id	sid	tid	token	lemma	pos
ap-2064	1	1	acta	acta	PROPN
ap-2064	1	2	polytechnica	polytechnica	PROPN
ap-2064	1	3	doi:10.14311	doi:10.14311	PROPN
ap-2064	1	4	/	/	SYM
ap-2064	1	5	ap.2015.55.0050	ap.2015.55.0050	PROPN
ap-2064	1	6	acta	acta	PROPN
ap-2064	1	7	polytechnica	polytechnica	PROPN
ap-2064	1	8	55(1):50–58	55(1):50–58	PROPN
ap-2064	1	9	,	,	PUNCT
ap-2064	1	10	2015	2015	NUM
ap-2064	1	11	©	©	PROPN
ap-2064	1	12	czech	czech	PROPN
ap-2064	1	13	technical	technical	PROPN
ap-2064	1	14	university	university	PROPN
ap-2064	1	15	in	in	ADP
ap-2064	1	16	prague	prague	PROPN
ap-2064	1	17	,	,	PUNCT
ap-2064	1	18	2015	2015	NUM
ap-2064	1	19	available	available	ADJ
ap-2064	1	20	online	online	ADV
ap-2064	1	21	at	at	ADP
ap-2064	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-2064	1	23	biperiodic	biperiodic	ADJ
ap-2064	1	24	fibonacci	fibonacci	NOUN
ap-2064	1	25	word	word	NOUN
ap-2064	1	26	and	and	CCONJ
ap-2064	1	27	its	its	PRON
ap-2064	1	28	fractal	fractal	ADJ
ap-2064	1	29	curve	curve	NOUN
ap-2064	1	30	josé	josé	PROPN
ap-2064	1	31	l.	l.	PROPN
ap-2064	1	32	ramíreza,∗	ramíreza,∗	PROPN
ap-2064	1	33	,	,	PUNCT
ap-2064	1	34	gustavo	gustavo	PROPN
ap-2064	1	35	n.	n.	PROPN
ap-2064	1	36	rubianob	rubianob	VERB
ap-2064	1	37	a	a	DET
ap-2064	1	38	departamento	departamento	NOUN
ap-2064	1	39	de	de	PROPN
ap-2064	1	40	matemáticas	matemáticas	NOUN
ap-2064	1	41	,	,	PUNCT
ap-2064	1	42	universidad	universidad	PROPN
ap-2064	1	43	sergio	sergio	PROPN
ap-2064	1	44	arboleda	arboleda	PROPN
ap-2064	1	45	,	,	PUNCT
ap-2064	1	46	bogotá	bogotá	NOUN
ap-2064	1	47	,	,	PUNCT
ap-2064	1	48	colombia	colombia	PROPN
ap-2064	1	49	b	b	PROPN
ap-2064	1	50	departamento	departamento	PROPN
ap-2064	1	51	de	de	PROPN
ap-2064	1	52	matemáticas	matemáticas	PROPN
ap-2064	1	53	,	,	PUNCT
ap-2064	1	54	universidad	universidad	PROPN
ap-2064	1	55	nacional	nacional	PROPN
ap-2064	1	56	de	de	X
ap-2064	1	57	colombia	colombia	PROPN
ap-2064	1	58	,	,	PUNCT
ap-2064	1	59	bogotá	bogotá	NOUN
ap-2064	1	60	,	,	PUNCT
ap-2064	1	61	colombia	colombia	PROPN
ap-2064	1	62	∗	∗	NOUN
ap-2064	1	63	corresponding	correspond	VERB
ap-2064	1	64	author	author	NOUN
ap-2064	1	65	:	:	PUNCT
ap-2064	1	66	josel.ramirez@ima.usergioarboleda.edu.co	josel.ramirez@ima.usergioarboleda.edu.co	ADJ
ap-2064	1	67	abstract	abstract	NOUN
ap-2064	1	68	.	.	PUNCT
ap-2064	2	1	in	in	ADP
ap-2064	2	2	the	the	DET
ap-2064	2	3	present	present	ADJ
ap-2064	2	4	article	article	NOUN
ap-2064	2	5	,	,	PUNCT
ap-2064	2	6	we	we	PRON
