3	n−	n−	NOUN
ejpam-4476	154	4	1)2	1)2	NUM
ejpam-4476	154	5	=	=	SYM
ejpam-4476	154	6	2[12	2[12	NUM
ejpam-4476	154	7	+	+	CCONJ
ejpam-4476	154	8	22	22	NUM
ejpam-4476	154	9	+	+	CCONJ
ejpam-4476	154	10	·	·	PUNCT
ejpam-4476	154	11	·	·	PUNCT
ejpam-4476	154	12	·	·	PUNCT
ejpam-4476	155	1	+	+	PUNCT
ejpam-4476	155	2	(	(	PUNCT
ejpam-4476	155	3	k	k	PROPN
ejpam-4476	155	4	−	−	PROPN
ejpam-4476	155	5	1)2	1)2	NUM
ejpam-4476	155	6	+	+	X
ejpam-4476	155	7	·	·	PUNCT
ejpam-4476	155	8	·	·	PUNCT
ejpam-4476	155	9	·	·	PUNCT
ejpam-4476	155	10	+	+	CCONJ
ejpam-4476	155	11	(	(	PUNCT
ejpam-4476	155	12	n−	n−	NOUN
ejpam-4476	155	13	1)2	1)2	NUM
ejpam-4476	155	14	]	]	PUNCT
ejpam-4476	155	15	+	+	CCONJ
ejpam-4476	156	1	[	[	X
ejpam-4476	156	2	1	1	NUM
ejpam-4476	156	3	+	+	NUM
ejpam-4476	156	4	2	2	NUM
ejpam-4476	156	5	+	+	CCONJ
ejpam-4476	156	6	·	·	PUNCT
ejpam-4476	156	7	·	·	PUNCT
ejpam-4476	156	8	·	·	PUNCT
ejpam-4476	156	9	+	+	PUNCT
ejpam-4476	156	10	(	(	PUNCT
ejpam-4476	156	11	k	k	NOUN
ejpam-4476	156	12	−	−	PROPN
ejpam-4476	156	13	1	1	NUM
ejpam-4476	156	14	)	)	PUNCT
ejpam-4476	156	15	+	+	CCONJ
ejpam-4476	156	16	·	·	PUNCT
ejpam-4476	156	17	·	·	PUNCT
ejpam-4476	156	18	·	·	PUNCT
ejpam-4476	156	19	+	+	CCONJ
ejpam-4476	156	20	(	(	PUNCT
ejpam-4476	156	21	n−	n−	NOUN
ejpam-4476	156	22	1	1	NUM
ejpam-4476	156	23	)	)	PUNCT
ejpam-4476	156	24	]	]	PUNCT
ejpam-4476	157	1	=	=	PUNCT
ejpam-4476	157	2	2[12	2[12	NUM
ejpam-4476	157	3	+	+	CCONJ
ejpam-4476	157	4	22	22	NUM
ejpam-4476	157	5	+	+	CCONJ
ejpam-4476	157	6	·	·	PUNCT
ejpam-4476	157	7	·	·	PUNCT
ejpam-4476	157	8	·	·	PUNCT
ejpam-4476	158	1	+	+	PUNCT
ejpam-4476	158	2	(	(	PUNCT
ejpam-4476	158	3	k	k	PROPN
ejpam-4476	158	4	−	−	PROPN
ejpam-4476	158	5	1)2	1)2	NUM
ejpam-4476	158	6	+	+	X
ejpam-4476	158	7	·	·	PUNCT
ejpam-4476	158	8	·	·	PUNCT
ejpam-4476	158	9	·	·	PUNCT
ejpam-4476	158	10	+	+	CCONJ
ejpam-4476	158	11	(	(	PUNCT
ejpam-4476	158	12	n−	n−	NOUN
ejpam-4476	158	13	1)2	1)2	NUM
ejpam-4476	158	14	+	+	CCONJ
ejpam-4476	158	15	n2]−	n2]−	ADV
ejpam-4476	158	16	2n2	2n2	NUM
ejpam-4476	158	17	+	+	NOUN
ejpam-4476	159	1	[	[	X
ejpam-4476	159	2	1	1	NUM
ejpam-4476	159	3	+	+	NUM
ejpam-4476	159	4	2	2	NUM
ejpam-4476	159	5	+	+	CCONJ
ejpam-4476	159	6	·	·	PUNCT
ejpam-4476	159	7	·	·	PUNCT
ejpam-4476	159	8	·	·	PUNCT
ejpam-4476	160	1	+	+	PUNCT
ejpam-4476	160	2	(	(	PUNCT
ejpam-4476	160	3	k	k	NOUN
ejpam-4476	160	4	−	−	PROPN
ejpam-4476	160	5	1	1	NUM
ejpam-4476	160	6	)	)	PUNCT
ejpam-4476	160	7	+	+	CCONJ
ejpam-4476	160	8	·	·	PUNCT
ejpam-4476	160	9	·	·	PUNCT
ejpam-4476	160	10	·	·	PUNCT
ejpam-4476	160	11	+	+	CCONJ
ejpam-4476	160	12	(	(	PUNCT
ejpam-4476	160	13	n−	n−	NOUN
ejpam-4476	160	14	1	1	NUM
ejpam-4476	160	15	)	)	PUNCT
ejpam-4476	160	16	+	+	CCONJ
ejpam-4476	160	17	n]−	n]−	ADV
ejpam-4476	160	18	n.	n.	NOUN
ejpam-4476	160	19	(	(	PUNCT
ejpam-4476	160	20	10	10	NUM
ejpam-4476	160	21	)	)	PUNCT
ejpam-4476	160	22	using	use	VERB
ejpam-4476	160	23	equations	equation	NOUN
ejpam-4476	160	24	(	(	PUNCT
ejpam-4476	160	25	9	9	NUM
ejpam-4476	160	26	)	)	PUNCT
ejpam-4476	160	27	and	and	CCONJ
ejpam-4476	160	28	(	(	PUNCT
ejpam-4476	160	29	10	10	NUM
ejpam-4476	160	30	)	)	PUNCT
ejpam-4476	160	31	instead	instead	ADV
ejpam-4476	160	32	of	of	ADP
ejpam-4476	160	33	equation	equation	NOUN
ejpam-4476	160	34	(	(	PUNCT
ejpam-4476	160	35	8)	8)	NUM
ejpam-4476	160	36	,	,	PUNCT
ejpam-4476	160	37	equation	equation	NOUN
ejpam-4476	160	38	(	(	PUNCT
ejpam-4476	160	39	7	7	X
ejpam-4476	160	40	)	)	PUNCT
ejpam-4476	160	41	can	can	AUX
ejpam-4476	160	42	be	be	AUX
ejpam-4476	160	43	written	write	VERB
ejpam-4476	160	44	as	as	ADP
ejpam-4476	160	45	12	12	NUM
ejpam-4476	160	46	+	+	CCONJ
ejpam-4476	160	47	22	22	NUM
ejpam-4476	160	48	+	+	NUM
ejpam-4476	160	49	32	32	NUM
ejpam-4476	160	50	+	+	CCONJ
ejpam-4476	160	51	·	·	PUNCT
ejpam-4476	160	52	·	·	PUNCT
ejpam-4476	160	53	·	·	PUNCT
ejpam-4476	160	54	+	+	NUM
ejpam-4476	160	55	k2	k2	X
ejpam-4476	160	56	+	+	X
ejpam-4476	160	57	·	·	PUNCT
ejpam-4476	160	58	·	·	PUNCT
ejpam-4476	160	59	·	·	PUNCT
ejpam-4476	160	60	+	+	NUM
ejpam-4476	160	61	n2	n2	NOUN
ejpam-4476	160	62	=	=	PROPN
ejpam-4476	160	63	n3	n3	NOUN
ejpam-4476	160	64	+	+	CCONJ
ejpam-4476	160	65	2n2	2n2	NUM
ejpam-4476	160	66	+	+	CCONJ
ejpam-4476	160	67	n−	n−	NOUN
ejpam-4476	160	68	2[12	2[12	NUM
ejpam-4476	160	69	+	+	CCONJ
ejpam-4476	160	70	22	22	NUM
ejpam-4476	160	71	+	+	CCONJ
ejpam-4476	160	72	·	·	PUNCT
ejpam-4476	160	73	·	·	PUNCT
ejpam-4476	160	74	·	·	PUNCT
ejpam-4476	161	1	+	+	PUNCT
ejpam-4476	161	2	(	(	PUNCT
ejpam-4476	161	3	k	k	PROPN
ejpam-4476	161	4	−	−	PROPN
ejpam-4476	161	5	1)2	1)2	NUM
ejpam-4476	161	6	+	+	X
ejpam-4476	161	7	·	·	PUNCT
ejpam-4476	161	8	·	·	PUNCT
ejpam-4476	161	9	·	·	PUNCT
ejpam-4476	161	10	+	+	CCONJ
ejpam-4476	161	11	(	(	PUNCT
ejpam-4476	161	12	n−	n−	NOUN
ejpam-4476	161	13	1)2	1)2	NUM
ejpam-4476	161	14	+	+	CCONJ
ejpam-4476	161	15	n2	n2	ADJ
ejpam-4476	161	16	]	]	X
ejpam-4476	161	17	−[1	−[1	NOUN
ejpam-4476	161	18	+	+	CCONJ
ejpam-4476	161	19	2	2	NUM
ejpam-4476	161	20	+	+	CCONJ
ejpam-4476	161	21	·	·	PUNCT
ejpam-4476	161	22	·	·	PUNCT
ejpam-4476	161	23	·	·	PUNCT
ejpam-4476	162	1	+	+	PUNCT
ejpam-4476	162	2	(	(	PUNCT
ejpam-4476	162	3	k	k	NOUN
ejpam-4476	162	4	−	−	PROPN
ejpam-4476	162	5	1	1	NUM
ejpam-4476	162	6	)	)	PUNCT
ejpam-4476	162	7	+	+	CCONJ
ejpam-4476	162	8	·	·	PUNCT
ejpam-4476	162	9	·	·	PUNCT
ejpam-4476	162	10	·	·	PUNCT
ejpam-4476	162	11	+	+	CCONJ
ejpam-4476	162	12	(	(	PUNCT
ejpam-4476	162	13	n−	n−	NOUN
ejpam-4476	162	14	1	1	NUM
ejpam-4476	162	15	)	)	PUNCT
ejpam-4476	162	16	+	+	NUM
ejpam-4476	162	17	n	n	CCONJ
ejpam-4476	162	18	]	]	PUNCT
ejpam-4476	162	19	.	.	PUNCT
ejpam-4476	163	1	(	(	PUNCT
ejpam-4476	163	2	11	11	NUM
ejpam-4476	163	3	)	)	PUNCT
ejpam-4476	163	4	with	with	ADP
ejpam-4476	163	5	slight	slight	ADJ
ejpam-4476	163	6	rearrangements	rearrangement	NOUN
ejpam-4476	163	7	,	,	PUNCT
ejpam-4476	163	8	we	we	PRON
ejpam-4476	163	9	obtain	obtain	VERB
ejpam-4476	163	10	3	3	NUM
ejpam-4476	163	11	n∑	n∑	NOUN
ejpam-4476	163	12	k=1	k=1	PROPN
ejpam-4476	163	13	k2	k2	PROPN
ejpam-4476	163	14	=	=	PUNCT
ejpam-4476	164	1	n(n+	n(n+	PROPN
ejpam-4476	165	1	1)2	1)2	NUM
ejpam-4476	165	2	−	−	PUNCT
ejpam-4476	165	3	n∑	n∑	INTJ
ejpam-4476	165	4	k=1	k=1	PROPN
ejpam-4476	166	1	k.	k.	PROPN
ejpam-4476	166	2	(	(	PUNCT
ejpam-4476	166	3	12	12	NUM
ejpam-4476	166	4	)	)	PUNCT
ejpam-4476	166	5	equation	equation	NOUN
ejpam-4476	166	6	(	(	PUNCT
ejpam-4476	166	7	6	6	NUM
ejpam-4476	166	8	)	)	PUNCT
ejpam-4476	166	9	can	can	AUX
ejpam-4476	166	10	also	also	ADV
ejpam-4476	166	11	be	be	AUX
ejpam-4476	166	12	derived	derive	VERB
ejpam-4476	166	13	from	from	ADP
ejpam-4476	166	14	equation	equation	NOUN
ejpam-4476	166	15	(	(	PUNCT
ejpam-4476	166	16	12	12	NUM
ejpam-4476	166	17	)	)	PUNCT
ejpam-4476	166	18	by	by	ADP
ejpam-4476	166	19	observing	observe	VERB
ejpam-4476	166	20	that	that	SCONJ
ejpam-4476	166	21	n(n	n(n	PROPN
ejpam-4476	166	22	+	+	CCONJ
ejpam-4476	166	23	1)2	1)2	NUM
ejpam-4476	166	24	=	=	SYM
ejpam-4476	166	25	2(n+	2(n+	NOUN
ejpam-4476	166	26	1	1	NUM
ejpam-4476	166	27	)	)	PUNCT
ejpam-4476	166	28	n∑	n∑	NOUN
ejpam-4476	167	1	k=1	k=1	PROPN
ejpam-4476	167	2	k.	k.	PROPN
ejpam-4476	167	3	3.2	3.2	NUM
ejpam-4476	167	4	.	.	PUNCT
ejpam-4476	168	1	visual	visual	ADJ
ejpam-4476	168	2	thinking	thinking	NOUN
ejpam-4476	168	3	it	it	PRON
ejpam-4476	168	4	is	be	AUX
ejpam-4476	168	5	important	important	ADJ
ejpam-4476	168	6	to	to	PART
ejpam-4476	168	7	illustrate	illustrate	VERB
ejpam-4476	168	8	how	how	SCONJ
ejpam-4476	168	9	the	the	DET
ejpam-4476	168	10	calculations	calculation	NOUN
ejpam-4476	168	11	are	be	AUX
ejpam-4476	168	12	performed	perform	VERB
ejpam-4476	168	13	in	in	ADP
ejpam-4476	168	14	the	the	DET
ejpam-4476	168	15	equations	equation	NOUN
ejpam-4476	168	16	and	and	CCONJ
ejpam-4476	168	17	to	to	PART
ejpam-4476	168	18	visually	visually	ADV
ejpam-4476	168	19	perceive	perceive	VERB
ejpam-4476	168	20	how	how	SCONJ
ejpam-4476	168	21	they	they	PRON
ejpam-4476	168	22	work	work	VERB
ejpam-4476	168	23	.	.	PUNCT
ejpam-4476	169	1	the	the	DET
ejpam-4476	169	2	relationship	relationship	NOUN
ejpam-4476	169	3	between	between	ADP
ejpam-4476	169	4	the	the	DET
ejpam-4476	169	5	sum	sum	NOUN
ejpam-4476	169	6	of	of	ADP
ejpam-4476	169	7	square	square	ADJ
ejpam-4476	169	8	numbers	number	NOUN
ejpam-4476	169	9	and	and	CCONJ
ejpam-4476	169	10	that	that	PRON
ejpam-4476	169	11	of	of	ADP
ejpam-4476	169	12	triangular	triangular	NOUN
ejpam-4476	169	13	numbers	number	NOUN
ejpam-4476	169	14	can	can	AUX
ejpam-4476	169	15	pictorially	pictorially	ADV
ejpam-4476	169	16	be	be	AUX
ejpam-4476	169	17	shown	show	VERB
ejpam-4476	169	18	by	by	ADP
ejpam-4476	169	19	expressing	express	VERB
ejpam-4476	169	20	the	the	DET
ejpam-4476	169	21	square	square	ADJ
ejpam-4476	169	22	numbers	number	NOUN
ejpam-4476	169	23	as	as	ADP
ejpam-4476	169	24	sums	sum	NOUN
ejpam-4476	169	25	of	of	ADP
ejpam-4476	169	26	consecutive	consecutive	ADJ
ejpam-4476	169	27	odd	odd	ADJ
ejpam-4476	169	28	numbers	number	NOUN
ejpam-4476	169	29	.	.	PUNCT
ejpam-4476	170	1	instructors	instructor	NOUN
ejpam-4476	170	2	should	should	AUX
ejpam-4476	170	3	ask	ask	VERB
ejpam-4476	170	4	students	student	NOUN
ejpam-4476	170	5	to	to	PART
ejpam-4476	170	6	illustrate	illustrate	VERB
ejpam-4476	170	7	the	the	DET
ejpam-4476	170	8	process	process	NOUN
ejpam-4476	170	9	of	of	ADP
ejpam-4476	170	10	deriving	derive	VERB
ejpam-4476	170	11	equations	equation	NOUN
ejpam-4476	170	12	(	(	PUNCT
ejpam-4476	170	13	6	6	NUM
ejpam-4476	170	14	)	)	PUNCT
ejpam-4476	170	15	and	and	CCONJ
ejpam-4476	170	16	(	(	PUNCT
ejpam-4476	170	17	7	7	NUM
ejpam-4476	170	18	)	)	PUNCT
ejpam-4476	170	19	from	from	ADP
ejpam-4476	170	20	table	table	NOUN
ejpam-4476	170	21	2	2	NUM
ejpam-4476	170	22	,	,	PUNCT
ejpam-4476	170	23	as	as	SCONJ
ejpam-4476	170	24	shown	show	VERB
ejpam-4476	170	25	in	in	ADP
ejpam-4476	170	26	figure	figure	NOUN
ejpam-4476	170	27	4	4	NUM
ejpam-4476	170	28	.	.	NOUN
ejpam-4476	170	29	12	12	NUM
ejpam-4476	170	30	22	22	NUM
ejpam-4476	170	31	32	32	NUM
ejpam-4476	170	32	1	1	NUM
ejpam-4476	170	33	1	1	NUM
ejpam-4476	170	34	3	3	NUM
ejpam-4476	170	35	1	1	NUM
ejpam-4476	170	36	3	3	NUM
ejpam-4476	170	37	5	5	NUM
ejpam-4476	170	38	32	32	NUM
ejpam-4476	170	39	32	32	NUM
ejpam-4476	170	40	22	22	NUM
ejpam-4476	170	41	22	22	NUM
ejpam-4476	170	42	12	12	NUM
ejpam-4476	170	43	12	12	NUM
ejpam-4476	170	44	2×	2×	NUM
ejpam-4476	170	45	3	3	NUM
ejpam-4476	170	46	+	+	CCONJ
ejpam-4476	170	47	1	1	NUM
ejpam-4476	170	48	1	1	NUM
ejpam-4476	170	49	+	+	NUM
ejpam-4476	170	50	2	2	NUM
ejpam-4476	170	51	+	+	SYM
ejpam-4476	170	52	3	3	NUM
ejpam-4476	170	53	1×	1×	NUM
ejpam-4476	170	54	3	3	NUM
ejpam-4476	170	55	3×	3×	NUM
ejpam-4476	170	56	2	2	NUM
ejpam-4476	170	57	5×	5×	NOUN
ejpam-4476	170	58	1	1	NUM
ejpam-4476	170	59	1	1	NUM
ejpam-4476	170	60	1	1	NUM
ejpam-4476	170	61	1	1	NUM
ejpam-4476	170	62	3	3	NUM
ejpam-4476	170	63	3	3	NUM