ap-2064	2	7	study	study	VERB
ap-2064	2	8	a	a	DET
ap-2064	2	9	word	word	NOUN
ap-2064	2	10	-	-	PUNCT
ap-2064	2	11	combinatorial	combinatorial	ADJ
ap-2064	2	12	interpretation	interpretation	NOUN
ap-2064	2	13	of	of	ADP
ap-2064	2	14	the	the	DET
ap-2064	2	15	biperiodic	biperiodic	ADJ
ap-2064	2	16	fibonacci	fibonacci	PROPN
ap-2064	2	17	sequence	sequence	NOUN
ap-2064	2	18	of	of	ADP
ap-2064	2	19	integer	integer	NOUN
ap-2064	2	20	numbers	number	NOUN
ap-2064	2	21	(	(	PUNCT
ap-2064	2	22	f	f	X
ap-2064	2	23	(	(	PUNCT
ap-2064	2	24	a	a	DET
ap-2064	2	25	,	,	PUNCT
ap-2064	2	26	b	b	NOUN
ap-2064	2	27	)	)	PUNCT
ap-2064	2	28	n	n	NOUN
ap-2064	2	29	)	)	PUNCT
ap-2064	2	30	.	.	PUNCT
ap-2064	3	1	this	this	DET
ap-2064	3	2	sequence	sequence	NOUN
ap-2064	3	3	is	be	AUX
ap-2064	3	4	defined	define	VERB
ap-2064	3	5	by	by	ADP
ap-2064	3	6	the	the	DET
ap-2064	3	7	recurrence	recurrence	NOUN
ap-2064	3	8	relation	relation	PROPN
ap-2064	3	9	af	af	PROPN
ap-2064	3	10	(	(	PUNCT
ap-2064	3	11	a	a	DET
ap-2064	3	12	,	,	PUNCT
ap-2064	3	13	b	b	NOUN
ap-2064	3	14	)	)	PUNCT
ap-2064	3	15	n−1	n−1	PROPN
ap-2064	4	1	+	+	NUM
ap-2064	4	2	f	f	X
ap-2064	4	3	(	(	PUNCT
ap-2064	4	4	a	a	DET
ap-2064	4	5	,	,	PUNCT
ap-2064	4	6	b	b	NOUN
ap-2064	4	7	)	)	PUNCT
ap-2064	4	8	n−2	n−2	PROPN
ap-2064	4	9	if	if	SCONJ
ap-2064	4	10	n	n	PRON
ap-2064	4	11	is	be	AUX
ap-2064	4	12	even	even	ADV
ap-2064	4	13	and	and	CCONJ
ap-2064	4	14	bf	bf	X
ap-2064	4	15	(	(	PUNCT
ap-2064	4	16	a	a	DET
ap-2064	4	17	,	,	PUNCT
ap-2064	4	18	b	b	NOUN
ap-2064	4	19	)	)	PUNCT
ap-2064	4	20	n−1	n−1	PROPN
ap-2064	5	1	+	+	NUM
ap-2064	5	2	f	f	X
ap-2064	5	3	(	(	PUNCT
ap-2064	5	4	a	a	DET
ap-2064	5	5	,	,	PUNCT
ap-2064	5	6	b	b	NOUN
ap-2064	5	7	)	)	PUNCT
ap-2064	5	8	n−2	n−2	PROPN
ap-2064	5	9	if	if	SCONJ
ap-2064	5	10	n	n	PRON
ap-2064	5	11	is	be	AUX
ap-2064	5	12	odd	odd	ADJ
ap-2064	5	13	,	,	PUNCT
ap-2064	5	14	where	where	SCONJ
ap-2064	5	15	a	a	PRON
ap-2064	5	16	and	and	CCONJ
ap-2064	5	17	b	b	NOUN
ap-2064	5	18	are	be	AUX
ap-2064	5	19	any	any	DET
ap-2064	5	20	real	real	ADJ
ap-2064	5	21	numbers	number	NOUN
ap-2064	5	22	.	.	PUNCT
ap-2064	6	1	this	this	DET
ap-2064	6	2	sequence	sequence	NOUN
ap-2064	6	3	of	of	ADP
ap-2064	6	4	integers	integer	NOUN
ap-2064	6	5	is	be	AUX
ap-2064	6	6	associated	associate	VERB
ap-2064	6	7	with	with	ADP
ap-2064	6	8	a	a	DET
ap-2064	6	9	family	family	NOUN
ap-2064	6	10	of	of	ADP
ap-2064	6	11	finite	finite	ADJ
ap-2064	6	12	binary	binary	ADJ
ap-2064	6	13	words	word	NOUN