ejpam-4476	170	64	5	5	NUM
ejpam-4476	170	65	figure	figure	NOUN
ejpam-4476	170	66	4	4	NUM
ejpam-4476	170	67	:	:	PUNCT
ejpam-4476	170	68	pictorial	pictorial	ADJ
ejpam-4476	170	69	proof	proof	NOUN
ejpam-4476	170	70	that	that	SCONJ
ejpam-4476	170	71	3(12	3(12	NUM
ejpam-4476	170	72	+	+	CCONJ
ejpam-4476	170	73	22	22	NUM
ejpam-4476	170	74	+	+	NUM
ejpam-4476	170	75	32	32	NUM
ejpam-4476	170	76	)	)	PUNCT
ejpam-4476	170	77	=	=	SYM
ejpam-4476	170	78	(	(	PUNCT
ejpam-4476	170	79	2×	2×	NUM
ejpam-4476	170	80	3	3	NUM
ejpam-4476	170	81	+	+	SYM
ejpam-4476	170	82	1)(1	1)(1	NUM
ejpam-4476	170	83	+	+	CCONJ
ejpam-4476	170	84	2	2	NUM
ejpam-4476	170	85	+	+	NUM
ejpam-4476	170	86	3	3	NUM
ejpam-4476	170	87	)	)	PUNCT
ejpam-4476	170	88	based	base	VERB
ejpam-4476	170	89	on	on	ADP
ejpam-4476	170	90	sums	sum	NOUN
ejpam-4476	170	91	of	of	ADP
ejpam-4476	170	92	odd	odd	ADJ
ejpam-4476	170	93	numbers	number	NOUN
ejpam-4476	170	94	.	.	PUNCT
ejpam-4476	171	1	y.	y.	PROPN
ejpam-4476	171	2	kobayashi	kobayashi	PROPN
ejpam-4476	171	3	/	/	SYM
ejpam-4476	171	4	eur	eur	PROPN
ejpam-4476	171	5	.	.	PUNCT
ejpam-4476	172	1	j.	j.	PROPN
ejpam-4476	172	2	pure	pure	PROPN
ejpam-4476	172	3	appl	appl	PROPN
ejpam-4476	172	4	.	.	PROPN
ejpam-4476	172	5	math	math	PROPN
ejpam-4476	172	6	,	,	PUNCT
ejpam-4476	172	7	15	15	NUM
ejpam-4476	172	8	(	(	PUNCT
ejpam-4476	172	9	3	3	NUM
ejpam-4476	172	10	)	)	PUNCT
ejpam-4476	172	11	(	(	PUNCT
ejpam-4476	172	12	2022	2022	NUM
ejpam-4476	172	13	)	)	PUNCT
ejpam-4476	172	14	,	,	PUNCT
ejpam-4476	172	15	864	864	NUM
ejpam-4476	172	16	-	-	SYM
ejpam-4476	172	17	877	877	NUM
ejpam-4476	172	18	873	873	NUM
ejpam-4476	172	19	the	the	DET
ejpam-4476	172	20	meaning	meaning	NOUN
ejpam-4476	172	21	of	of	ADP
ejpam-4476	172	22	the	the	DET
ejpam-4476	172	23	sum	sum	NOUN
ejpam-4476	172	24	of	of	ADP
ejpam-4476	172	25	square	square	ADJ
ejpam-4476	172	26	numbers	number	NOUN
ejpam-4476	172	27	can	can	AUX
ejpam-4476	172	28	also	also	ADV
ejpam-4476	172	29	be	be	AUX
ejpam-4476	172	30	illustrated	illustrate	VERB
ejpam-4476	172	31	with	with	ADP
ejpam-4476	172	32	reference	reference	NOUN
ejpam-4476	172	33	to	to	ADP
ejpam-4476	172	34	table	table	NOUN
ejpam-4476	172	35	3	3	NUM
ejpam-4476	172	36	:	:	SYM
ejpam-4476	172	37	12	12	NUM
ejpam-4476	172	38	+	+	NUM
ejpam-4476	172	39	22	22	NUM
ejpam-4476	173	1	+	+	NUM
ejpam-4476	173	2	32	32	NUM
ejpam-4476	173	3	+	+	NUM
ejpam-4476	173	4	42	42	NUM
ejpam-4476	173	5	=	=	NOUN
ejpam-4476	174	1	[	[	X
ejpam-4476	174	2	1	1	NUM
ejpam-4476	174	3	+	+	CCONJ
ejpam-4476	174	4	(	(	PUNCT
ejpam-4476	174	5	1	1	NUM
ejpam-4476	174	6	+	+	NUM
ejpam-4476	174	7	2	2	NUM
ejpam-4476	174	8	)	)	PUNCT
ejpam-4476	174	9	+	+	CCONJ
ejpam-4476	174	10	(	(	PUNCT
ejpam-4476	174	11	1	1	NUM
ejpam-4476	174	12	+	+	NUM
ejpam-4476	174	13	2	2	NUM
ejpam-4476	174	14	+	+	NUM
ejpam-4476	174	15	3	3	NUM
ejpam-4476	174	16	)	)	PUNCT
ejpam-4476	174	17	]	]	PUNCT
ejpam-4476	175	1	+	+	PUNCT
ejpam-4476	175	2	[	[	X
ejpam-4476	175	3	1	1	NUM
ejpam-4476	175	4	+	+	CCONJ
ejpam-4476	175	5	(	(	PUNCT
ejpam-4476	175	6	1	1	NUM
ejpam-4476	175	7	+	+	NUM
ejpam-4476	175	8	2	2	NUM
ejpam-4476	175	9	)	)	PUNCT
ejpam-4476	175	10	+	+	CCONJ
ejpam-4476	175	11	(	(	PUNCT
ejpam-4476	175	12	1	1	NUM
ejpam-4476	175	13	+	+	NUM
ejpam-4476	175	14	2	2	NUM
ejpam-4476	175	15	+	+	NUM
ejpam-4476	175	16	3	3	NUM
ejpam-4476	175	17	)	)	PUNCT
ejpam-4476	175	18	+	+	CCONJ
ejpam-4476	175	19	(	(	PUNCT
ejpam-4476	175	20	1	1	NUM
ejpam-4476	175	21	+	+	NUM
ejpam-4476	175	22	2	2	NUM
ejpam-4476	175	23	+	+	NUM
ejpam-4476	175	24	3	3	NUM
ejpam-4476	175	25	+	+	NUM
ejpam-4476	175	26	4	4	NUM
ejpam-4476	175	27	)	)	PUNCT
ejpam-4476	175	28	]	]	PUNCT
ejpam-4476	175	29	.	.	PUNCT
ejpam-4476	176	1	(	(	PUNCT
ejpam-4476	176	2	13	13	NUM
ejpam-4476	176	3	)	)	PUNCT
ejpam-4476	176	4	the	the	DET
ejpam-4476	176	5	right	right	ADJ
ejpam-4476	176	6	-	-	PUNCT
ejpam-4476	176	7	hand	hand	NOUN
ejpam-4476	176	8	side	side	NOUN
ejpam-4476	176	9	is	be	AUX
ejpam-4476	176	10	the	the	DET
ejpam-4476	176	11	sum	sum	NOUN
ejpam-4476	176	12	of	of	ADP
ejpam-4476	176	13	the	the	DET
ejpam-4476	176	14	sums	sum	NOUN
ejpam-4476	176	15	of	of	ADP
ejpam-4476	176	16	triangular	triangular	NOUN
ejpam-4476	176	17	numbers	number	NOUN
ejpam-4476	176	18	.	.	PUNCT
ejpam-4476	177	1	this	this	DET
ejpam-4476	177	2	relationship	relationship	NOUN
ejpam-4476	177	3	can	can	AUX
ejpam-4476	177	4	also	also	ADV
ejpam-4476	177	5	be	be	AUX
ejpam-4476	177	6	derived	derive	VERB
ejpam-4476	177	7	from	from	ADP
ejpam-4476	177	8	other	other	ADJ
ejpam-4476	177	9	expressions	expression	NOUN
ejpam-4476	177	10	of	of	ADP
ejpam-4476	177	11	square	square	ADJ
ejpam-4476	177	12	numbers	number	NOUN
ejpam-4476	177	13	,	,	PUNCT
ejpam-4476	177	14	as	as	SCONJ
ejpam-4476	177	15	shown	show	VERB
ejpam-4476	177	16	in	in	ADP
ejpam-4476	177	17	table	table	NOUN
ejpam-4476	177	18	4	4	NUM
ejpam-4476	177	19	.	.	NOUN
ejpam-4476	177	20	12	12	NUM
ejpam-4476	177	21	+	+	NUM
ejpam-4476	177	22	22	22	NUM
ejpam-4476	177	23	+	+	NUM
ejpam-4476	177	24	32	32	NUM
ejpam-4476	177	25	+	+	NUM
ejpam-4476	177	26	42	42	NUM
ejpam-4476	177	27	=	=	SYM
ejpam-4476	177	28	2[0	2[0	NUM
ejpam-4476	177	29	+	+	NOUN
ejpam-4476	177	30	1	1	NUM
ejpam-4476	177	31	+	+	CCONJ
ejpam-4476	177	32	(	(	PUNCT
ejpam-4476	177	33	1	1	NUM
ejpam-4476	177	34	+	+	NUM
ejpam-4476	177	35	2	2	NUM
ejpam-4476	177	36	)	)	PUNCT
ejpam-4476	177	37	+	+	CCONJ
ejpam-4476	177	38	(	(	PUNCT
ejpam-4476	177	39	1	1	NUM
ejpam-4476	177	40	+	+	NUM
ejpam-4476	177	41	2	2	NUM
ejpam-4476	177	42	+	+	NUM
ejpam-4476	177	43	3	3	NUM
ejpam-4476	177	44	)	)	PUNCT
ejpam-4476	177	45	]	]	PUNCT
ejpam-4476	178	1	+	+	CCONJ
ejpam-4476	178	2	(	(	PUNCT
ejpam-4476	178	3	1	1	NUM
ejpam-4476	178	4	+	+	NUM
ejpam-4476	178	5	2	2	NUM
ejpam-4476	178	6	+	+	NUM
ejpam-4476	178	7	3	3	NUM
ejpam-4476	178	8	+	+	NUM
ejpam-4476	178	9	4	4	NUM
ejpam-4476	178	10	)	)	PUNCT
ejpam-4476	178	11	,	,	PUNCT
ejpam-4476	178	12	(	(	PUNCT
ejpam-4476	178	13	14	14	NUM
ejpam-4476	178	14	)	)	PUNCT
ejpam-4476	178	15	which	which	PRON
ejpam-4476	178	16	coincides	coincide	VERB
ejpam-4476	178	17	with	with	ADP
ejpam-4476	178	18	equation	equation	NOUN
ejpam-4476	178	19	(	(	PUNCT
ejpam-4476	178	20	13	13	NUM
ejpam-4476	178	21	)	)	PUNCT
ejpam-4476	178	22	.	.	PUNCT
ejpam-4476	179	1	the	the	DET
ejpam-4476	179	2	right	right	ADJ
ejpam-4476	179	3	-	-	PUNCT
ejpam-4476	179	4	hand	hand	NOUN
ejpam-4476	179	5	sides	side	NOUN
ejpam-4476	179	6	of	of	ADP
ejpam-4476	179	7	equations	equation	NOUN
ejpam-4476	179	8	(	(	PUNCT
ejpam-4476	179	9	13	13	NUM
ejpam-4476	179	10	)	)	PUNCT
ejpam-4476	179	11	and	and	CCONJ
ejpam-4476	179	12	(	(	PUNCT
ejpam-4476	179	13	14	14	NUM
ejpam-4476	179	14	)	)	PUNCT
ejpam-4476	179	15	can	can	AUX
ejpam-4476	179	16	be	be	AUX
ejpam-4476	179	17	pictorially	pictorially	ADV
ejpam-4476	179	18	represented	represent	VERB
ejpam-4476	179	19	,	,	PUNCT
ejpam-4476	179	20	as	as	SCONJ
ejpam-4476	179	21	shown	show	VERB
ejpam-4476	179	22	in	in	ADP
ejpam-4476	179	23	(	(	PUNCT
ejpam-4476	179	24	a	a	NOUN
ejpam-4476	179	25	)	)	PUNCT
ejpam-4476	179	26	and	and	CCONJ
ejpam-4476	179	27	(	(	PUNCT
ejpam-4476	179	28	b	b	NOUN
ejpam-4476	179	29	)	)	PUNCT
ejpam-4476	179	30	in	in	ADP
ejpam-4476	179	31	figure	figure	NOUN
ejpam-4476	179	32	5	5	NUM
ejpam-4476	179	33	,	,	PUNCT
ejpam-4476	179	34	respectively	respectively	ADV
ejpam-4476	179	35	.	.	PUNCT
ejpam-4476	180	1	we	we	PRON
ejpam-4476	180	2	can	can	AUX
ejpam-4476	180	3	draw	draw	VERB
ejpam-4476	180	4	the	the	DET
ejpam-4476	180	5	usage	usage	NOUN
ejpam-4476	180	6	of	of	ADP
ejpam-4476	180	7	(	(	PUNCT
ejpam-4476	180	8	b	b	NOUN
ejpam-4476	180	9	)	)	PUNCT
ejpam-4476	180	10	as	as	SCONJ
ejpam-4476	180	11	shown	show	VERB
ejpam-4476	180	12	in	in	ADP
ejpam-4476	180	13	(	(	PUNCT
ejpam-4476	180	14	c	c	NOUN
ejpam-4476	180	15	)	)	PUNCT
ejpam-4476	180	16	,	,	PUNCT
ejpam-4476	180	17	and	and	CCONJ
ejpam-4476	180	18	thus	thus	ADV
ejpam-4476	180	19	2[0	2[0	NUM
ejpam-4476	180	20	+	+	SYM
ejpam-4476	180	21	1	1	NUM
ejpam-4476	180	22	+	+	CCONJ
ejpam-4476	180	23	(	(	PUNCT
ejpam-4476	180	24	1	1	NUM
ejpam-4476	180	25	+	+	NUM
ejpam-4476	180	26	2	2	NUM
ejpam-4476	180	27	)	)	PUNCT
ejpam-4476	180	28	]	]	PUNCT
ejpam-4476	181	1	=	=	SYM
ejpam-4476	181	2	2	2	NUM
ejpam-4476	181	3	3	3	NUM
ejpam-4476	181	4	(	(	PUNCT
ejpam-4476	181	5	1	1	NUM
ejpam-4476	181	6	+	+	NUM
ejpam-4476	181	7	2)(1	2)(1	NUM
ejpam-4476	181	8	+	+	CCONJ
ejpam-4476	181	9	2	2	NUM
ejpam-4476	181	10	+	+	NUM
ejpam-4476	181	11	1	1	NUM
ejpam-4476	181	12	)	)	PUNCT
ejpam-4476	181	13	.	.	PUNCT
ejpam-4476	182	1	(	(	PUNCT
ejpam-4476	182	2	15	15	X
ejpam-4476	182	3	)	)	PUNCT
ejpam-4476	182	4	generalising	generalising	NOUN
ejpam-4476	182	5	,	,	PUNCT
ejpam-4476	182	6	we	we	PRON
ejpam-4476	182	7	can	can	AUX
ejpam-4476	182	8	obtain	obtain	VERB
ejpam-4476	182	9	2[0	2[0	NUM
ejpam-4476	182	10	+	+	NUM
ejpam-4476	182	11	1	1	NUM
ejpam-4476	182	12	+	+	CCONJ
ejpam-4476	182	13	(	(	PUNCT
ejpam-4476	182	14	1	1	NUM
ejpam-4476	182	15	+	+	NUM
ejpam-4476	182	16	2	2	NUM
ejpam-4476	182	17	)	)	PUNCT
ejpam-4476	182	18	+	+	CCONJ
ejpam-4476	182	19	(	(	PUNCT
ejpam-4476	182	20	1	1	NUM
ejpam-4476	182	21	+	+	NUM
ejpam-4476	182	22	2	2	NUM
ejpam-4476	182	23	+	+	NUM
ejpam-4476	182	24	3	3	NUM
ejpam-4476	182	25	)	)	PUNCT
ejpam-4476	182	26	+	+	NUM
ejpam-4476	182	27	·	·	PUNCT
ejpam-4476	182	28	·	·	PUNCT
ejpam-4476	182	29	·	·	PUNCT
ejpam-4476	183	1	+	+	CCONJ
ejpam-4476	183	2	(	(	PUNCT
ejpam-4476	183	3	1	1	NUM
ejpam-4476	183	4	+	+	NUM
ejpam-4476	183	5	2	2	NUM
ejpam-4476	183	6	+	+	CCONJ
ejpam-4476	183	7	·	·	PUNCT
ejpam-4476	183	8	·	·	PUNCT
ejpam-4476	183	9	·	·	PUNCT
ejpam-4476	183	10	+	+	NUM
ejpam-4476	183	11	n	n	CCONJ
ejpam-4476	183	12	)	)	PUNCT
ejpam-4476	183	13	]	]	PUNCT
ejpam-4476	183	14	=	=	SYM
ejpam-4476	184	1	2	2	NUM
ejpam-4476	184	2	3	3	NUM
ejpam-4476	185	1	[	[	X
ejpam-4476	185	2	1	1	NUM
ejpam-4476	185	3	+	+	NUM
ejpam-4476	185	4	2	2	NUM
ejpam-4476	185	5	+	+	CCONJ
ejpam-4476	185	6	·	·	PUNCT
ejpam-4476	185	7	·	·	PUNCT
ejpam-4476	185	8	·	·	PUNCT
ejpam-4476	185	9	+	+	CCONJ
ejpam-4476	185	10	(	(	PUNCT
ejpam-4476	185	11	n−	n−	NOUN
ejpam-4476	185	12	1)](1	1)](1	NUM
ejpam-4476	185	13	+	+	CCONJ
ejpam-4476	185	14	n+	n+	NUM
ejpam-4476	185	15	1	1	X
ejpam-4476	185	16	)	)	PUNCT
ejpam-4476	185	17	=	=	SYM
ejpam-4476	185	18	2	2	NUM
ejpam-4476	185	19	3	3	NUM
ejpam-4476	185	20	·	·	PUNCT
ejpam-4476	185	21	(	(	PUNCT
ejpam-4476	185	22	n−	n−	NOUN
ejpam-4476	185	23	1)n	1)n	X
ejpam-4476	185	24	2	2	NUM
ejpam-4476	185	25	·	·	PUNCT
ejpam-4476	185	26	(	(	PUNCT
ejpam-4476	185	27	n+	n+	NOUN
ejpam-4476	185	28	1	1	NUM
ejpam-4476	185	29	)	)	PUNCT
ejpam-4476	185	30	,	,	PUNCT
ejpam-4476	185	31	(	(	PUNCT
ejpam-4476	185	32	16	16	NUM
ejpam-4476	185	33	)	)	PUNCT
ejpam-4476	185	34	and	and	CCONJ
ejpam-4476	185	35	equation	equation	NOUN
ejpam-4476	185	36	(	(	PUNCT
ejpam-4476	185	37	14	14	NUM
ejpam-4476	185	38	)	)	PUNCT
ejpam-4476	185	39	can	can	AUX
ejpam-4476	185	40	be	be	AUX
ejpam-4476	185	41	extended	extend	VERB
ejpam-4476	185	42	to	to	ADP
ejpam-4476	185	43	12	12	NUM
ejpam-4476	185	44	+	+	NUM
ejpam-4476	185	45	22	22	NUM
ejpam-4476	185	46	+	+	CCONJ
ejpam-4476	185	47	·	·	PUNCT
ejpam-4476	185	48	·	·	PUNCT
ejpam-4476	185	49	·	·	PUNCT
ejpam-4476	186	1	+	+	NUM
ejpam-4476	186	2	n2	n2	ADJ
ejpam-4476	186	3	=	=	SYM
ejpam-4476	186	4	2	2	NUM
ejpam-4476	186	5	3	3	NUM
ejpam-4476	186	6	·	·	PUNCT
ejpam-4476	186	7	(	(	PUNCT
ejpam-4476	186	8	n−	n−	NOUN
ejpam-4476	186	9	1)n	1)n	X
ejpam-4476	186	10	2	2	NUM
ejpam-4476	186	11	·	·	PUNCT
ejpam-4476	186	12	(	(	PUNCT
ejpam-4476	186	13	n+	n+	NOUN
ejpam-4476	186	14	1	1	X
ejpam-4476	186	15	)	)	PUNCT
ejpam-4476	186	16	+	+	NUM
ejpam-4476	186	17	n(n+	n(n+	NUM
ejpam-4476	186	18	1	1	NUM
ejpam-4476	186	19	)	)	SYM
ejpam-4476	186	20	2	2	NUM
ejpam-4476	186	21	=	=	SYM
ejpam-4476	186	22	n(n+	n(n+	X
ejpam-4476	186	23	1)(2n+	1)(2n+	NUM
ejpam-4476	186	24	1	1	NUM
ejpam-4476	186	25	)	)	PUNCT
ejpam-4476	186	26	6	6	NUM
ejpam-4476	186	27	.	.	PUNCT
ejpam-4476	187	1	(	(	PUNCT
ejpam-4476	187	2	17	17	NUM
ejpam-4476	187	3	)	)	PUNCT
ejpam-4476	187	4	table	table	NOUN
ejpam-4476	187	5	3	3	NUM
ejpam-4476	187	6	.	.	PUNCT
ejpam-4476	188	1	square	square	ADJ
ejpam-4476	188	2	numbers	number	NOUN
ejpam-4476	188	3	expressed	express	VERB
ejpam-4476	188	4	as	as	ADP
ejpam-4476	188	5	sums	sum	NOUN
ejpam-4476	188	6	of	of	ADP
ejpam-4476	188	7	two	two	NUM
ejpam-4476	188	8	consecutive	consecutive	ADJ
ejpam-4476	188	9	triangular	triangular	NOUN
ejpam-4476	188	10	numbers	number	NOUN
ejpam-4476	188	11	.	.	PUNCT
ejpam-4476	189	1	12	12	NUM
ejpam-4476	189	2	=	=	SYM
ejpam-4476	189	3	1	1	NUM
ejpam-4476	189	4	.	.	NOUN
ejpam-4476	189	5	22	22	NUM
ejpam-4476	189	6	=	=	SYM
ejpam-4476	189	7	1	1	NUM
ejpam-4476	189	8	+	+	CCONJ
ejpam-4476	189	9	(	(	PUNCT
ejpam-4476	189	10	1	1	NUM
ejpam-4476	189	11	+	+	NUM
ejpam-4476	189	12	2	2	NUM
ejpam-4476	189	13	)	)	PUNCT
ejpam-4476	189	14	.	.	PUNCT
ejpam-4476	190	1	32	32	NUM
ejpam-4476	190	2	=	=	SYM
ejpam-4476	190	3	1	1	NUM
ejpam-4476	190	4	+	+	CCONJ
ejpam-4476	190	5	(	(	PUNCT
ejpam-4476	190	6	1	1	NUM
ejpam-4476	190	7	+	+	NUM
ejpam-4476	190	8	2	2	NUM
ejpam-4476	190	9	)	)	PUNCT
ejpam-4476	190	10	+	+	CCONJ
ejpam-4476	190	11	(	(	PUNCT
ejpam-4476	190	12	2	2	NUM
ejpam-4476	190	13	+	+	NUM
ejpam-4476	190	14	3	3	NUM
ejpam-4476	190	15	)	)	PUNCT
ejpam-4476	190	16	=	=	NOUN
ejpam-4476	190	17	(	(	PUNCT
ejpam-4476	190	18	1	1	NUM
ejpam-4476	190	19	+	+	NUM
ejpam-4476	190	20	2	2	NUM
ejpam-4476	190	21	)	)	PUNCT
ejpam-4476	190	22	+	+	CCONJ
ejpam-4476	190	23	(	(	PUNCT
ejpam-4476	190	24	1	1	NUM
ejpam-4476	190	25	+	+	NUM
ejpam-4476	190	26	2	2	NUM
ejpam-4476	190	27	+	+	NUM
ejpam-4476	190	28	3	3	NUM
ejpam-4476	190	29	)	)	PUNCT
ejpam-4476	190	30	.	.	PUNCT
ejpam-4476	191	1	42	42	NUM
ejpam-4476	191	2	=	=	SYM
ejpam-4476	191	3	1	1	NUM
ejpam-4476	191	4	+	+	CCONJ
ejpam-4476	191	5	(	(	PUNCT
ejpam-4476	191	6	1	1	NUM
ejpam-4476	191	7	+	+	NUM
ejpam-4476	191	8	2	2	NUM
ejpam-4476	191	9	)	)	PUNCT
ejpam-4476	191	10	+	+	CCONJ
ejpam-4476	191	11	(	(	PUNCT
ejpam-4476	191	12	2	2	NUM
ejpam-4476	191	13	+	+	NUM
ejpam-4476	191	14	3	3	NUM
ejpam-4476	191	15	)	)	PUNCT
ejpam-4476	191	16	+	+	CCONJ
ejpam-4476	191	17	(	(	PUNCT
ejpam-4476	191	18	3	3	NUM
ejpam-4476	191	19	+	+	NUM
ejpam-4476	191	20	4	4	NUM
ejpam-4476	191	21	)	)	PUNCT
ejpam-4476	191	22	=	=	NOUN
ejpam-4476	191	23	(	(	PUNCT
ejpam-4476	191	24	1	1	NUM
ejpam-4476	191	25	+	+	NUM
ejpam-4476	191	26	2	2	NUM
ejpam-4476	191	27	+	+	NUM
ejpam-4476	191	28	3	3	NUM
ejpam-4476	191	29	)	)	PUNCT
ejpam-4476	191	30	+	+	CCONJ
ejpam-4476	191	31	(	(	PUNCT
ejpam-4476	191	32	1	1	NUM
ejpam-4476	191	33	+	+	NUM
ejpam-4476	191	34	2	2	NUM
ejpam-4476	191	35	+	+	NUM
ejpam-4476	191	36	3	3	NUM
ejpam-4476	191	37	+	+	NUM
ejpam-4476	191	38	4	4	NUM
ejpam-4476	191	39	)	)	PUNCT
ejpam-4476	191	40	.	.	PUNCT
ejpam-4476	192	1	this	this	DET
ejpam-4476	192	2	table	table	NOUN
ejpam-4476	192	3	is	be	AUX
ejpam-4476	192	4	a	a	DET
ejpam-4476	192	5	revised	revise	VERB
ejpam-4476	192	6	version	version	NOUN
ejpam-4476	192	7	of	of	ADP
ejpam-4476	192	8	table	table	NOUN
ejpam-4476	192	9	2	2	NUM
ejpam-4476	192	10	.	.	PUNCT
ejpam-4476	192	11	table	table	NOUN
ejpam-4476	192	12	4	4	NUM
ejpam-4476	192	13	.	.	PUNCT
ejpam-4476	193	1	square	square	ADJ
ejpam-4476	193	2	numbers	number	NOUN
ejpam-4476	193	3	expressed	express	VERB
ejpam-4476	193	4	as	as	ADP
ejpam-4476	193	5	triangular	triangular	NOUN
ejpam-4476	193	6	numbers	number	NOUN
ejpam-4476	193	7	.	.	PUNCT
ejpam-4476	194	1	12	12	NUM
ejpam-4476	194	2	=	=	SYM
ejpam-4476	194	3	2×	2×	NUM
ejpam-4476	194	4	0	0	PUNCT
ejpam-4476	195	1	+	+	CCONJ
ejpam-4476	196	1	1	1	NUM
ejpam-4476	196	2	.	.	X
ejpam-4476	196	3	22	22	NUM
ejpam-4476	196	4	=	=	SYM
ejpam-4476	196	5	2×	2×	NUM
ejpam-4476	196	6	1	1	NUM
ejpam-4476	196	7	+	+	CCONJ
ejpam-4476	196	8	2	2	NUM
ejpam-4476	196	9	.	.	X
ejpam-4476	196	10	32	32	NUM
ejpam-4476	196	11	=	=	SYM
ejpam-4476	196	12	2×	2×	NOUN
ejpam-4476	196	13	(	(	PUNCT
ejpam-4476	196	14	1	1	NUM
ejpam-4476	196	15	+	+	NUM
ejpam-4476	196	16	2	2	NUM
ejpam-4476	196	17	)	)	PUNCT
ejpam-4476	196	18	+	+	NOUN
ejpam-4476	197	1	3	3	X
ejpam-4476	197	2	.	.	X
ejpam-4476	197	3	42	42	NUM
ejpam-4476	197	4	=	=	SYM
ejpam-4476	197	5	2×	2×	NOUN
ejpam-4476	197	6	(	(	PUNCT
ejpam-4476	197	7	1	1	NUM
ejpam-4476	197	8	+	+	NUM
ejpam-4476	197	9	2	2	NUM
ejpam-4476	197	10	+	+	NUM
ejpam-4476	197	11	3	3	NUM
ejpam-4476	197	12	)	)	PUNCT
ejpam-4476	197	13	+	+	NOUN
ejpam-4476	197	14	4	4	X
ejpam-4476	197	15	.	.	PUNCT
ejpam-4476	197	16	y.	y.	PROPN
ejpam-4476	197	17	kobayashi	kobayashi	PROPN
ejpam-4476	197	18	/	/	SYM
ejpam-4476	197	19	eur	eur	PROPN
ejpam-4476	197	20	.	.	PUNCT
ejpam-4476	198	1	j.	j.	PROPN
ejpam-4476	198	2	pure	pure	PROPN
ejpam-4476	198	3	appl	appl	PROPN
ejpam-4476	198	4	.	.	PROPN
ejpam-4476	198	5	math	math	PROPN
ejpam-4476	198	6	,	,	PUNCT
ejpam-4476	198	7	15	15	NUM
ejpam-4476	198	8	(	(	PUNCT
ejpam-4476	198	9	3	3	NUM
ejpam-4476	198	10	)	)	PUNCT
ejpam-4476	198	11	(	(	PUNCT
ejpam-4476	198	12	2022	2022	NUM
ejpam-4476	198	13	)	)	PUNCT
ejpam-4476	198	14	,	,	PUNCT
ejpam-4476	198	15	864	864	NUM
ejpam-4476	198	16	-	-	SYM
ejpam-4476	198	17	877	877	NUM
ejpam-4476	198	18	874	874	NUM
ejpam-4476	198	19	1	1	NUM
ejpam-4476	198	20	2	2	NUM
ejpam-4476	198	21	+	+	CCONJ
ejpam-4476	198	22	1	1	NUM
ejpam-4476	198	23	1	1	NUM
ejpam-4476	198	24	3	3	NUM
ejpam-4476	198	25	+	+	CCONJ
ejpam-4476	198	26	2	2	NUM
ejpam-4476	198	27	+	+	CCONJ
ejpam-4476	198	28	1	1	NUM
ejpam-4476	198	29	1	1	NUM
ejpam-4476	198	30	+	+	NUM
ejpam-4476	198	31	2	2	NUM
ejpam-4476	198	32	(	(	PUNCT
ejpam-4476	198	33	a	a	NOUN
ejpam-4476	198	34	)	)	PUNCT
ejpam-4476	198	35	1	1	NUM
ejpam-4476	198	36	2	2	NUM
ejpam-4476	198	37	1	1	NUM
ejpam-4476	198	38	1	1	NUM
ejpam-4476	198	39	3	3	NUM
ejpam-4476	198	40	2	2	NUM
ejpam-4476	198	41	+	+	CCONJ
ejpam-4476	198	42	1	1	NUM
ejpam-4476	198	43	2	2	NUM
ejpam-4476	198	44	+	+	NUM
ejpam-4476	198	45	1	1	NUM
ejpam-4476	198	46	(	(	PUNCT
ejpam-4476	198	47	b	b	NOUN
ejpam-4476	198	48	)	)	PUNCT
ejpam-4476	199	1	+	+	NUM
ejpam-4476	199	2	2×	2×	NUM
ejpam-4476	199	3	1	1	NUM
ejpam-4476	199	4	2×	2×	NOUN
ejpam-4476	199	5	(	(	PUNCT
ejpam-4476	199	6	2	2	NUM
ejpam-4476	199	7	+	+	CCONJ
ejpam-4476	199	8	1	1	NUM
ejpam-4476	199	9	)	)	SYM
ejpam-4476	199	10	1	1	NUM
ejpam-4476	199	11	2	2	NUM
ejpam-4476	199	12	3	3	NUM
ejpam-4476	199	13	2	2	NUM
ejpam-4476	199	14	3	3	NUM
ejpam-4476	199	15	(	(	PUNCT
ejpam-4476	199	16	1	1	NUM
ejpam-4476	199	17	+	+	NUM
ejpam-4476	199	18	2)(1	2)(1	NUM
ejpam-4476	199	19	+	+	CCONJ
ejpam-4476	199	20	2	2	NUM
ejpam-4476	199	21	+	+	CCONJ
ejpam-4476	199	22	1	1	NUM
ejpam-4476	199	23	)	)	SYM
ejpam-4476	199	24	1	1	NUM
ejpam-4476	199	25	2	2	NUM
ejpam-4476	199	26	1	1	NUM
ejpam-4476	199	27	2	2	NUM
ejpam-4476	199	28	1	1	NUM
ejpam-4476	199	29	(	(	PUNCT
ejpam-4476	199	30	c	c	NOUN
ejpam-4476	199	31	)	)	PUNCT
ejpam-4476	199	32	figure	figure	NOUN
ejpam-4476	199	33	5	5	NUM
ejpam-4476	199	34	:	:	PUNCT
ejpam-4476	199	35	square	square	ADJ
ejpam-4476	199	36	numbers	number	NOUN
ejpam-4476	199	37	expressed	express	VERB
ejpam-4476	199	38	as	as	ADP
ejpam-4476	199	39	triangular	triangular	NOUN
ejpam-4476	199	40	numbers	number	NOUN
ejpam-4476	199	41	.	.	PUNCT
ejpam-4476	200	1	for	for	ADP
ejpam-4476	200	2	simplicity	simplicity	NOUN
ejpam-4476	200	3	,	,	PUNCT
ejpam-4476	200	4	the	the	DET
ejpam-4476	200	5	cases	case	NOUN
ejpam-4476	200	6	of	of	ADP
ejpam-4476	200	7	12	12	NUM
ejpam-4476	200	8	,	,	PUNCT
ejpam-4476	200	9	22	22	NUM
ejpam-4476	200	10	,	,	PUNCT
ejpam-4476	200	11	and	and	CCONJ
ejpam-4476	200	12	32	32	NUM
ejpam-4476	200	13	are	be	AUX
ejpam-4476	200	14	presented	present	VERB
ejpam-4476	200	15	.	.	PUNCT
ejpam-4476	201	1	diagrams	diagram	NOUN
ejpam-4476	201	2	(	(	PUNCT
ejpam-4476	201	3	a	a	X
ejpam-4476	201	4	)	)	PUNCT
ejpam-4476	201	5	and	and	CCONJ
ejpam-4476	201	6	(	(	PUNCT
ejpam-4476	201	7	b	b	X
ejpam-4476	201	8	)	)	PUNCT
ejpam-4476	201	9	represent	represent	VERB
ejpam-4476	201	10	the	the	DET
ejpam-4476	201	11	sums	sum	NOUN
ejpam-4476	201	12	in	in	ADP
ejpam-4476	201	13	tables	table	NOUN
ejpam-4476	201	14	3	3	NUM
ejpam-4476	201	15	and	and	CCONJ
ejpam-4476	201	16	4	4	NUM
ejpam-4476	201	17	,	,	PUNCT
ejpam-4476	201	18	respectively	respectively	ADV
ejpam-4476	201	19	.	.	PUNCT
ejpam-4476	202	1	these	these	DET
ejpam-4476	202	2	diagrams	diagram	NOUN
ejpam-4476	202	3	illustrate	illustrate	VERB
ejpam-4476	202	4	the	the	DET
ejpam-4476	202	5	different	different	ADJ
ejpam-4476	202	6	ways	way	NOUN
ejpam-4476	202	7	of	of	ADP
ejpam-4476	202	8	counting	count	VERB
ejpam-4476	202	9	the	the	DET
ejpam-4476	202	10	squares	square	NOUN
ejpam-4476	202	11	.	.	PUNCT
ejpam-4476	203	1	diagram	diagram	NOUN
ejpam-4476	203	2	(	(	PUNCT
ejpam-4476	203	3	c	c	NOUN
ejpam-4476	203	4	)	)	PUNCT
ejpam-4476	203	5	shows	show	VERB
ejpam-4476	203	6	how	how	SCONJ
ejpam-4476	203	7	to	to	PART
ejpam-4476	203	8	find	find	VERB
ejpam-4476	203	9	12	12	NUM
ejpam-4476	203	10	+	+	SYM
ejpam-4476	203	11	22	22	NUM
ejpam-4476	203	12	+	+	SYM
ejpam-4476	203	13	32	32	NUM
ejpam-4476	203	14	using	use	VERB
ejpam-4476	203	15	(	(	PUNCT
ejpam-4476	203	16	b	b	NOUN
ejpam-4476	203	17	)	)	PUNCT
ejpam-4476	203	18	.	.	PUNCT
ejpam-4476	204	1	4	4	X
ejpam-4476	204	2	.	.	X
ejpam-4476	204	3	algebraic	algebraic	ADJ
ejpam-4476	204	4	meanings	meaning	NOUN
ejpam-4476	204	5	of	of	ADP
ejpam-4476	204	6	n∑	n∑	NOUN
ejpam-4476	204	7	k=1	k=1	PROPN
ejpam-4476	204	8	k3	k3	VERB
ejpam-4476	204	9	and	and	CCONJ
ejpam-4476	204	10	n∑	n∑	NOUN
ejpam-4476	204	11	k=1	k=1	PROPN
ejpam-4476	204	12	k2	k2	PROPN
ejpam-4476	204	13	in	in	ADP
ejpam-4476	204	14	terms	term	NOUN
ejpam-4476	204	15	of	of	ADP
ejpam-4476	204	16	multiplication	multiplication	NOUN
ejpam-4476	204	17	of	of	ADP
ejpam-4476	204	18	a	a	DET
ejpam-4476	204	19	vertical	vertical	ADJ
ejpam-4476	204	20	vector	vector	NOUN
ejpam-4476	204	21	with	with	ADP