ap-2064	6	14	,	,	PUNCT
ap-2064	6	15	called	call	VERB
ap-2064	6	16	finite	finite	PROPN
ap-2064	6	17	biperiodic	biperiodic	PROPN
ap-2064	6	18	fibonacci	fibonacci	PROPN
ap-2064	6	19	words	word	NOUN
ap-2064	6	20	.	.	PUNCT
ap-2064	7	1	we	we	PRON
ap-2064	7	2	study	study	VERB
ap-2064	7	3	several	several	ADJ
ap-2064	7	4	properties	property	NOUN
ap-2064	7	5	,	,	PUNCT
ap-2064	7	6	such	such	ADJ
ap-2064	7	7	as	as	ADP
ap-2064	7	8	the	the	DET
ap-2064	7	9	number	number	NOUN
ap-2064	7	10	of	of	ADP
ap-2064	7	11	occurrences	occurrence	NOUN
ap-2064	7	12	of	of	ADP
ap-2064	7	13	0	0	NUM
ap-2064	7	14	and	and	CCONJ
ap-2064	7	15	1	1	NUM
ap-2064	7	16	,	,	PUNCT
ap-2064	7	17	and	and	CCONJ
ap-2064	7	18	the	the	DET
ap-2064	7	19	concatenation	concatenation	NOUN
ap-2064	7	20	of	of	ADP
ap-2064	7	21	these	these	DET
ap-2064	7	22	words	word	NOUN
ap-2064	7	23	,	,	PUNCT
ap-2064	7	24	among	among	ADP
ap-2064	7	25	others	other	NOUN
ap-2064	7	26	.	.	PUNCT
ap-2064	8	1	we	we	PRON
ap-2064	8	2	also	also	ADV
ap-2064	8	3	study	study	VERB
ap-2064	8	4	the	the	DET
ap-2064	8	5	infinite	infinite	PROPN
ap-2064	8	6	biperiodic	biperiodic	ADJ
ap-2064	8	7	fibonacci	fibonacci	NOUN
ap-2064	8	8	word	word	NOUN
ap-2064	8	9	,	,	PUNCT
ap-2064	8	10	which	which	PRON
ap-2064	8	11	is	be	AUX
ap-2064	8	12	the	the	DET
ap-2064	8	13	limiting	limit	VERB
ap-2064	8	14	sequence	sequence	NOUN
ap-2064	8	15	of	of	ADP
ap-2064	8	16	finite	finite	PROPN
ap-2064	8	17	biperiodic	biperiodic	PROPN
ap-2064	8	18	fibonacci	fibonacci	NOUN
ap-2064	8	19	words	word	NOUN
ap-2064	8	20	.	.	PUNCT
ap-2064	9	1	it	it	PRON
ap-2064	9	2	turns	turn	VERB
ap-2064	9	3	out	out	ADP
ap-2064	9	4	that	that	SCONJ
ap-2064	9	5	this	this	DET
ap-2064	9	6	family	family	NOUN
ap-2064	9	7	of	of	ADP
ap-2064	9	8	infinite	infinite	ADJ
ap-2064	9	9	words	word	NOUN
ap-2064	9	10	are	be	AUX
ap-2064	9	11	sturmian	sturmian	ADJ
ap-2064	9	12	words	word	NOUN
ap-2064	9	13	of	of	ADP
ap-2064	9	14	the	the	DET
ap-2064	9	15	slope	slope	NOUN
ap-2064	10	1	[	[	X
ap-2064	10	2	0	0	NUM
ap-2064	10	3	,	,	PUNCT
ap-2064	10	4	a	a	DET
ap-2064	10	5	,	,	PUNCT
ap-2064	10	6	b	b	NOUN
ap-2064	10	7	]	]	X
ap-2064	10	8	.	.	PUNCT
ap-2064	11	1	finally	finally	ADV
ap-2064	11	2	,	,	PUNCT
ap-2064	11	3	we	we	PRON
ap-2064	11	4	associate	associate	VERB
ap-2064	11	5	to	to	ADP
ap-2064	11	6	this	this	DET
ap-2064	11	7	family	family	NOUN
ap-2064	11	8	of	of	ADP
ap-2064	11	9	words	word	NOUN
ap-2064	11	10	a	a	DET
ap-2064	11	11	family	family	NOUN
ap-2064	11	12	of	of	ADP