ejpam-4476	204	22	a	a	DET
ejpam-4476	204	23	horizontal	horizontal	ADJ
ejpam-4476	204	24	one	one	NOUN
ejpam-4476	204	25	the	the	DET
ejpam-4476	204	26	sum	sum	NOUN
ejpam-4476	204	27	n∑	n∑	PROPN
ejpam-4476	205	1	k=1	k=1	PROPN
ejpam-4476	205	2	k3	k3	PROPN
ejpam-4476	205	3	can	can	AUX
ejpam-4476	205	4	be	be	AUX
ejpam-4476	205	5	expressed	express	VERB
ejpam-4476	205	6	as	as	ADP
ejpam-4476	205	7	the	the	DET
ejpam-4476	205	8	sum	sum	NOUN
ejpam-4476	205	9	n∑	n∑	PROPN
ejpam-4476	206	1	k=1	k=1	PUNCT
ejpam-4476	207	1	k	k	X
ejpam-4476	207	2	in	in	ADP
ejpam-4476	207	3	terms	term	NOUN
ejpam-4476	207	4	of	of	ADP
ejpam-4476	207	5	matrix	matrix	NOUN
ejpam-4476	207	6	multiplication	multiplication	NOUN
ejpam-4476	207	7	.	.	PUNCT
ejpam-4476	208	1	for	for	ADP
ejpam-4476	208	2	simplicity	simplicity	NOUN
ejpam-4476	208	3	,	,	PUNCT
ejpam-4476	208	4	we	we	PRON
ejpam-4476	208	5	illustrate	illustrate	VERB
ejpam-4476	208	6	(	(	PUNCT
ejpam-4476	208	7	1	1	NUM
ejpam-4476	208	8	+	+	NUM
ejpam-4476	208	9	2	2	NUM
ejpam-4476	208	10	+	+	CCONJ
ejpam-4476	208	11	3)2	3)2	NUM
ejpam-4476	208	12	in	in	ADP
ejpam-4476	208	13	the	the	DET
ejpam-4476	208	14	following	following	NOUN
ejpam-4476	208	15	.	.	PUNCT
ejpam-4476	209	1	the	the	DET
ejpam-4476	209	2	right	right	ADJ
ejpam-4476	209	3	-	-	PUNCT
ejpam-4476	209	4	hand	hand	NOUN
ejpam-4476	209	5	side	side	NOUN
ejpam-4476	209	6	of	of	ADP
ejpam-4476	209	7	(	(	PUNCT
ejpam-4476	209	8	1	1	NUM
ejpam-4476	209	9	+	+	NUM
ejpam-4476	209	10	2	2	NUM
ejpam-4476	209	11	+	+	CCONJ
ejpam-4476	209	12	3)2	3)2	NUM
ejpam-4476	209	13	=	=	SYM
ejpam-4476	210	1	[	[	X
ejpam-4476	210	2	1	1	NUM
ejpam-4476	210	3	+	+	CCONJ
ejpam-4476	210	4	(	(	PUNCT
ejpam-4476	210	5	1	1	NUM
ejpam-4476	210	6	+	+	NUM
ejpam-4476	210	7	1	1	NUM
ejpam-4476	210	8	)	)	PUNCT
ejpam-4476	210	9	+	+	CCONJ
ejpam-4476	210	10	(	(	PUNCT
ejpam-4476	210	11	1	1	NUM
ejpam-4476	210	12	+	+	NUM
ejpam-4476	210	13	1	1	NUM
ejpam-4476	210	14	+	+	SYM
ejpam-4476	210	15	1)]×	1)]×	NUM
ejpam-4476	210	16	[	[	SYM
ejpam-4476	210	17	1	1	NUM
ejpam-4476	210	18	+	+	CCONJ
ejpam-4476	210	19	(	(	PUNCT
ejpam-4476	210	20	1	1	NUM
ejpam-4476	210	21	+	+	NUM
ejpam-4476	210	22	1	1	NUM
ejpam-4476	210	23	)	)	PUNCT
ejpam-4476	210	24	+	+	CCONJ
ejpam-4476	210	25	(	(	PUNCT
ejpam-4476	210	26	1	1	NUM
ejpam-4476	210	27	+	+	SYM
ejpam-4476	210	28	1	1	NUM
ejpam-4476	210	29	+	+	NUM
ejpam-4476	210	30	1	1	NUM
ejpam-4476	210	31	)	)	PUNCT
ejpam-4476	210	32	]	]	PUNCT
ejpam-4476	210	33	(	(	PUNCT
ejpam-4476	210	34	18	18	NUM
ejpam-4476	210	35	)	)	PUNCT
ejpam-4476	210	36	is	be	AUX
ejpam-4476	210	37	the	the	DET
ejpam-4476	210	38	total	total	NOUN
ejpam-4476	210	39	of	of	ADP
ejpam-4476	210	40	36	36	NUM
ejpam-4476	210	41	1×	1×	NUM
ejpam-4476	210	42	1	1	NUM
ejpam-4476	210	43	,	,	PUNCT
ejpam-4476	210	44	which	which	PRON
ejpam-4476	210	45	can	can	AUX
ejpam-4476	210	46	be	be	AUX
ejpam-4476	210	47	regarded	regard	VERB
ejpam-4476	210	48	as	as	ADP
ejpam-4476	210	49	the	the	DET
ejpam-4476	210	50	sum	sum	NOUN
ejpam-4476	210	51	of	of	ADP
ejpam-4476	210	52	all	all	DET
ejpam-4476	210	53	the	the	DET
ejpam-4476	210	54	components	component	NOUN
ejpam-4476	210	55	of	of	ADP
ejpam-4476	210	56	t	t	PROPN
ejpam-4476	210	57	(	(	PUNCT
ejpam-4476	210	58	1	1	NUM
ejpam-4476	210	59	1	1	NUM
ejpam-4476	210	60	1	1	NUM
ejpam-4476	210	61	1	1	NUM
ejpam-4476	210	62	1	1	NUM
ejpam-4476	210	63	1	1	NUM
ejpam-4476	210	64	)	)	PUNCT
ejpam-4476	210	65	(	(	PUNCT
ejpam-4476	210	66	1	1	NUM
ejpam-4476	210	67	1	1	NUM
ejpam-4476	210	68	1	1	NUM
ejpam-4476	210	69	1	1	NUM
ejpam-4476	210	70	1	1	NUM
ejpam-4476	210	71	1	1	NUM
ejpam-4476	210	72	)	)	PUNCT
ejpam-4476	210	73	.	.	PUNCT
ejpam-4476	211	1	the	the	DET
ejpam-4476	211	2	product	product	NOUN
ejpam-4476	211	3	of	of	ADP
ejpam-4476	211	4	these	these	DET
ejpam-4476	211	5	two	two	NUM
ejpam-4476	211	6	vectors	vector	NOUN
ejpam-4476	211	7	can	can	AUX
ejpam-4476	211	8	be	be	AUX
ejpam-4476	211	9	expressed	express	VERB
ejpam-4476	211	10	in	in	ADP
ejpam-4476	211	11	a	a	DET
ejpam-4476	211	12	multiplication	multiplication	NOUN
ejpam-4476	211	13	table	table	NOUN
ejpam-4476	211	14	,	,	PUNCT
ejpam-4476	211	15	as	as	SCONJ
ejpam-4476	211	16	shown	show	VERB
ejpam-4476	211	17	in	in	ADP
ejpam-4476	211	18	figure	figure	NOUN
ejpam-4476	211	19	6	6	NUM
ejpam-4476	211	20	.	.	PUNCT
ejpam-4476	212	1	the	the	DET
ejpam-4476	212	2	sum	sum	NOUN
ejpam-4476	212	3	of	of	ADP
ejpam-4476	212	4	36	36	NUM
ejpam-4476	212	5	1×	1×	NUM
ejpam-4476	212	6	1	1	NUM
ejpam-4476	212	7	can	can	AUX
ejpam-4476	212	8	be	be	AUX
ejpam-4476	212	9	written	write	VERB
ejpam-4476	212	10	as	as	ADP
ejpam-4476	212	11	(	(	PUNCT
ejpam-4476	212	12	1	1	NUM
ejpam-4476	212	13	+	+	NUM
ejpam-4476	212	14	2	2	NUM
ejpam-4476	212	15	+	+	CCONJ
ejpam-4476	212	16	3)2	3)2	NUM
ejpam-4476	212	17	=	=	SYM
ejpam-4476	212	18	12	12	NUM
ejpam-4476	212	19	×	×	NOUN
ejpam-4476	212	20	1	1	NUM
ejpam-4476	212	21	+	+	CCONJ
ejpam-4476	212	22	22	22	NUM
ejpam-4476	212	23	×	×	NOUN
ejpam-4476	212	24	2	2	NUM
ejpam-4476	212	25	+	+	CCONJ
ejpam-4476	212	26	32	32	NUM
ejpam-4476	212	27	×	×	NOUN
ejpam-4476	212	28	3	3	NUM
ejpam-4476	212	29	(	(	PUNCT
ejpam-4476	212	30	19	19	NUM
ejpam-4476	212	31	)	)	PUNCT
ejpam-4476	212	32	in	in	ADP
ejpam-4476	212	33	the	the	DET
ejpam-4476	212	34	left	left	ADJ
ejpam-4476	212	35	figure	figure	NOUN
ejpam-4476	212	36	,	,	PUNCT
ejpam-4476	212	37	while	while	SCONJ
ejpam-4476	212	38	it	it	PRON
ejpam-4476	212	39	can	can	AUX
ejpam-4476	212	40	be	be	AUX
ejpam-4476	212	41	written	write	VERB
ejpam-4476	212	42	as	as	ADP
ejpam-4476	212	43	the	the	DET
ejpam-4476	212	44	sum	sum	NOUN
ejpam-4476	212	45	of	of	ADP
ejpam-4476	212	46	six	six	NUM
ejpam-4476	212	47	consecutive	consecutive	ADJ
ejpam-4476	212	48	odd	odd	ADJ
ejpam-4476	212	49	numbers	number	NOUN
ejpam-4476	212	50	(	(	PUNCT
ejpam-4476	212	51	1	1	NUM
ejpam-4476	212	52	+	+	NUM
ejpam-4476	212	53	2	2	NUM
ejpam-4476	213	1	+	+	CCONJ
ejpam-4476	213	2	3)2	3)2	NUM
ejpam-4476	213	3	=	=	SYM
ejpam-4476	213	4	1	1	NUM
ejpam-4476	213	5	+	+	NUM
ejpam-4476	213	6	3	3	NUM
ejpam-4476	213	7	+	+	SYM
ejpam-4476	213	8	5	5	NUM
ejpam-4476	213	9	+	+	CCONJ
ejpam-4476	213	10	7	7	NUM
ejpam-4476	213	11	+	+	CCONJ
ejpam-4476	213	12	9	9	NUM
ejpam-4476	213	13	+	+	NUM
ejpam-4476	213	14	11	11	NUM
ejpam-4476	213	15	(	(	PUNCT
ejpam-4476	213	16	20	20	NUM
ejpam-4476	213	17	)	)	PUNCT
ejpam-4476	213	18	in	in	ADP
ejpam-4476	213	19	the	the	DET
ejpam-4476	213	20	right	right	ADJ
ejpam-4476	213	21	figure	figure	NOUN
ejpam-4476	213	22	.	.	PUNCT
ejpam-4476	214	1	thus	thus	ADV
ejpam-4476	214	2	,	,	PUNCT
ejpam-4476	214	3	the	the	DET
ejpam-4476	214	4	sum	sum	NOUN
ejpam-4476	214	5	of	of	ADP
ejpam-4476	214	6	the	the	DET
ejpam-4476	214	7	cubic	cubic	ADJ
ejpam-4476	214	8	numbers	number	NOUN
ejpam-4476	214	9	13	13	NUM
ejpam-4476	214	10	+	+	NUM
ejpam-4476	214	11	23	23	NUM
ejpam-4476	214	12	+	+	SYM
ejpam-4476	214	13	33	33	NUM
ejpam-4476	214	14	is	be	AUX
ejpam-4476	214	15	equal	equal	ADJ
ejpam-4476	214	16	to	to	ADP
ejpam-4476	214	17	the	the	DET
ejpam-4476	214	18	sum	sum	NOUN
ejpam-4476	214	19	of	of	ADP
ejpam-4476	214	20	(	(	PUNCT
ejpam-4476	214	21	1	1	NUM
ejpam-4476	214	22	+	+	NOUN
ejpam-4476	214	23	2	2	NUM
ejpam-4476	214	24	+	+	NOUN
ejpam-4476	214	25	3	3	NUM
ejpam-4476	214	26	)	)	PUNCT
ejpam-4476	214	27	consecutive	consecutive	ADJ
ejpam-4476	214	28	odd	odd	ADJ
ejpam-4476	214	29	numbers	number	NOUN
ejpam-4476	214	30	and	and	CCONJ
ejpam-4476	214	31	the	the	DET
ejpam-4476	214	32	square	square	NOUN
ejpam-4476	214	33	of	of	ADP
ejpam-4476	214	34	the	the	DET
ejpam-4476	214	35	sum	sum	NOUN
ejpam-4476	214	36	of	of	ADP
ejpam-4476	214	37	the	the	DET
ejpam-4476	214	38	natural	natural	ADJ
ejpam-4476	214	39	numbers	number	NOUN
ejpam-4476	214	40	,	,	PUNCT
ejpam-4476	214	41	(	(	PUNCT
ejpam-4476	214	42	1	1	NUM
ejpam-4476	214	43	+	+	SYM
ejpam-4476	214	44	2	2	NUM
ejpam-4476	214	45	+	+	SYM
ejpam-4476	214	46	3)2	3)2	NUM
ejpam-4476	214	47	.	.	PUNCT
ejpam-4476	215	1	another	another	DET
ejpam-4476	215	2	relationship	relationship	NOUN
ejpam-4476	215	3	between	between	ADP
ejpam-4476	215	4	the	the	DET
ejpam-4476	215	5	sum	sum	NOUN
ejpam-4476	215	6	of	of	ADP
ejpam-4476	215	7	square	square	ADJ
ejpam-4476	215	8	numbers	number	NOUN
ejpam-4476	215	9	and	and	CCONJ
ejpam-4476	215	10	that	that	PRON
ejpam-4476	215	11	of	of	ADP
ejpam-4476	215	12	natural	natural	ADJ
ejpam-4476	215	13	numbers	number	NOUN
ejpam-4476	215	14	can	can	AUX
ejpam-4476	215	15	also	also	ADV
ejpam-4476	215	16	be	be	AUX
ejpam-4476	215	17	interpreted	interpret	VERB
ejpam-4476	215	18	with	with	ADP
ejpam-4476	215	19	reference	reference	NOUN
ejpam-4476	215	20	to	to	ADP
ejpam-4476	215	21	the	the	DET
ejpam-4476	215	22	revised	revise	VERB
ejpam-4476	215	23	version	version	NOUN
ejpam-4476	215	24	of	of	ADP
ejpam-4476	215	25	figure	figure	NOUN
ejpam-4476	215	26	6	6	NUM
ejpam-4476	215	27	.	.	PUNCT
ejpam-4476	216	1	the	the	DET
ejpam-4476	216	2	sum	sum	NOUN
ejpam-4476	216	3	of	of	ADP
ejpam-4476	216	4	the	the	DET
ejpam-4476	216	5	y.	y.	PROPN
ejpam-4476	216	6	kobayashi	kobayashi	PROPN
ejpam-4476	216	7	/	/	SYM
ejpam-4476	216	8	eur	eur	PROPN
ejpam-4476	216	9	.	.	PUNCT
ejpam-4476	217	1	j.	j.	PROPN
ejpam-4476	217	2	pure	pure	PROPN
ejpam-4476	217	3	appl	appl	PROPN
ejpam-4476	217	4	.	.	PROPN
ejpam-4476	217	5	math	math	PROPN
ejpam-4476	217	6	,	,	PUNCT
ejpam-4476	217	7	15	15	NUM
ejpam-4476	217	8	(	(	PUNCT
ejpam-4476	217	9	3	3	NUM
ejpam-4476	217	10	)	)	PUNCT
ejpam-4476	217	11	(	(	PUNCT
ejpam-4476	217	12	2022	2022	NUM
ejpam-4476	217	13	)	)	PUNCT
ejpam-4476	217	14	,	,	PUNCT
ejpam-4476	217	15	864	864	NUM
ejpam-4476	217	16	-	-	SYM
ejpam-4476	217	17	877	877	NUM
ejpam-4476	217	18	875	875	NUM
ejpam-4476	217	19	×	×	NOUN
ejpam-4476	217	20	1	1	NUM
ejpam-4476	217	21	1	1	NUM
ejpam-4476	217	22	1	1	NUM
ejpam-4476	217	23	1	1	NUM
ejpam-4476	217	24	1	1	NUM
ejpam-4476	217	25	1	1	NUM
ejpam-4476	217	26	1	1	NUM
ejpam-4476	217	27	1	1	NUM
ejpam-4476	217	28	1	1	NUM
ejpam-4476	217	29	1	1	NUM
ejpam-4476	217	30	1	1	NUM
ejpam-4476	217	31	1	1	NUM
ejpam-4476	217	32	1	1	NUM
ejpam-4476	217	33	1	1	NUM
ejpam-4476	217	34	1	1	NUM
ejpam-4476	217	35	1	1	NUM
ejpam-4476	217	36	1	1	NUM
ejpam-4476	217	37	1	1	NUM
ejpam-4476	217	38	1	1	NUM
ejpam-4476	217	39	1	1	NUM
ejpam-4476	217	40	1	1	NUM
ejpam-4476	217	41	1	1	NUM
ejpam-4476	217	42	1	1	NUM
ejpam-4476	217	43	1	1	NUM
ejpam-4476	217	44	1	1	NUM
ejpam-4476	217	45	1	1	NUM
ejpam-4476	217	46	1	1	NUM
ejpam-4476	217	47	1	1	NUM
ejpam-4476	217	48	1	1	NUM
ejpam-4476	217	49	1	1	NUM
ejpam-4476	217	50	1	1	NUM
ejpam-4476	217	51	1	1	NUM
ejpam-4476	217	52	1	1	NUM
ejpam-4476	217	53	1	1	NUM
ejpam-4476	217	54	1	1	NUM
ejpam-4476	217	55	1	1	NUM
ejpam-4476	217	56	1	1	NUM
ejpam-4476	217	57	1	1	NUM
ejpam-4476	217	58	1	1	NUM
ejpam-4476	217	59	1	1	NUM
ejpam-4476	217	60	1	1	NUM
ejpam-4476	217	61	1	1	NUM
ejpam-4476	217	62	1	1	NUM
ejpam-4476	217	63	1	1	NUM
ejpam-4476	217	64	1	1	NUM
ejpam-4476	217	65	1	1	NUM
ejpam-4476	217	66	1	1	NUM
ejpam-4476	217	67	1	1	NUM
ejpam-4476	217	68	1	1	NUM
ejpam-4476	217	69	3	3	NUM
ejpam-4476	217	70	5	5	NUM
ejpam-4476	217	71	7	7	NUM
ejpam-4476	217	72	9	9	NUM
ejpam-4476	217	73	11	11	NUM
ejpam-4476	217	74	1	1	NUM
ejpam-4476	217	75	1	1	NUM
ejpam-4476	217	76	1	1	NUM
ejpam-4476	217	77	1	1	NUM
ejpam-4476	217	78	1	1	NUM
ejpam-4476	217	79	1	1	NUM
ejpam-4476	217	80	1	1	NUM
ejpam-4476	217	81	1	1	NUM
ejpam-4476	217	82	1	1	NUM
ejpam-4476	217	83	1	1	NUM
ejpam-4476	217	84	1	1	NUM
ejpam-4476	217	85	1	1	NUM
ejpam-4476	217	86	1	1	NUM
ejpam-4476	217	87	1	1	NUM
ejpam-4476	217	88	1	1	NUM
ejpam-4476	217	89	1	1	NUM
ejpam-4476	217	90	1	1	NUM
ejpam-4476	217	91	1	1	NUM
ejpam-4476	217	92	1	1	NUM
ejpam-4476	217	93	1	1	NUM
ejpam-4476	217	94	1	1	NUM
ejpam-4476	217	95	1	1	NUM
ejpam-4476	217	96	1	1	NUM
ejpam-4476	217	97	1	1	NUM
ejpam-4476	217	98	1	1	NUM
ejpam-4476	217	99	1	1	NUM
ejpam-4476	217	100	1	1	NUM
ejpam-4476	217	101	1	1	NUM
ejpam-4476	217	102	1	1	NUM
ejpam-4476	217	103	1	1	NUM
ejpam-4476	217	104	1	1	NUM
ejpam-4476	217	105	1	1	NUM
ejpam-4476	217	106	1	1	NUM
ejpam-4476	217	107	1	1	NUM
ejpam-4476	217	108	1	1	NUM
ejpam-4476	217	109	1	1	NUM
ejpam-4476	217	110	figure	figure	NOUN
ejpam-4476	217	111	6	6	NUM
ejpam-4476	217	112	:	:	PUNCT
ejpam-4476	217	113	(	(	PUNCT
ejpam-4476	217	114	1	1	NUM
ejpam-4476	217	115	+	+	SYM
ejpam-4476	217	116	2	2	NUM
ejpam-4476	217	117	+	+	SYM
ejpam-4476	217	118	3)2	3)2	NUM
ejpam-4476	217	119	expressed	express	VERB
ejpam-4476	217	120	as	as	ADP
ejpam-4476	217	121	36	36	NUM
ejpam-4476	217	122	1×	1×	NUM
ejpam-4476	217	123	1	1	NUM
ejpam-4476	217	124	.	.	PUNCT
ejpam-4476	218	1	the	the	DET
ejpam-4476	218	2	left	left	ADJ
ejpam-4476	218	3	and	and	CCONJ
ejpam-4476	218	4	right	right	ADJ
ejpam-4476	218	5	figures	figure	NOUN
ejpam-4476	218	6	illustrate	illustrate	VERB
ejpam-4476	218	7	the	the	DET
ejpam-4476	218	8	different	different	ADJ
ejpam-4476	218	9	ways	way	NOUN
ejpam-4476	218	10	of	of	ADP
ejpam-4476	218	11	counting	count	VERB
ejpam-4476	218	12	1	1	NUM
ejpam-4476	218	13	.	.	PUNCT
ejpam-4476	219	1	components	component	NOUN
ejpam-4476	219	2	of	of	ADP
ejpam-4476	219	3	the	the	DET
ejpam-4476	219	4	right	right	ADJ
ejpam-4476	219	5	-	-	PUNCT
ejpam-4476	219	6	hand	hand	NOUN
ejpam-4476	219	7	side	side	NOUN
ejpam-4476	219	8	of	of	ADP
ejpam-4476	219	9	(	(	PUNCT
ejpam-4476	219	10	1	1	NUM
ejpam-4476	219	11	2	2	NUM
ejpam-4476	219	12	3	3	NUM
ejpam-4476	219	13	)	)	PUNCT
ejpam-4476	219	14	(	(	PUNCT
ejpam-4476	219	15	1	1	NUM
ejpam-4476	219	16	2	2	NUM
ejpam-4476	219	17	3	3	NUM
ejpam-4476	219	18	)	)	PUNCT
ejpam-4476	219	19	=	=	SYM
ejpam-4476	220	1	(	(	PUNCT
ejpam-4476	220	2	12	12	NUM
ejpam-4476	220	3	1	1	NUM
ejpam-4476	220	4	·	·	SYM
ejpam-4476	220	5	2	2	NUM
ejpam-4476	220	6	1	1	NUM
ejpam-4476	220	7	·	·	SYM
ejpam-4476	220	8	3	3	NUM
ejpam-4476	220	9	2	2	NUM
ejpam-4476	220	10	·	·	SYM
ejpam-4476	220	11	1	1	NUM
ejpam-4476	220	12	22	22	NUM
ejpam-4476	220	13	2	2	NUM
ejpam-4476	220	14	·	·	SYM
ejpam-4476	220	15	3	3	NUM
ejpam-4476	220	16	3	3	NUM
ejpam-4476	220	17	·	·	SYM
ejpam-4476	220	18	1	1	NUM
ejpam-4476	220	19	3	3	NUM
ejpam-4476	220	20	·	·	SYM
ejpam-4476	220	21	2	2	NUM
ejpam-4476	220	22	32	32	NUM
ejpam-4476	220	23	)	)	PUNCT
ejpam-4476	220	24	(	(	PUNCT
ejpam-4476	220	25	21	21	NUM
ejpam-4476	220	26	)	)	PUNCT
ejpam-4476	220	27	can	can	AUX
ejpam-4476	220	28	be	be	AUX
ejpam-4476	220	29	tabulated	tabulate	VERB
ejpam-4476	220	30	as	as	SCONJ
ejpam-4476	220	31	shown	show	VERB
ejpam-4476	220	32	in	in	ADP
ejpam-4476	220	33	figure	figure	NOUN
ejpam-4476	220	34	7	7	NUM
ejpam-4476	220	35	,	,	PUNCT
ejpam-4476	220	36	and	and	CCONJ
ejpam-4476	220	37	thus	thus	ADV
ejpam-4476	220	38	12	12	NUM
ejpam-4476	220	39	+	+	SYM
ejpam-4476	220	40	22	22	NUM
ejpam-4476	220	41	+	+	NUM
ejpam-4476	220	42	32	32	NUM
ejpam-4476	220	43	=	=	SYM
ejpam-4476	220	44	(	(	PUNCT
ejpam-4476	220	45	1	1	NUM
ejpam-4476	220	46	+	+	NUM
ejpam-4476	220	47	2	2	NUM
ejpam-4476	220	48	+	+	CCONJ
ejpam-4476	220	49	3)2	3)2	NUM
ejpam-4476	220	50	−	−	PROPN
ejpam-4476	220	51	2(1	2(1	NUM
ejpam-4476	220	52	·	·	PUNCT
ejpam-4476	220	53	2	2	NUM
ejpam-4476	220	54	+	+	CCONJ
ejpam-4476	220	55	2	2	NUM
ejpam-4476	220	56	·	·	SYM
ejpam-4476	220	57	3	3	NUM
ejpam-4476	220	58	+	+	CCONJ
ejpam-4476	220	59	3	3	NUM
ejpam-4476	220	60	·	·	SYM
ejpam-4476	220	61	1	1	NUM
ejpam-4476	220	62	)	)	PUNCT
ejpam-4476	220	63	.	.	PUNCT
ejpam-4476	221	1	(	(	PUNCT
ejpam-4476	221	2	22	22	NUM
ejpam-4476	221	3	)	)	PUNCT
ejpam-4476	221	4	×	×	NOUN
ejpam-4476	221	5	1	1	NUM
ejpam-4476	221	6	1	1	NUM
ejpam-4476	221	7	1	1	NUM
ejpam-4476	221	8	1	1	NUM
ejpam-4476	221	9	1	1	NUM
ejpam-4476	221	10	1	1	NUM
ejpam-4476	221	11	1	1	NUM
ejpam-4476	221	12	1	1	NUM
ejpam-4476	221	13	1	1	NUM
ejpam-4476	221	14	1	1	NUM
ejpam-4476	221	15	1	1	NUM
ejpam-4476	221	16	1	1	NUM
ejpam-4476	221	17	1	1	NUM
ejpam-4476	221	18	1	1	NUM
ejpam-4476	221	19	1	1	NUM
ejpam-4476	221	20	1	1	NUM
ejpam-4476	221	21	1	1	NUM
ejpam-4476	221	22	1	1	NUM
ejpam-4476	221	23	1	1	NUM
ejpam-4476	221	24	1	1	NUM
ejpam-4476	221	25	1	1	NUM
ejpam-4476	221	26	1	1	NUM
ejpam-4476	221	27	1	1	NUM
ejpam-4476	221	28	1	1	NUM
ejpam-4476	221	29	1	1	NUM
ejpam-4476	221	30	1	1	NUM
ejpam-4476	221	31	1	1	NUM
ejpam-4476	221	32	1	1	NUM
ejpam-4476	221	33	1	1	NUM
ejpam-4476	221	34	1	1	NUM
ejpam-4476	221	35	1	1	NUM
ejpam-4476	221	36	1	1	NUM
ejpam-4476	221	37	1	1	NUM
ejpam-4476	221	38	1	1	NUM
ejpam-4476	221	39	1	1	NUM
ejpam-4476	221	40	1	1	NUM
ejpam-4476	221	41	1	1	NUM
ejpam-4476	221	42	1	1	NUM
ejpam-4476	221	43	1	1	NUM
ejpam-4476	221	44	1	1	NUM
ejpam-4476	221	45	1	1	NUM
ejpam-4476	221	46	1	1	NUM
ejpam-4476	221	47	1	1	NUM
ejpam-4476	221	48	1	1	NUM
ejpam-4476	221	49	1	1	NUM
ejpam-4476	221	50	1	1	NUM
ejpam-4476	221	51	1	1	NUM
ejpam-4476	221	52	1	1	NUM
ejpam-4476	221	53	figure	figure	NOUN
ejpam-4476	221	54	7	7	NUM
ejpam-4476	221	55	:	:	PUNCT
ejpam-4476	221	56	(	(	PUNCT
ejpam-4476	221	57	1	1	NUM
ejpam-4476	221	58	+	+	SYM
ejpam-4476	221	59	2	2	NUM
ejpam-4476	221	60	+	+	CCONJ
ejpam-4476	221	61	3)2	3)2	NUM
ejpam-4476	221	62	expressed	express	VERB
ejpam-4476	221	63	as	as	ADP
ejpam-4476	221	64	36	36	NUM
ejpam-4476	221	65	1×	1×	NUM
ejpam-4476	221	66	1	1	NUM
ejpam-4476	221	67	.	.	PUNCT
ejpam-4476	222	1	the	the	DET
ejpam-4476	222	2	sum	sum	NOUN
ejpam-4476	222	3	of	of	ADP
ejpam-4476	222	4	the	the	DET
ejpam-4476	222	5	numbers	number	NOUN
ejpam-4476	222	6	shaded	shade	VERB
ejpam-4476	222	7	in	in	ADP
ejpam-4476	222	8	grey	grey	PROPN
ejpam-4476	222	9	is	be	AUX
ejpam-4476	222	10	1×	1×	NUM
ejpam-4476	222	11	2	2	NUM
ejpam-4476	222	12	,	,	PUNCT
ejpam-4476	222	13	the	the	DET
ejpam-4476	222	14	sum	sum	NOUN
ejpam-4476	222	15	of	of	ADP
ejpam-4476	222	16	the	the	DET
ejpam-4476	222	17	numbers	number	NOUN
ejpam-4476	222	18	in	in	ADP
ejpam-4476	222	19	the	the	DET
ejpam-4476	222	20	shaded	shaded	ADJ
ejpam-4476	222	21	area	area	NOUN
ejpam-4476	222	22	is	be	AUX
ejpam-4476	222	23	2×	2×	NUM
ejpam-4476	222	24	3	3	NUM
ejpam-4476	222	25	,	,	PUNCT
ejpam-4476	222	26	and	and	CCONJ
ejpam-4476	222	27	the	the	DET
ejpam-4476	222	28	sum	sum	NOUN
ejpam-4476	222	29	of	of	ADP
ejpam-4476	222	30	the	the	DET
ejpam-4476	222	31	numbers	number	NOUN
ejpam-4476	222	32	shaded	shade	VERB
ejpam-4476	222	33	in	in	ADP
ejpam-4476	222	34	light	light	ADJ
ejpam-4476	222	35	grey	grey	NOUN
ejpam-4476	222	36	is	be	AUX
ejpam-4476	222	37	3×	3×	NUM
ejpam-4476	222	38	1	1	NUM
ejpam-4476	222	39	.	.	PUNCT
ejpam-4476	223	1	the	the	DET
ejpam-4476	223	2	sum	sum	NOUN
ejpam-4476	223	3	n∑	n∑	PROPN
ejpam-4476	224	1	k=1	k=1	PROPN
ejpam-4476	224	2	k2	k2	PROPN
ejpam-4476	224	3	can	can	AUX
ejpam-4476	224	4	also	also	ADV
ejpam-4476	224	5	be	be	AUX
ejpam-4476	224	6	represented	represent	VERB
ejpam-4476	224	7	by	by	ADP
ejpam-4476	224	8	the	the	DET
ejpam-4476	224	9	sum	sum	NOUN
ejpam-4476	224	10	n∑	n∑	PROPN
ejpam-4476	225	1	k=1	k=1	PUNCT
ejpam-4476	226	1	k	k	X
ejpam-4476	226	2	in	in	ADP
ejpam-4476	226	3	terms	term	NOUN
ejpam-4476	226	4	of	of	ADP
ejpam-4476	226	5	matrix	matrix	NOUN
ejpam-4476	226	6	multiplication	multiplication	NOUN
ejpam-4476	226	7	.	.	PUNCT
ejpam-4476	227	1	for	for	ADP
ejpam-4476	227	2	simplicity	simplicity	NOUN
ejpam-4476	227	3	,	,	PUNCT
ejpam-4476	227	4	we	we	PRON
ejpam-4476	227	5	illustrate	illustrate	VERB
ejpam-4476	227	6	12	12	NUM
ejpam-4476	227	7	+	+	NUM
ejpam-4476	227	8	22	22	NUM
ejpam-4476	227	9	+	+	NUM
ejpam-4476	227	10	32	32	NUM
ejpam-4476	227	11	in	in	ADP
ejpam-4476	227	12	the	the	DET
ejpam-4476	227	13	following	following	NOUN
ejpam-4476	227	14	.	.	PUNCT
ejpam-4476	228	1	to	to	PART
ejpam-4476	228	2	adopt	adopt	VERB
ejpam-4476	228	3	a	a	DET
ejpam-4476	228	4	heuristic	heuristic	ADJ
ejpam-4476	228	5	method	method	NOUN
ejpam-4476	228	6	[	[	X
ejpam-4476	228	7	6	6	NUM
ejpam-4476	228	8	]	]	PUNCT
ejpam-4476	228	9	,	,	PUNCT
ejpam-4476	228	10	we	we	PRON
ejpam-4476	228	11	replace	replace	VERB
ejpam-4476	228	12	the	the	DET
ejpam-4476	228	13	square	square	ADV
ejpam-4476	228	14	-	-	PUNCT
ejpam-4476	228	15	shaped	shape	VERB
ejpam-4476	228	16	table	table	NOUN
ejpam-4476	228	17	in	in	ADP
ejpam-4476	228	18	figure	figure	NOUN
ejpam-4476	228	19	6	6	NUM
ejpam-4476	228	20	with	with	ADP
ejpam-4476	228	21	a	a	DET
ejpam-4476	228	22	rectangular	rectangular	ADJ
ejpam-4476	228	23	one	one	NOUN
ejpam-4476	228	24	,	,	PUNCT
ejpam-4476	228	25	as	as	SCONJ
ejpam-4476	228	26	shown	show	VERB
ejpam-4476	228	27	in	in	ADP
ejpam-4476	228	28	figure	figure	NOUN
ejpam-4476	228	29	8	8	NUM
ejpam-4476	228	30	,	,	PUNCT
ejpam-4476	228	31	which	which	PRON
ejpam-4476	228	32	can	can	AUX
ejpam-4476	228	33	be	be	AUX
ejpam-4476	228	34	regarded	regard	VERB
ejpam-4476	228	35	as	as	ADP
ejpam-4476	228	36	the	the	DET
ejpam-4476	228	37	sum	sum	NOUN
ejpam-4476	228	38	of	of	ADP
ejpam-4476	228	39	all	all	DET
ejpam-4476	228	40	the	the	DET
ejpam-4476	228	41	components	component	NOUN
ejpam-4476	228	42	of	of	ADP
ejpam-4476	228	43	t	t	PROPN
ejpam-4476	228	44	(	(	PUNCT
ejpam-4476	228	45	1	1	NUM
ejpam-4476	228	46	1	1	NUM
ejpam-4476	228	47	1	1	NUM
ejpam-4476	228	48	1	1	NUM
ejpam-4476	228	49	1	1	NUM
ejpam-4476	228	50	1	1	NUM
ejpam-4476	228	51	1	1	NUM
ejpam-4476	228	52	)	)	PUNCT
ejpam-4476	228	53	(	(	PUNCT
ejpam-4476	228	54	1	1	NUM
ejpam-4476	228	55	1	1	NUM
ejpam-4476	228	56	1	1	NUM
ejpam-4476	228	57	1	1	NUM
ejpam-4476	228	58	1	1	NUM
ejpam-4476	228	59	1	1	NUM
ejpam-4476	228	60	)	)	PUNCT
ejpam-4476	228	61	.	.	PUNCT
ejpam-4476	229	1	the	the	DET
ejpam-4476	229	2	product	product	NOUN
ejpam-4476	229	3	of	of	ADP
ejpam-4476	229	4	these	these	DET
ejpam-4476	229	5	two	two	NUM
ejpam-4476	229	6	matrices	matrix	NOUN
ejpam-4476	229	7	can	can	AUX
ejpam-4476	229	8	be	be	AUX
ejpam-4476	229	9	expressed	express	VERB
ejpam-4476	229	10	in	in	ADP
ejpam-4476	229	11	a	a	DET
ejpam-4476	229	12	multiplication	multiplication	NOUN
ejpam-4476	229	13	table	table	NOUN
ejpam-4476	229	14	,	,	PUNCT