ap-2064	11	13	curves	curve	NOUN
ap-2064	11	14	with	with	ADP
ap-2064	11	15	interesting	interesting	ADJ
ap-2064	11	16	patterns	pattern	NOUN
ap-2064	11	17	.	.	PUNCT
ap-2064	12	1	keywords	keyword	NOUN
ap-2064	12	2	:	:	PUNCT
ap-2064	12	3	fibonacci	fibonacci	NOUN
ap-2064	12	4	word	word	NOUN
ap-2064	12	5	,	,	PUNCT
ap-2064	12	6	biperiodic	biperiodic	ADJ
ap-2064	12	7	fibonacci	fibonacci	NOUN
ap-2064	12	8	word	word	NOUN
ap-2064	12	9	,	,	PUNCT
ap-2064	12	10	biperiodic	biperiodic	PROPN
ap-2064	12	11	fibonacci	fibonacci	NOUN
ap-2064	12	12	curve	curve	NOUN
ap-2064	12	13	.	.	PUNCT
ap-2064	13	1	1	1	X
ap-2064	13	2	.	.	X
ap-2064	13	3	introduction	introduction	NOUN
ap-2064	13	4	the	the	DET
ap-2064	13	5	fibonacci	fibonacci	NOUN
ap-2064	13	6	numbers	number	NOUN
ap-2064	13	7	and	and	CCONJ
ap-2064	13	8	their	their	PRON
ap-2064	13	9	generalizations	generalization	NOUN
ap-2064	13	10	have	have	VERB
ap-2064	13	11	many	many	ADJ
ap-2064	13	12	interesting	interesting	ADJ
ap-2064	13	13	properties	property	NOUN
ap-2064	13	14	and	and	CCONJ
ap-2064	13	15	combinatorial	combinatorial	ADJ
ap-2064	13	16	interpretations	interpretation	NOUN
ap-2064	13	17	,	,	PUNCT
ap-2064	13	18	see	see	VERB
ap-2064	13	19	,	,	PUNCT
ap-2064	13	20	e.g.	e.g.	ADV
ap-2064	13	21	,	,	PUNCT
ap-2064	13	22	[	[	X
ap-2064	13	23	13	13	NUM
ap-2064	13	24	]	]	PUNCT
ap-2064	13	25	.	.	PUNCT
ap-2064	14	1	the	the	DET
ap-2064	14	2	fibonacci	fibonacci	NOUN
ap-2064	14	3	numbers	number	NOUN
ap-2064	14	4	fn	fn	VERB
ap-2064	14	5	are	be	AUX
ap-2064	14	6	defined	define	VERB
ap-2064	14	7	by	by	ADP
ap-2064	14	8	the	the	DET
ap-2064	14	9	recurrence	recurrence	NOUN
ap-2064	14	10	relation	relation	NOUN
ap-2064	14	11	fn	fn	PROPN
ap-2064	15	1	=	=	PUNCT
ap-2064	15	2	fn−1	fn−1	PROPN
ap-2064	15	3	+	+	CCONJ
ap-2064	15	4	fn−2	fn−2	ADJ
ap-2064	15	5	,	,	PUNCT
ap-2064	15	6	for	for	ADP
ap-2064	15	7	all	all	DET
ap-2064	15	8	integer	integer	NOUN
ap-2064	15	9	n	n	PRON
ap-2064	15	10	≥	≥	NOUN
ap-2064	15	11	2	2	NUM
ap-2064	15	12	,	,	PUNCT
ap-2064	15	13	and	and	CCONJ
ap-2064	15	14	with	with	ADP
ap-2064	15	15	initial	initial	ADJ
ap-2064	15	16	values	value	NOUN
ap-2064	15	17	f0	f0	PROPN
ap-2064	15	18	=	=	SYM
ap-2064	15	19	1	1	NUM
ap-2064	15	20	=	=	SYM
ap-2064	15	21	f1	f1	NOUN
ap-2064	15	22	.	.	PUNCT
ap-2064	16	1	many	many	ADJ
ap-2064	16	2	kinds	kind	NOUN
ap-2064	16	3	of	of	ADP
ap-2064	16	4	generalizations	generalization	NOUN
ap-2064	16	5	of	of	ADP
ap-2064	16	6	the	the	DET
ap-2064	16	7	fibonacci	fibonacci	NOUN