ejpam-4476	229	15	as	as	SCONJ
ejpam-4476	229	16	shown	show	VERB
ejpam-4476	229	17	in	in	ADP
ejpam-4476	229	18	figure	figure	NOUN
ejpam-4476	229	19	8	8	NUM
ejpam-4476	229	20	.	.	PUNCT
ejpam-4476	230	1	the	the	DET
ejpam-4476	230	2	sum	sum	NOUN
ejpam-4476	230	3	of	of	ADP
ejpam-4476	230	4	42	42	NUM
ejpam-4476	230	5	1×	1×	NUM
ejpam-4476	230	6	1	1	NUM
ejpam-4476	230	7	can	can	AUX
ejpam-4476	230	8	be	be	AUX
ejpam-4476	230	9	written	write	VERB
ejpam-4476	230	10	as	as	ADP
ejpam-4476	230	11	42×	42×	NUM
ejpam-4476	230	12	(	(	PUNCT
ejpam-4476	230	13	1×	1×	NUM
ejpam-4476	230	14	1	1	NUM
ejpam-4476	230	15	)	)	PUNCT
ejpam-4476	230	16	=	=	PUNCT
ejpam-4476	231	1	3×	3×	NUM
ejpam-4476	232	1	[	[	X
ejpam-4476	232	2	1	1	NUM
ejpam-4476	232	3	+	+	CCONJ
ejpam-4476	232	4	(	(	PUNCT
ejpam-4476	232	5	1	1	NUM
ejpam-4476	232	6	+	+	NUM
ejpam-4476	232	7	3	3	NUM
ejpam-4476	232	8	)	)	PUNCT
ejpam-4476	232	9	+	+	CCONJ
ejpam-4476	232	10	(	(	PUNCT
ejpam-4476	232	11	1	1	NUM
ejpam-4476	232	12	+	+	NUM
ejpam-4476	232	13	3	3	NUM
ejpam-4476	232	14	+	+	NUM
ejpam-4476	232	15	5	5	NUM
ejpam-4476	232	16	)	)	PUNCT
ejpam-4476	232	17	]	]	PUNCT
ejpam-4476	233	1	=	=	PUNCT
ejpam-4476	233	2	3×	3×	NUM
ejpam-4476	233	3	(	(	PUNCT
ejpam-4476	233	4	12	12	NUM
ejpam-4476	233	5	+	+	NUM
ejpam-4476	233	6	22	22	NUM
ejpam-4476	233	7	+	+	NUM
ejpam-4476	233	8	32	32	NUM
ejpam-4476	233	9	)	)	PUNCT
ejpam-4476	233	10	(	(	PUNCT
ejpam-4476	233	11	23	23	X
ejpam-4476	233	12	)	)	PUNCT
ejpam-4476	233	13	y.	y.	PROPN
ejpam-4476	233	14	kobayashi	kobayashi	PROPN
ejpam-4476	233	15	/	/	SYM
ejpam-4476	233	16	eur	eur	PROPN
ejpam-4476	233	17	.	.	PUNCT
ejpam-4476	234	1	j.	j.	PROPN
ejpam-4476	234	2	pure	pure	PROPN
ejpam-4476	234	3	appl	appl	PROPN
ejpam-4476	234	4	.	.	PROPN
ejpam-4476	234	5	math	math	PROPN
ejpam-4476	234	6	,	,	PUNCT
ejpam-4476	234	7	15	15	NUM
ejpam-4476	234	8	(	(	PUNCT
ejpam-4476	234	9	3	3	NUM
ejpam-4476	234	10	)	)	PUNCT
ejpam-4476	234	11	(	(	PUNCT
ejpam-4476	234	12	2022	2022	NUM
ejpam-4476	234	13	)	)	PUNCT
ejpam-4476	234	14	,	,	PUNCT
ejpam-4476	234	15	864	864	NUM
ejpam-4476	234	16	-	-	SYM
ejpam-4476	234	17	877	877	NUM
ejpam-4476	234	18	876	876	NUM
ejpam-4476	234	19	in	in	ADP
ejpam-4476	234	20	the	the	DET
ejpam-4476	234	21	left	left	ADJ
ejpam-4476	234	22	figure	figure	NOUN
ejpam-4476	234	23	,	,	PUNCT
ejpam-4476	234	24	while	while	SCONJ
ejpam-4476	234	25	it	it	PRON
ejpam-4476	234	26	can	can	AUX
ejpam-4476	234	27	be	be	AUX
ejpam-4476	234	28	written	write	VERB
ejpam-4476	234	29	as	as	ADP
ejpam-4476	234	30	a	a	DET
ejpam-4476	234	31	sum	sum	NOUN
ejpam-4476	234	32	of	of	ADP
ejpam-4476	234	33	the	the	DET
ejpam-4476	234	34	sums	sum	NOUN
ejpam-4476	234	35	of	of	ADP
ejpam-4476	234	36	natural	natural	ADJ
ejpam-4476	234	37	numbers	number	NOUN
ejpam-4476	234	38	,	,	PUNCT
ejpam-4476	234	39	1	1	NUM
ejpam-4476	234	40	+	+	SYM
ejpam-4476	234	41	2	2	NUM
ejpam-4476	234	42	+	+	SYM
ejpam-4476	234	43	3	3	NUM
ejpam-4476	234	44	,	,	PUNCT
ejpam-4476	234	45	42×	42×	NUM
ejpam-4476	234	46	(	(	PUNCT
ejpam-4476	234	47	1×	1×	NUM
ejpam-4476	234	48	1	1	NUM
ejpam-4476	234	49	)	)	PUNCT
ejpam-4476	234	50	=	=	SYM
ejpam-4476	234	51	(	(	PUNCT
ejpam-4476	234	52	2×	2×	NUM
ejpam-4476	234	53	3	3	NUM
ejpam-4476	234	54	+	+	SYM
ejpam-4476	234	55	1)×	1)×	NUM
ejpam-4476	234	56	(	(	PUNCT
ejpam-4476	234	57	1	1	NUM
ejpam-4476	234	58	+	+	NUM
ejpam-4476	234	59	2	2	NUM
ejpam-4476	234	60	+	+	NUM
ejpam-4476	234	61	3	3	NUM
ejpam-4476	234	62	)	)	PUNCT
ejpam-4476	234	63	,	,	PUNCT
ejpam-4476	234	64	(	(	PUNCT
ejpam-4476	234	65	24	24	NUM
ejpam-4476	234	66	)	)	PUNCT
ejpam-4476	234	67	in	in	ADP
ejpam-4476	234	68	the	the	DET
ejpam-4476	234	69	right	right	ADJ
ejpam-4476	234	70	figure	figure	NOUN
ejpam-4476	234	71	.	.	PUNCT
ejpam-4476	235	1	thus	thus	ADV
ejpam-4476	235	2	,	,	PUNCT
ejpam-4476	235	3	the	the	DET
ejpam-4476	235	4	sum	sum	NOUN
ejpam-4476	235	5	of	of	ADP
ejpam-4476	235	6	the	the	DET
ejpam-4476	235	7	square	square	ADJ
ejpam-4476	235	8	numbers	number	NOUN
ejpam-4476	235	9	,	,	PUNCT
ejpam-4476	235	10	12	12	NUM
ejpam-4476	235	11	+	+	SYM
ejpam-4476	235	12	22	22	NUM
ejpam-4476	235	13	+	+	NUM
ejpam-4476	235	14	32	32	NUM
ejpam-4476	235	15	,	,	PUNCT
ejpam-4476	235	16	is	be	AUX
ejpam-4476	235	17	equal	equal	ADJ
ejpam-4476	235	18	to	to	ADP
ejpam-4476	235	19	1	1	NUM
ejpam-4476	235	20	3	3	NUM
ejpam-4476	235	21	(	(	PUNCT
ejpam-4476	235	22	2×	2×	NUM
ejpam-4476	235	23	3	3	NUM
ejpam-4476	235	24	+	+	SYM
ejpam-4476	235	25	1)×	1)×	NUM
ejpam-4476	235	26	(	(	PUNCT
ejpam-4476	235	27	1	1	NUM
ejpam-4476	235	28	+	+	NUM
ejpam-4476	235	29	2	2	NUM
ejpam-4476	235	30	+	+	NOUN
ejpam-4476	235	31	3	3	NUM
ejpam-4476	235	32	)	)	PUNCT
ejpam-4476	235	33	.	.	PUNCT
ejpam-4476	236	1	adding	add	VERB
ejpam-4476	236	2	two	two	NUM
ejpam-4476	236	3	rows	row	NOUN
ejpam-4476	236	4	and	and	CCONJ
ejpam-4476	236	5	four	four	NUM
ejpam-4476	236	6	columns	column	NOUN
ejpam-4476	236	7	to	to	ADP
ejpam-4476	236	8	the	the	DET
ejpam-4476	236	9	multiplication	multiplication	NOUN
ejpam-4476	236	10	tables	table	NOUN
ejpam-4476	236	11	represents	represent	VERB
ejpam-4476	236	12	the	the	DET
ejpam-4476	236	13	case	case	NOUN
ejpam-4476	236	14	of	of	ADP
ejpam-4476	236	15	12	12	NUM
ejpam-4476	236	16	+	+	NOUN
ejpam-4476	236	17	22	22	NUM
ejpam-4476	236	18	+	+	SYM
ejpam-4476	236	19	32	32	NUM
ejpam-4476	236	20	+	+	SYM
ejpam-4476	236	21	42	42	NUM
ejpam-4476	236	22	.	.	PUNCT
ejpam-4476	237	1	in	in	ADP
ejpam-4476	237	2	this	this	DET
ejpam-4476	237	3	case	case	NOUN
ejpam-4476	237	4	,	,	PUNCT
ejpam-4476	237	5	the	the	DET
ejpam-4476	237	6	vertical	vertical	ADJ
ejpam-4476	237	7	direction	direction	NOUN
ejpam-4476	237	8	in	in	ADP
ejpam-4476	237	9	the	the	DET
ejpam-4476	237	10	left	left	ADJ
ejpam-4476	237	11	figure	figure	NOUN
ejpam-4476	237	12	can	can	AUX
ejpam-4476	237	13	be	be	AUX
ejpam-4476	237	14	trisected	trisect	VERB
ejpam-4476	237	15	,	,	PUNCT
ejpam-4476	237	16	in	in	ADP
ejpam-4476	237	17	contrast	contrast	NOUN
ejpam-4476	237	18	to	to	ADP
ejpam-4476	237	19	12	12	NUM
ejpam-4476	237	20	+	+	NUM
ejpam-4476	237	21	22	22	NUM
ejpam-4476	237	22	+	+	NUM
ejpam-4476	237	23	32	32	NUM
ejpam-4476	237	24	.	.	PUNCT
ejpam-4476	238	1	×	×	NOUN
ejpam-4476	238	2	1	1	NUM
ejpam-4476	238	3	1	1	NUM
ejpam-4476	238	4	1	1	NUM
ejpam-4476	238	5	1	1	NUM
ejpam-4476	238	6	1	1	NUM
ejpam-4476	238	7	1	1	NUM
ejpam-4476	238	8	1	1	NUM
ejpam-4476	238	9	1	1	NUM
ejpam-4476	238	10	1	1	NUM
ejpam-4476	238	11	1	1	NUM
ejpam-4476	238	12	1	1	NUM
ejpam-4476	238	13	1	1	NUM
ejpam-4476	238	14	1	1	NUM
ejpam-4476	238	15	1	1	NUM
ejpam-4476	238	16	1	1	NUM
ejpam-4476	238	17	1	1	NUM
ejpam-4476	238	18	1	1	NUM
ejpam-4476	238	19	1	1	NUM
ejpam-4476	238	20	1	1	NUM
ejpam-4476	238	21	1	1	NUM
ejpam-4476	238	22	1	1	NUM
ejpam-4476	238	23	1	1	NUM
ejpam-4476	238	24	1	1	NUM
ejpam-4476	238	25	1	1	NUM
ejpam-4476	238	26	1	1	NUM
ejpam-4476	238	27	1	1	NUM
ejpam-4476	238	28	1	1	NUM
ejpam-4476	238	29	1	1	NUM
ejpam-4476	238	30	1	1	NUM
ejpam-4476	238	31	1	1	NUM
ejpam-4476	238	32	1	1	NUM
ejpam-4476	238	33	1	1	NUM
ejpam-4476	238	34	1	1	NUM
ejpam-4476	238	35	1	1	NUM
ejpam-4476	238	36	1	1	NUM
ejpam-4476	238	37	1	1	NUM
ejpam-4476	238	38	1	1	NUM
ejpam-4476	238	39	1	1	NUM
ejpam-4476	238	40	1	1	NUM
ejpam-4476	238	41	1	1	NUM
ejpam-4476	238	42	1	1	NUM
ejpam-4476	238	43	1	1	NUM
ejpam-4476	238	44	1	1	NUM
ejpam-4476	238	45	1	1	NUM
ejpam-4476	238	46	1	1	NUM
ejpam-4476	238	47	1	1	NUM
ejpam-4476	238	48	1	1	NUM
ejpam-4476	238	49	1	1	NUM
ejpam-4476	238	50	1	1	NUM
ejpam-4476	238	51	1	1	NUM
ejpam-4476	238	52	1	1	NUM
ejpam-4476	238	53	1	1	NUM
ejpam-4476	238	54	1	1	NUM
ejpam-4476	238	55	1	1	NUM
ejpam-4476	238	56	1	1	NUM
ejpam-4476	238	57	1	1	NUM
ejpam-4476	238	58	1	1	NUM
ejpam-4476	238	59	1	1	NUM
ejpam-4476	238	60	1	1	NUM
ejpam-4476	238	61	1	1	NUM
ejpam-4476	238	62	1	1	NUM
ejpam-4476	238	63	1	1	NUM
ejpam-4476	238	64	1	1	NUM
ejpam-4476	238	65	1	1	NUM
ejpam-4476	238	66	1	1	NUM
ejpam-4476	238	67	1	1	NUM
ejpam-4476	238	68	1	1	NUM
ejpam-4476	238	69	1	1	NUM
ejpam-4476	238	70	1	1	NUM
ejpam-4476	238	71	1	1	NUM
ejpam-4476	238	72	1	1	NUM
ejpam-4476	238	73	1	1	NUM
ejpam-4476	238	74	1	1	NUM
ejpam-4476	238	75	1	1	NUM
ejpam-4476	238	76	1	1	NUM
ejpam-4476	238	77	1	1	NUM
ejpam-4476	238	78	1	1	NUM
ejpam-4476	238	79	1	1	NUM
ejpam-4476	238	80	1	1	NUM
ejpam-4476	238	81	1	1	NUM
ejpam-4476	238	82	1	1	NUM
ejpam-4476	238	83	1	1	NUM
ejpam-4476	238	84	1	1	NUM
ejpam-4476	238	85	1	1	NUM
ejpam-4476	238	86	1	1	NUM
ejpam-4476	238	87	1	1	NUM
ejpam-4476	238	88	1	1	NUM
ejpam-4476	238	89	1	1	NUM
ejpam-4476	238	90	1	1	NUM
ejpam-4476	238	91	1	1	NUM
ejpam-4476	238	92	1	1	NUM
ejpam-4476	238	93	1	1	NUM
ejpam-4476	238	94	1	1	NUM
ejpam-4476	238	95	1	1	NUM
ejpam-4476	238	96	1	1	NUM
ejpam-4476	238	97	1	1	NUM
ejpam-4476	238	98	1	1	NUM
ejpam-4476	238	99	3	3	NUM
ejpam-4476	238	100	3	3	NUM
ejpam-4476	238	101	1	1	NUM
ejpam-4476	238	102	1	1	NUM
ejpam-4476	238	103	2	2	NUM
ejpam-4476	238	104	3	3	NUM
ejpam-4476	238	105	figure	figure	NOUN
ejpam-4476	238	106	8	8	NUM
ejpam-4476	238	107	:	:	PUNCT
ejpam-4476	238	108	(	(	PUNCT
ejpam-4476	238	109	2×	2×	NUM
ejpam-4476	238	110	3	3	NUM
ejpam-4476	238	111	+	+	SYM
ejpam-4476	238	112	1)×	1)×	NUM
ejpam-4476	238	113	(	(	PUNCT
ejpam-4476	238	114	1	1	NUM
ejpam-4476	238	115	+	+	SYM
ejpam-4476	238	116	2	2	NUM
ejpam-4476	238	117	+	+	NUM
ejpam-4476	238	118	3	3	NUM
ejpam-4476	238	119	)	)	PUNCT
ejpam-4476	238	120	expressed	express	VERB
ejpam-4476	238	121	as	as	ADP
ejpam-4476	238	122	42	42	NUM
ejpam-4476	238	123	1×	1×	NUM
ejpam-4476	238	124	1	1	NUM
ejpam-4476	238	125	.	.	PUNCT
ejpam-4476	239	1	the	the	DET
ejpam-4476	239	2	left	left	ADJ
ejpam-4476	239	3	and	and	CCONJ
ejpam-4476	239	4	right	right	ADJ
ejpam-4476	239	5	figures	figure	NOUN
ejpam-4476	239	6	show	show	VERB
ejpam-4476	239	7	the	the	DET
ejpam-4476	239	8	different	different	ADJ
ejpam-4476	239	9	ways	way	NOUN
ejpam-4476	239	10	of	of	ADP
ejpam-4476	239	11	counting	count	VERB
ejpam-4476	239	12	1	1	NUM
ejpam-4476	239	13	.	.	PUNCT
ejpam-4476	240	1	the	the	DET
ejpam-4476	240	2	meaning	meaning	NOUN
ejpam-4476	240	3	of	of	ADP
ejpam-4476	240	4	(	(	PUNCT
ejpam-4476	240	5	n∑	n∑	INTJ
ejpam-4476	240	6	k=1	k=1	PUNCT
ejpam-4476	240	7	k	k	X
ejpam-4476	240	8	)	)	PUNCT
ejpam-4476	240	9	2	2	NUM
ejpam-4476	240	10	can	can	AUX
ejpam-4476	240	11	also	also	ADV
ejpam-4476	240	12	be	be	AUX
ejpam-4476	240	13	interpreted	interpret	VERB
ejpam-4476	240	14	in	in	ADP
ejpam-4476	240	15	terms	term	NOUN
ejpam-4476	240	16	of	of	ADP
ejpam-4476	240	17	combinatorics	combinatoric	NOUN
ejpam-4476	240	18	because	because	SCONJ
ejpam-4476	240	19	the	the	DET
ejpam-4476	240	20	left	left	ADJ
ejpam-4476	240	21	figure	figure	NOUN
ejpam-4476	240	22	can	can	AUX
ejpam-4476	240	23	be	be	AUX
ejpam-4476	240	24	regarded	regard	VERB
ejpam-4476	240	25	as	as	ADP
ejpam-4476	240	26	a	a	DET
ejpam-4476	240	27	league	league	NOUN
ejpam-4476	240	28	matchup	matchup	NOUN
ejpam-4476	240	29	chart	chart	NOUN
ejpam-4476	241	1	[	[	X
ejpam-4476	241	2	7	7	NUM
ejpam-4476	241	3	,	,	PUNCT
ejpam-4476	241	4	8	8	NUM
ejpam-4476	241	5	]	]	PUNCT
ejpam-4476	241	6	.	.	PUNCT
ejpam-4476	242	1	5	5	X
ejpam-4476	242	2	.	.	X
ejpam-4476	242	3	concluding	conclude	VERB
ejpam-4476	242	4	remarks	remark	VERB
ejpam-4476	242	5	the	the	DET
ejpam-4476	242	6	fundamental	fundamental	ADJ
ejpam-4476	242	7	idea	idea	NOUN
ejpam-4476	242	8	discussed	discuss	VERB
ejpam-4476	242	9	in	in	ADP
ejpam-4476	242	10	this	this	DET
ejpam-4476	242	11	paper	paper	NOUN
ejpam-4476	242	12	is	be	AUX
ejpam-4476	242	13	to	to	PART
ejpam-4476	242	14	express	express	VERB
ejpam-4476	242	15	square	square	ADJ
ejpam-4476	242	16	and	and	CCONJ
ejpam-4476	242	17	cubic	cubic	ADJ
ejpam-4476	242	18	numbers	number	NOUN
ejpam-4476	242	19	as	as	ADP
ejpam-4476	242	20	sums	sum	NOUN
ejpam-4476	242	21	of	of	ADP
ejpam-4476	242	22	odd	odd	ADJ
ejpam-4476	242	23	numbers	number	NOUN
ejpam-4476	242	24	.	.	PUNCT
ejpam-4476	243	1	the	the	DET
ejpam-4476	243	2	important	important	ADJ
ejpam-4476	243	3	point	point	NOUN
ejpam-4476	243	4	is	be	AUX
ejpam-4476	243	5	to	to	PART
ejpam-4476	243	6	reassemble	reassemble	VERB
ejpam-4476	243	7	these	these	DET
ejpam-4476	243	8	sums	sum	NOUN
ejpam-4476	243	9	for	for	ADP
ejpam-4476	243	10	computation	computation	NOUN
ejpam-4476	243	11	and	and	CCONJ
ejpam-4476	243	12	devise	devise	VERB
ejpam-4476	243	13	ways	way	NOUN
ejpam-4476	243	14	of	of	ADP
ejpam-4476	243	15	counting	counting	NOUN
ejpam-4476	243	16	.	.	PUNCT
ejpam-4476	244	1	to	to	PART
ejpam-4476	244	2	realise	realise	VERB
ejpam-4476	244	3	this	this	DET
ejpam-4476	244	4	idea	idea	NOUN
ejpam-4476	244	5	,	,	PUNCT
ejpam-4476	244	6	it	it	PRON
ejpam-4476	244	7	is	be	AUX
ejpam-4476	244	8	essential	essential	ADJ
ejpam-4476	244	9	not	not	PART
ejpam-4476	244	10	only	only	ADV
ejpam-4476	244	11	to	to	PART
ejpam-4476	244	12	manipulate	manipulate	VERB
ejpam-4476	244	13	the	the	DET
ejpam-4476	244	14	equations	equation	NOUN
ejpam-4476	244	15	,	,	PUNCT
ejpam-4476	244	16	but	but	CCONJ
ejpam-4476	244	17	also	also	ADV
ejpam-4476	244	18	to	to	PART
ejpam-4476	244	19	pictorially	pictorially	ADV
ejpam-4476	244	20	illustrate	illustrate	VERB
ejpam-4476	244	21	the	the	DET
ejpam-4476	244	22	calculation	calculation	NOUN
ejpam-4476	244	23	process	process	NOUN
ejpam-4476	244	24	.	.	PUNCT
ejpam-4476	245	1	although	although	SCONJ
ejpam-4476	245	2	different	different	ADJ
ejpam-4476	245	3	from	from	ADP
ejpam-4476	245	4	the	the	DET
ejpam-4476	245	5	theme	theme	NOUN
ejpam-4476	245	6	of	of	ADP
ejpam-4476	245	7	this	this	DET
ejpam-4476	245	8	paper	paper	NOUN
ejpam-4476	245	9	,	,	PUNCT
ejpam-4476	245	10	the	the	DET
ejpam-4476	245	11	idea	idea	NOUN
ejpam-4476	245	12	of	of	ADP
ejpam-4476	245	13	the	the	DET
ejpam-4476	245	14	sectional	sectional	ADJ
ejpam-4476	245	15	quadrature	quadrature	NOUN
ejpam-4476	245	16	method	method	NOUN
ejpam-4476	245	17	,	,	PUNCT
ejpam-4476	245	18	for	for	ADP
ejpam-4476	245	19	example	example	NOUN
ejpam-4476	245	20	,	,	PUNCT
ejpam-4476	245	21	is	be	AUX
ejpam-4476	245	22	similar	similar	ADJ
ejpam-4476	245	23	to	to	ADP
ejpam-4476	245	24	this	this	DET
ejpam-4476	245	25	one	one	NOUN
ejpam-4476	245	26	.	.	PUNCT
ejpam-4476	246	1	in	in	ADP
ejpam-4476	246	2	cartesian	cartesian	ADJ
ejpam-4476	246	3	terms	term	NOUN
ejpam-4476	246	4	,	,	PUNCT
ejpam-4476	246	5	the	the	DET
ejpam-4476	246	6	idea	idea	NOUN
ejpam-4476	246	7	of	of	ADP
ejpam-4476	246	8	decomposing	decompose	VERB
ejpam-4476	246	9	into	into	ADP
ejpam-4476	246	10	odd	odd	ADJ
ejpam-4476	246	11	numbers	number	NOUN
ejpam-4476	246	12	and	and	CCONJ
ejpam-4476	246	13	then	then	ADV
ejpam-4476	246	14	summing	sum	VERB
ejpam-4476	246	15	corresponds	correspond	NOUN
ejpam-4476	246	16	to	to	ADP
ejpam-4476	246	17	the	the	DET
ejpam-4476	246	18	procedures	procedure	NOUN
ejpam-4476	246	19	of	of	ADP
ejpam-4476	246	20	analysis	analysis	NOUN
ejpam-4476	246	21	and	and	CCONJ
ejpam-4476	246	22	synthesis	synthesis	NOUN
ejpam-4476	246	23	.	.	PUNCT
ejpam-4476	247	1	if	if	SCONJ
ejpam-4476	247	2	natural	natural	ADJ
ejpam-4476	247	3	numbers	number	NOUN
ejpam-4476	247	4	are	be	AUX
ejpam-4476	247	5	expressed	express	VERB
ejpam-4476	247	6	as	as	ADP
ejpam-4476	247	7	sums	sum	NOUN
ejpam-4476	247	8	of	of	ADP
ejpam-4476	247	9	only	only	ADV
ejpam-4476	247	10	1	1	NUM
ejpam-4476	247	11	,	,	PUNCT
ejpam-4476	247	12	the	the	DET
ejpam-4476	247	13	sums	sum	NOUN
ejpam-4476	247	14	of	of	ADP
ejpam-4476	247	15	square	square	ADJ
ejpam-4476	247	16	numbers	number	NOUN
ejpam-4476	247	17	and	and	CCONJ
ejpam-4476	247	18	of	of	ADP
ejpam-4476	247	19	cubic	cubic	ADJ
ejpam-4476	247	20	numbers	number	NOUN
ejpam-4476	247	21	can	can	AUX
ejpam-4476	247	22	be	be	AUX
ejpam-4476	247	23	related	relate	VERB
ejpam-4476	247	24	to	to	ADP
ejpam-4476	247	25	the	the	DET
ejpam-4476	247	26	multiplication	multiplication	NOUN
ejpam-4476	247	27	of	of	ADP
ejpam-4476	247	28	a	a	DET
ejpam-4476	247	29	vertical	vertical	ADJ
ejpam-4476	247	30	vector	vector	NOUN
ejpam-4476	247	31	with	with	ADP
ejpam-4476	247	32	a	a	DET
ejpam-4476	247	33	horizontal	horizontal	ADJ
ejpam-4476	247	34	one	one	NUM
ejpam-4476	247	35	whose	whose	DET
ejpam-4476	247	36	components	component	NOUN
ejpam-4476	247	37	are	be	AUX
ejpam-4476	247	38	all	all	PRON
ejpam-4476	247	39	1	1	NUM
ejpam-4476	247	40	’s	’s	NOUN
ejpam-4476	247	41	.	.	PUNCT
ejpam-4476	248	1	the	the	DET
ejpam-4476	248	2	sums	sum	NOUN
ejpam-4476	248	3	of	of	ADP
ejpam-4476	248	4	square	square	ADJ
ejpam-4476	248	5	and	and	CCONJ
ejpam-4476	248	6	cubic	cubic	ADJ
ejpam-4476	248	7	numbers	number	NOUN
ejpam-4476	248	8	are	be	AUX
ejpam-4476	248	9	the	the	DET
ejpam-4476	248	10	sums	sum	NOUN
ejpam-4476	248	11	of	of	ADP
ejpam-4476	248	12	the	the	DET
ejpam-4476	248	13	components	component	NOUN
ejpam-4476	248	14	of	of	ADP
ejpam-4476	248	15	the	the	DET
ejpam-4476	248	16	matrices	matrix	NOUN
ejpam-4476	248	17	that	that	PRON
ejpam-4476	248	18	represent	represent	VERB
ejpam-4476	248	19	the	the	DET
ejpam-4476	248	20	products	product	NOUN
ejpam-4476	248	21	of	of	ADP
ejpam-4476	248	22	these	these	DET
ejpam-4476	248	23	multiplications	multiplication	NOUN
ejpam-4476	248	24	.	.	PUNCT
ejpam-4476	249	1	if	if	SCONJ
ejpam-4476	249	2	natural	natural	ADJ
ejpam-4476	249	3	numbers	number	NOUN
ejpam-4476	249	4	are	be	AUX
ejpam-4476	249	5	expressed	express	VERB
ejpam-4476	249	6	as	as	ADP
ejpam-4476	249	7	sums	sum	NOUN
ejpam-4476	249	8	of	of	ADP
ejpam-4476	249	9	only	only	ADV
ejpam-4476	249	10	1	1	NUM
ejpam-4476	249	11	in	in	ADP
ejpam-4476	249	12	the	the	DET
ejpam-4476	249	13	analytical	analytical	ADJ
ejpam-4476	249	14	approach	approach	NOUN
ejpam-4476	249	15	,	,	PUNCT
ejpam-4476	249	16	the	the	DET
ejpam-4476	249	17	sums	sum	NOUN
ejpam-4476	249	18	of	of	ADP
ejpam-4476	249	19	square	square	ADJ
ejpam-4476	249	20	numbers	number	NOUN
ejpam-4476	249	21	and	and	CCONJ
ejpam-4476	249	22	of	of	ADP
ejpam-4476	249	23	cubic	cubic	ADJ
ejpam-4476	249	24	numbers	number	NOUN
ejpam-4476	249	25	can	can	AUX
ejpam-4476	249	26	be	be	AUX
ejpam-4476	249	27	related	relate	VERB
ejpam-4476	249	28	to	to	ADP
ejpam-4476	249	29	the	the	DET
ejpam-4476	249	30	multiplication	multiplication	NOUN
ejpam-4476	249	31	of	of	ADP
ejpam-4476	249	32	a	a	DET
ejpam-4476	249	33	vertical	vertical	ADJ
ejpam-4476	249	34	vector	vector	NOUN
ejpam-4476	249	35	with	with	ADP
ejpam-4476	249	36	a	a	DET
ejpam-4476	249	37	horizontal	horizontal	ADJ
ejpam-4476	249	38	one	one	NUM
ejpam-4476	249	39	whose	whose	DET
ejpam-4476	249	40	components	component	NOUN
ejpam-4476	249	41	are	be	AUX
ejpam-4476	249	42	all	all	PRON
ejpam-4476	249	43	1	1	NUM
ejpam-4476	249	44	’s	’s	NOUN
ejpam-4476	249	45	.	.	PUNCT
ejpam-4476	250	1	the	the	DET
ejpam-4476	250	2	sums	sum	NOUN
ejpam-4476	250	3	of	of	ADP
ejpam-4476	250	4	square	square	ADJ
ejpam-4476	250	5	and	and	CCONJ
ejpam-4476	250	6	of	of	ADP
ejpam-4476	250	7	cubic	cubic	ADJ
ejpam-4476	250	8	numbers	number	NOUN
ejpam-4476	250	9	are	be	AUX
ejpam-4476	250	10	the	the	DET
ejpam-4476	250	11	sums	sum	NOUN
ejpam-4476	250	12	of	of	ADP
ejpam-4476	250	13	the	the	DET
ejpam-4476	250	14	components	component	NOUN
ejpam-4476	250	15	of	of	ADP
ejpam-4476	250	16	the	the	DET
ejpam-4476	250	17	matrices	matrix	NOUN
ejpam-4476	250	18	that	that	PRON
ejpam-4476	250	19	represent	represent	VERB
ejpam-4476	250	20	the	the	DET
ejpam-4476	250	21	products	product	NOUN
ejpam-4476	250	22	references	reference	NOUN
ejpam-4476	250	23	877	877	NUM
ejpam-4476	250	24	of	of	ADP
ejpam-4476	250	25	the	the	DET
ejpam-4476	250	26	above	above	ADJ
ejpam-4476	250	27	multiplications	multiplication	NOUN
ejpam-4476	250	28	.	.	PUNCT
ejpam-4476	251	1	if	if	SCONJ
ejpam-4476	251	2	we	we	PRON
ejpam-4476	251	3	regard	regard	VERB
ejpam-4476	251	4	these	these	DET
ejpam-4476	251	5	products	product	NOUN
ejpam-4476	251	6	as	as	ADP
ejpam-4476	251	7	multiplication	multiplication	NOUN
ejpam-4476	251	8	tables	table	NOUN
ejpam-4476	251	9	,	,	PUNCT
ejpam-4476	251	10	we	we	PRON
ejpam-4476	251	11	can	can	AUX
ejpam-4476	251	12	compute	compute	VERB
ejpam-4476	251	13	the	the	DET
ejpam-4476	251	14	sum	sum	NOUN
ejpam-4476	251	15	of	of	ADP
ejpam-4476	251	16	the	the	DET
ejpam-4476	251	17	components	component	NOUN
ejpam-4476	251	18	by	by	ADP
ejpam-4476	251	19	devising	devise	VERB
ejpam-4476	251	20	a	a	DET
ejpam-4476	251	21	way	way	NOUN
ejpam-4476	251	22	to	to	PART
ejpam-4476	251	23	determine	determine	VERB
ejpam-4476	251	24	the	the	DET
ejpam-4476	251	25	sum	sum	NOUN
ejpam-4476	251	26	of	of	ADP
ejpam-4476	251	27	the	the	DET
ejpam-4476	251	28	odd	odd	ADJ
ejpam-4476	251	29	numbers	number	NOUN
ejpam-4476	251	30	.	.	PUNCT
ejpam-4476	252	1	it	it	PRON
ejpam-4476	252	2	is	be	AUX
ejpam-4476	252	3	educationally	educationally	ADV
ejpam-4476	252	4	effective	effective	ADJ
ejpam-4476	252	5	in	in	ADP
ejpam-4476	252	6	broadening	broaden	VERB
ejpam-4476	252	7	the	the	DET
ejpam-4476	252	8	view	view	NOUN
ejpam-4476	252	9	of	of	ADP
ejpam-4476	252	10	mathematics	mathematic	NOUN
ejpam-4476	252	11	to	to	PART
ejpam-4476	252	12	grasp	grasp	VERB
ejpam-4476	252	13	that	that	SCONJ
ejpam-4476	252	14	matrix	matrix	NOUN
ejpam-4476	252	15	operations	operation	NOUN
ejpam-4476	252	16	in	in	ADP
ejpam-4476	252	17	linear	linear	ADJ
ejpam-4476	252	18	algebra	algebra	NOUN
ejpam-4476	252	19	are	be	AUX
ejpam-4476	252	20	related	relate	VERB
ejpam-4476	252	21	to	to	ADP
ejpam-4476	252	22	the	the	DET
ejpam-4476	252	23	computation	computation	NOUN
ejpam-4476	252	24	of	of	ADP
ejpam-4476	252	25	series	series	NOUN
ejpam-4476	252	26	.	.	PUNCT
ejpam-4476	253	1	acknowledgements	acknowledgement	VERB
ejpam-4476	253	2	the	the	DET
ejpam-4476	253	3	author	author	NOUN
ejpam-4476	253	4	would	would	AUX
ejpam-4476	253	5	like	like	VERB
ejpam-4476	253	6	to	to	PART
ejpam-4476	253	7	thank	thank	VERB
ejpam-4476	253	8	editage	editage	NOUN
ejpam-4476	253	9	(	(	PUNCT