ap-2064	16	8	sequence	sequence	NOUN
ap-2064	16	9	have	have	AUX
ap-2064	16	10	been	be	AUX
ap-2064	16	11	presented	present	VERB
ap-2064	16	12	in	in	ADP
ap-2064	16	13	the	the	DET
ap-2064	16	14	literature	literature	NOUN
ap-2064	16	15	.	.	PUNCT
ap-2064	17	1	for	for	ADP
ap-2064	17	2	example	example	NOUN
ap-2064	17	3	,	,	PUNCT
ap-2064	17	4	edson	edson	PROPN
ap-2064	17	5	and	and	CCONJ
ap-2064	17	6	yayenie	yayenie	PROPN
ap-2064	18	1	[	[	X
ap-2064	18	2	12	12	NUM
ap-2064	18	3	]	]	PUNCT
ap-2064	18	4	introduced	introduce	VERB
ap-2064	18	5	the	the	DET
ap-2064	18	6	biperiodic	biperiodic	ADJ
ap-2064	18	7	fibonacci	fibonacci	NOUN
ap-2064	18	8	sequence	sequence	NOUN
ap-2064	18	9	{	{	PUNCT
ap-2064	18	10	f	f	PROPN
ap-2064	18	11	(	(	PUNCT
ap-2064	18	12	a	a	DET
ap-2064	18	13	,	,	PUNCT
ap-2064	18	14	b	b	NOUN
ap-2064	18	15	)	)	PUNCT
ap-2064	18	16	n	n	CCONJ
ap-2064	18	17	}	}	PUNCT
ap-2064	18	18	n∈n	n∈n	PROPN
ap-2064	18	19	.	.	PUNCT
ap-2064	19	1	for	for	ADP
ap-2064	19	2	any	any	DET
ap-2064	19	3	two	two	NUM
ap-2064	19	4	nonzero	nonzero	ADJ
ap-2064	19	5	real	real	ADJ
ap-2064	19	6	numbers	number	NOUN
ap-2064	19	7	a	a	PRON
ap-2064	19	8	and	and	CCONJ
ap-2064	19	9	b	b	NOUN
ap-2064	19	10	,	,	PUNCT
ap-2064	19	11	this	this	PRON
ap-2064	19	12	is	be	AUX
ap-2064	19	13	defined	define	VERB
ap-2064	19	14	recursively	recursively	ADV
ap-2064	19	15	by	by	ADP
ap-2064	19	16	f	f	PROPN
ap-2064	19	17	(	(	PUNCT
ap-2064	19	18	a	a	DET
ap-2064	19	19	,	,	PUNCT
ap-2064	19	20	b	b	NOUN
ap-2064	19	21	)	)	PUNCT
ap-2064	19	22	0	0	NUM
ap-2064	20	1	=	=	SYM
ap-2064	20	2	0	0	PROPN
ap-2064	20	3	,	,	PUNCT
ap-2064	20	4	f	f	PROPN
ap-2064	20	5	(	(	PUNCT
ap-2064	20	6	a	a	DET
ap-2064	20	7	,	,	PUNCT
ap-2064	20	8	b	b	NOUN
ap-2064	20	9	)	)	PUNCT
ap-2064	20	10	1	1	NUM
ap-2064	20	11	=	=	SYM
ap-2064	20	12	1	1	NUM
ap-2064	20	13	,	,	PUNCT
ap-2064	20	14	f	f	PROPN
ap-2064	20	15	(	(	PUNCT
ap-2064	20	16	a	a	DET
ap-2064	20	17	,	,	PUNCT
ap-2064	20	18	b	b	NOUN
ap-2064	20	19	)	)	PUNCT
ap-2064	20	20	n	n	NOUN
ap-2064	20	21	=	=	PUNCT
ap-2064	20	22	{	{	PUNCT
ap-2064	20	23	af	af	PROPN
ap-2064	20	24	(	(	PUNCT
ap-2064	20	25	a	a	DET
ap-2064	20	26	,	,	PUNCT
ap-2064	20	27	b	b	NOUN
ap-2064	20	28	)	)	PUNCT
ap-2064	20	29	n−1	n−1	PROPN
ap-2064	21	1	+	+	NUM
ap-2064	21	2	f	f	X
ap-2064	21	3	(	(	PUNCT
ap-2064	21	4	a	a	DET
ap-2064	21	5	,	,	PUNCT
ap-2064	21	6	b	b	NOUN
ap-2064	21	7	)	)	PUNCT
ap-2064	21	8	n−2	n−2	PROPN
ap-2064	21	9	,	,	PUNCT
ap-2064	21	10	if	if	SCONJ