ejpam-4476	253	10	www.editage.jp	www.editage.jp	ADV
ejpam-4476	253	11	)	)	PUNCT
ejpam-4476	253	12	for	for	ADP
ejpam-4476	253	13	english	english	ADJ
ejpam-4476	253	14	language	language	NOUN
ejpam-4476	253	15	editing	editing	NOUN
ejpam-4476	253	16	.	.	PUNCT
ejpam-4476	254	1	references	reference	NOUN
ejpam-4476	254	2	[	[	X
ejpam-4476	254	3	1	1	NUM
ejpam-4476	254	4	]	]	PUNCT
ejpam-4476	254	5	c.	c.	PROPN
ejpam-4476	254	6	alsina	alsina	PROPN
ejpam-4476	254	7	and	and	CCONJ
ejpam-4476	254	8	r.	r.	PROPN
ejpam-4476	254	9	b.	b.	PROPN
ejpam-4476	254	10	nelsen	nelsen	PROPN
ejpam-4476	254	11	.	.	PUNCT
ejpam-4476	255	1	an	an	DET
ejpam-4476	255	2	invitation	invitation	NOUN
ejpam-4476	255	3	to	to	AUX
ejpam-4476	255	4	proofs	proof	NOUN
ejpam-4476	255	5	without	without	ADP
ejpam-4476	255	6	words	word	NOUN
ejpam-4476	255	7	.	.	PUNCT
ejpam-4476	256	1	european	european	ADJ
ejpam-4476	256	2	journal	journal	PROPN
ejpam-4476	256	3	of	of	ADP
ejpam-4476	256	4	pure	pure	ADJ
ejpam-4476	256	5	and	and	CCONJ
ejpam-4476	256	6	applied	applied	ADJ
ejpam-4476	256	7	mathematic	mathematic	ADJ
ejpam-4476	256	8	,	,	PUNCT
ejpam-4476	256	9	3:118–127	3:118–127	NUM
ejpam-4476	256	10	,	,	PUNCT
ejpam-4476	256	11	2010	2010	NUM
ejpam-4476	256	12	.	.	PUNCT
ejpam-4476	257	1	[	[	X
ejpam-4476	257	2	2	2	X
ejpam-4476	257	3	]	]	PUNCT
ejpam-4476	257	4	p.	p.	NOUN
ejpam-4476	257	5	holmes	holme	NOUN
ejpam-4476	257	6	.	.	PUNCT
ejpam-4476	258	1	proof	proof	NOUN
ejpam-4476	258	2	without	without	ADP
ejpam-4476	258	3	words	word	NOUN
ejpam-4476	258	4	:	:	PUNCT
ejpam-4476	258	5	n∑	n∑	NOUN
ejpam-4476	258	6	k=1	k=1	PROPN
ejpam-4476	259	1	k3	k3	PROPN
ejpam-4476	259	2	=	=	SYM
ejpam-4476	259	3	(	(	PUNCT
ejpam-4476	259	4	n∑	n∑	INTJ
ejpam-4476	259	5	k=1	k=1	PUNCT
ejpam-4476	259	6	k	k	X
ejpam-4476	259	7	)	)	PUNCT
ejpam-4476	259	8	2	2	X
ejpam-4476	259	9	.	.	PUNCT
ejpam-4476	260	1	mathematical	mathematical	ADJ
ejpam-4476	260	2	gazette	gazette	NOUN
ejpam-4476	260	3	,	,	PUNCT
ejpam-4476	260	4	86:267	86:267	NUM
ejpam-4476	260	5	–	–	PUNCT
ejpam-4476	260	6	268	268	NUM
ejpam-4476	260	7	,	,	PUNCT
ejpam-4476	260	8	2002	2002	NUM
ejpam-4476	260	9	.	.	PUNCT
ejpam-4476	261	1	[	[	X
ejpam-4476	261	2	3	3	X
ejpam-4476	261	3	]	]	X
ejpam-4476	261	4	y	y	PROPN
ejpam-4476	261	5	kobayashi	kobayashi	PROPN
ejpam-4476	261	6	.	.	PUNCT
ejpam-4476	262	1	geometrical	geometrical	ADJ
ejpam-4476	262	2	meaning	meaning	NOUN
ejpam-4476	262	3	of	of	ADP
ejpam-4476	262	4	arithmetic	arithmetic	ADJ
ejpam-4476	262	5	series	series	NOUN
ejpam-4476	262	6	n∑	n∑	PROPN
ejpam-4476	262	7	k=1	k=1	PROPN
ejpam-4476	263	1	k	k	X
ejpam-4476	263	2	,	,	PUNCT
ejpam-4476	263	3	n∑	n∑	NOUN
ejpam-4476	263	4	k=1	k=1	PROPN
ejpam-4476	263	5	k2	k2	PROPN
ejpam-4476	263	6	,	,	PUNCT
ejpam-4476	263	7	and	and	CCONJ
ejpam-4476	263	8	n∑	n∑	NOUN
ejpam-4476	263	9	k=1	k=1	PROPN
ejpam-4476	264	1	k3	k3	VERB
ejpam-4476	264	2	in	in	ADP
ejpam-4476	264	3	terms	term	NOUN
ejpam-4476	264	4	of	of	ADP
ejpam-4476	264	5	the	the	DET
ejpam-4476	264	6	elementary	elementary	ADJ
ejpam-4476	264	7	combinatorics	combinatoric	NOUN
ejpam-4476	264	8	.	.	PUNCT
ejpam-4476	265	1	international	international	ADJ
ejpam-4476	265	2	journal	journal	PROPN
ejpam-4476	265	3	of	of	ADP
ejpam-4476	265	4	mathematical	mathematical	ADJ
ejpam-4476	265	5	education	education	NOUN
ejpam-4476	265	6	in	in	ADP
ejpam-4476	265	7	science	science	NOUN
ejpam-4476	265	8	and	and	CCONJ
ejpam-4476	265	9	technology	technology	NOUN
ejpam-4476	265	10	,	,	PUNCT
ejpam-4476	265	11	42:657–664	42:657–664	PROPN
ejpam-4476	265	12	,	,	PUNCT
ejpam-4476	265	13	2011	2011	NUM
ejpam-4476	265	14	.	.	PUNCT
ejpam-4476	266	1	[	[	X
ejpam-4476	266	2	4	4	X
ejpam-4476	266	3	]	]	X
ejpam-4476	266	4	y.	y.	PROPN
ejpam-4476	266	5	kobayashi	kobayashi	PROPN
ejpam-4476	266	6	.	.	PUNCT
ejpam-4476	267	1	integral	integral	ADJ
ejpam-4476	267	2	representation	representation	NOUN
ejpam-4476	267	3	of	of	ADP
ejpam-4476	267	4	the	the	DET
ejpam-4476	267	5	pictorial	pictorial	ADJ
ejpam-4476	267	6	proof	proof	NOUN
ejpam-4476	267	7	of	of	ADP
ejpam-4476	267	8	n∑	n∑	PROPN
ejpam-4476	267	9	k=1	k=1	PROPN
ejpam-4476	267	10	k2	k2	PROPN
ejpam-4476	267	11	=	=	NOUN
ejpam-4476	267	12	1	1	NUM
ejpam-4476	267	13	6	6	NUM
ejpam-4476	267	14	n(n+1)(2n+	n(n+1)(2n+	ADP
ejpam-4476	267	15	1	1	NUM
ejpam-4476	267	16	)	)	PUNCT
ejpam-4476	267	17	.	.	PUNCT
ejpam-4476	268	1	international	international	ADJ
ejpam-4476	268	2	journal	journal	PROPN
ejpam-4476	268	3	of	of	ADP
ejpam-4476	268	4	mathematical	mathematical	ADJ
ejpam-4476	268	5	education	education	NOUN
ejpam-4476	268	6	in	in	ADP
ejpam-4476	268	7	science	science	NOUN
ejpam-4476	268	8	and	and	CCONJ
ejpam-4476	268	9	technology	technology	NOUN
ejpam-4476	268	10	,	,	PUNCT
ejpam-4476	268	11	42:235–239	42:235–239	NUM
ejpam-4476	268	12	,	,	PUNCT
ejpam-4476	268	13	2011	2011	NUM
ejpam-4476	268	14	.	.	PUNCT
ejpam-4476	269	1	[	[	X
ejpam-4476	269	2	5	5	X
ejpam-4476	269	3	]	]	X
ejpam-4476	269	4	y.	y.	PROPN
ejpam-4476	269	5	kobayashi	kobayashi	PROPN
ejpam-4476	269	6	.	.	PUNCT
ejpam-4476	270	1	recursion	recursion	NOUN
ejpam-4476	270	2	formulae	formulae	NOUN
ejpam-4476	270	3	of	of	ADP
ejpam-4476	270	4	1ℓ+2ℓ+	1ℓ+2ℓ+	NUM
ejpam-4476	270	5	·	·	PUNCT
ejpam-4476	270	6	·	·	PUNCT
ejpam-4476	271	1	·	·	PUNCT
ejpam-4476	271	2	+	+	NOUN
ejpam-4476	271	3	nℓ	nℓ	NOUN
ejpam-4476	271	4	and	and	CCONJ
ejpam-4476	271	5	their	their	PRON
ejpam-4476	271	6	combinatoric	combinatoric	ADJ
ejpam-4476	271	7	meaning	meaning	NOUN
ejpam-4476	271	8	in	in	ADP
ejpam-4476	271	9	terms	term	NOUN
ejpam-4476	271	10	of	of	ADP
ejpam-4476	271	11	the	the	DET
ejpam-4476	271	12	tree	tree	NOUN
ejpam-4476	271	13	diagrams	diagram	NOUN
ejpam-4476	271	14	.	.	PUNCT
ejpam-4476	272	1	international	international	ADJ
ejpam-4476	272	2	journal	journal	PROPN
ejpam-4476	272	3	of	of	ADP
ejpam-4476	272	4	mathematical	mathematical	ADJ
ejpam-4476	272	5	education	education	NOUN
ejpam-4476	272	6	in	in	ADP
ejpam-4476	272	7	science	science	NOUN
ejpam-4476	272	8	and	and	CCONJ
ejpam-4476	272	9	technology	technology	NOUN
ejpam-4476	272	10	,	,	PUNCT
ejpam-4476	272	11	44:132–142	44:132–142	PROPN
ejpam-4476	272	12	,	,	PUNCT
ejpam-4476	272	13	2013	2013	NUM
ejpam-4476	272	14	.	.	PUNCT
ejpam-4476	273	1	[	[	X
ejpam-4476	273	2	6	6	NUM
ejpam-4476	273	3	]	]	X
ejpam-4476	273	4	y.	y.	PROPN
ejpam-4476	273	5	kobayashi	kobayashi	PROPN
ejpam-4476	273	6	.	.	PUNCT
ejpam-4476	274	1	derivation	derivation	NOUN
ejpam-4476	274	2	of	of	ADP
ejpam-4476	274	3	some	some	DET
ejpam-4476	274	4	formulae	formulae	NOUN
ejpam-4476	274	5	in	in	ADP
ejpam-4476	274	6	combinatorics	combinatoric	NOUN
ejpam-4476	274	7	by	by	ADP
ejpam-4476	274	8	heuristic	heuristic	ADJ
ejpam-4476	274	9	method	method	NOUN
ejpam-4476	274	10	.	.	PUNCT
ejpam-4476	275	1	international	international	ADJ
ejpam-4476	275	2	journal	journal	PROPN
ejpam-4476	275	3	of	of	ADP
ejpam-4476	275	4	mathematical	mathematical	ADJ
ejpam-4476	275	5	education	education	NOUN
ejpam-4476	275	6	in	in	ADP
ejpam-4476	275	7	science	science	NOUN
ejpam-4476	275	8	and	and	CCONJ
ejpam-4476	275	9	technology	technology	NOUN
ejpam-4476	275	10	,	,	PUNCT
ejpam-4476	275	11	46:469	46:469	NUM
ejpam-4476	275	12	–	–	PUNCT
ejpam-4476	275	13	476	476	NUM
ejpam-4476	275	14	,	,	PUNCT
ejpam-4476	275	15	2015	2015	NUM
ejpam-4476	275	16	.	.	PUNCT
ejpam-4476	276	1	[	[	X
ejpam-4476	276	2	erratum	erratum	PROPN
ejpam-4476	276	3	46:795	46:795	NUM
ejpam-4476	276	4	,	,	PUNCT
ejpam-4476	276	5	2015	2015	NUM
ejpam-4476	276	6	]	]	PUNCT
ejpam-4476	276	7	.	.	PUNCT
ejpam-4476	277	1	[	[	X
ejpam-4476	277	2	7	7	X
ejpam-4476	277	3	]	]	X
ejpam-4476	277	4	y.	y.	PROPN
ejpam-4476	277	5	kobayashi	kobayashi	PROPN
ejpam-4476	277	6	.	.	PUNCT
ejpam-4476	278	1	proof	proof	NOUN
ejpam-4476	278	2	without	without	ADP
ejpam-4476	278	3	words	word	NOUN
ejpam-4476	278	4	:	:	PUNCT
ejpam-4476	278	5	relationship	relationship	NOUN
ejpam-4476	278	6	between	between	ADP
ejpam-4476	278	7	combination	combination	NOUN
ejpam-4476	278	8	nc2	nc2	NOUN
ejpam-4476	278	9	and	and	CCONJ
ejpam-4476	278	10	arithmetic	arithmetic	ADJ
ejpam-4476	278	11	series	series	NOUN
ejpam-4476	278	12	.	.	PUNCT
ejpam-4476	279	1	far	far	PROPN
ejpam-4476	279	2	east	east	PROPN
ejpam-4476	279	3	journal	journal	PROPN
ejpam-4476	279	4	of	of	ADP
ejpam-4476	279	5	mathematical	mathematical	ADJ
ejpam-4476	279	6	education	education	NOUN
ejpam-4476	279	7	,	,	PUNCT
ejpam-4476	279	8	19:57–58	19:57–58	NUM
ejpam-4476	279	9	,	,	PUNCT
ejpam-4476	279	10	2019	2019	NUM
ejpam-4476	279	11	.	.	PUNCT
ejpam-4476	280	1	[	[	X
ejpam-4476	280	2	8	8	NUM
ejpam-4476	280	3	]	]	X
ejpam-4476	280	4	y.	y.	PROPN
ejpam-4476	280	5	kobayashi	kobayashi	PROPN
ejpam-4476	280	6	.	.	PUNCT
ejpam-4476	281	1	proof	proof	NOUN
ejpam-4476	281	2	without	without	ADP
ejpam-4476	281	3	words	word	NOUN
ejpam-4476	281	4	:	:	PUNCT
ejpam-4476	281	5	relationship	relationship	NOUN
ejpam-4476	281	6	between	between	ADP
ejpam-4476	281	7	combination	combination	NOUN
ejpam-4476	281	8	(	(	PUNCT
ejpam-4476	281	9	nc2	nc2	NOUN
ejpam-4476	281	10	)	)	PUNCT
ejpam-4476	281	11	2	2	NUM
ejpam-4476	281	12	and	and	CCONJ
ejpam-4476	281	13	sum	sum	NOUN
ejpam-4476	281	14	of	of	ADP
ejpam-4476	281	15	cubic	cubic	ADJ
ejpam-4476	281	16	numbers	number	NOUN
ejpam-4476	281	17	.	.	PUNCT
ejpam-4476	282	1	far	far	PROPN
ejpam-4476	282	2	east	east	PROPN
ejpam-4476	282	3	journal	journal	PROPN
ejpam-4476	282	4	of	of	ADP
ejpam-4476	282	5	mathematical	mathematical	ADJ
ejpam-4476	282	6	education	education	NOUN
ejpam-4476	282	7	,	,	PUNCT
ejpam-4476	282	8	20:79–82	20:79–82	NUM
ejpam-4476	282	9	,	,	PUNCT
ejpam-4476	282	10	2020	2020	NUM
ejpam-4476	282	11	.	.	PUNCT
ejpam-4476	283	1	[	[	X
ejpam-4476	283	2	9	9	NUM
ejpam-4476	283	3	]	]	PUNCT
ejpam-4476	283	4	r.	r.	PROPN
ejpam-4476	283	5	b.	b.	PROPN
ejpam-4476	283	6	nelsen	nelsen	PROPN
ejpam-4476	283	7	.	.	PUNCT
ejpam-4476	284	1	proofs	proof	NOUN
ejpam-4476	284	2	without	without	ADP
ejpam-4476	284	3	words	word	NOUN
ejpam-4476	284	4	.	.	PUNCT
ejpam-4476	285	1	the	the	DET
ejpam-4476	285	2	mathematical	mathematical	ADJ
ejpam-4476	285	3	association	association	PROPN
ejpam-4476	285	4	of	of	ADP
ejpam-4476	285	5	america	america	PROPN
ejpam-4476	285	6	,	,	PUNCT
ejpam-4476	285	7	1993	1993	NUM
ejpam-4476	285	8	.	.	PUNCT
ejpam-4476	286	1	[	[	X
ejpam-4476	286	2	10	10	NUM
ejpam-4476	286	3	]	]	X
ejpam-4476	286	4	r.	r.	PROPN
ejpam-4476	286	5	b.	b.	PROPN
ejpam-4476	286	6	nelsen	nelsen	PROPN
ejpam-4476	286	7	.	.	PUNCT
ejpam-4476	287	1	proofs	proof	NOUN
ejpam-4476	287	2	without	without	ADP
ejpam-4476	287	3	words	word	NOUN
ejpam-4476	287	4	ii	ii	PROPN
ejpam-4476	287	5	.	.	PUNCT
ejpam-4476	288	1	the	the	DET
ejpam-4476	288	2	mathematical	mathematical	ADJ
ejpam-4476	288	3	association	association	PROPN
ejpam-4476	288	4	of	of	ADP
ejpam-4476	288	5	america	america	PROPN
ejpam-4476	288	6	,	,	PUNCT
ejpam-4476	288	7	2000	2000	NUM
ejpam-4476	288	8	.	.	PUNCT