ap-2064	21	11	n	n	PRON
ap-2064	21	12	≥	≥	NOUN
ap-2064	21	13	2	2	NUM
ap-2064	21	14	is	be	AUX
ap-2064	21	15	even	even	ADV
ap-2064	21	16	,	,	PUNCT
ap-2064	21	17	bf	bf	INTJ
ap-2064	21	18	(	(	PUNCT
ap-2064	21	19	a	a	DET
ap-2064	21	20	,	,	PUNCT
ap-2064	21	21	b	b	NOUN
ap-2064	21	22	)	)	PUNCT
ap-2064	21	23	n−1	n−1	PROPN
ap-2064	22	1	+	+	NUM
ap-2064	22	2	f	f	X
ap-2064	22	3	(	(	PUNCT
ap-2064	22	4	a	a	DET
ap-2064	22	5	,	,	PUNCT
ap-2064	22	6	b	b	NOUN
ap-2064	22	7	)	)	PUNCT
ap-2064	22	8	n−2	n−2	PROPN
ap-2064	22	9	,	,	PUNCT
ap-2064	22	10	if	if	SCONJ
ap-2064	22	11	n	n	PRON
ap-2064	22	12	≥	≥	NOUN
ap-2064	22	13	2	2	NUM
ap-2064	22	14	is	be	AUX
ap-2064	22	15	odd	odd	ADJ
ap-2064	22	16	.	.	PUNCT
ap-2064	23	1	to	to	PART
ap-2064	23	2	avoid	avoid	VERB
ap-2064	23	3	cumbersome	cumbersome	ADJ
ap-2064	23	4	notation	notation	NOUN
ap-2064	23	5	,	,	PUNCT
ap-2064	23	6	let	let	VERB
ap-2064	23	7	us	we	PRON
ap-2064	23	8	denote	denote	VERB
ap-2064	23	9	f	f	PROPN
ap-2064	23	10	(	(	PUNCT
ap-2064	23	11	a	a	DET
ap-2064	23	12	,	,	PUNCT
ap-2064	23	13	b	b	NOUN
ap-2064	23	14	)	)	PUNCT
ap-2064	23	15	n	n	CCONJ
ap-2064	23	16	by	by	ADP
ap-2064	23	17	qn	qn	PROPN
ap-2064	23	18	.	.	PUNCT
ap-2064	24	1	the	the	DET
ap-2064	24	2	first	first	ADJ
ap-2064	24	3	few	few	ADJ
ap-2064	24	4	terms	term	NOUN
ap-2064	24	5	are	be	AUX
ap-2064	24	6	{	{	PUNCT
ap-2064	24	7	qn}∞n=0	qn}∞n=0	X
ap-2064	24	8	=	=	SYM
ap-2064	24	9	{	{	PUNCT
ap-2064	24	10	0	0	NUM
ap-2064	24	11	,	,	PUNCT
ap-2064	24	12	1	1	NUM
ap-2064	24	13	,	,	PUNCT
ap-2064	24	14	a	a	PRON
ap-2064	24	15	,	,	PUNCT
ap-2064	24	16	ab+	ab+	NOUN
ap-2064	24	17	1	1	NUM
ap-2064	24	18	,	,	PUNCT
ap-2064	24	19	a2b+	a2b+	PROPN
ap-2064	24	20	2a	2a	NUM
ap-2064	24	21	,	,	PUNCT
ap-2064	24	22	a2b2	a2b2	PROPN
ap-2064	25	1	+	+	CCONJ
ap-2064	25	2	3ab+	3ab+	PROPN
ap-2064	25	3	1	1	NUM
ap-2064	25	4	,	,	PUNCT
ap-2064	25	5	a3b2	a3b2	PUNCT
ap-2064	25	6	+	+	CCONJ
ap-2064	25	7	4a2b+	4a2b+	NUM
ap-2064	25	8	3a	3a	NUM
ap-2064	25	9	,	,	PUNCT
ap-2064	25	10	.	.	PUNCT
ap-2064	25	11	.	.	PUNCT
ap-2064	25	12	.	.	PUNCT
ap-2064	26	1	}	}	PUNCT
ap-2064	26	2	.	.	PUNCT
ap-2064	27	1	note	note	VERB
ap-2064	27	2	that	that	SCONJ
ap-2064	27	3	if	if	SCONJ
ap-2064	27	4	a	a	DET
ap-2064	27	5	=	=	SYM
ap-2064	27	6	b	b	NOUN
ap-2064	27	7	=	=	SYM
ap-2064	27	8	1	1	NUM
ap-2064	27	9	,	,	PUNCT
ap-2064	27	10	then	then	ADV
ap-2064	27	11	qn	qn	PROPN
ap-2064	27	12	is	be	AUX
ap-2064	27	13	the	the	DET
ap-2064	27	14	nth	nth	PROPN
ap-2064	27	15	fibonacci	fibonacci	NOUN
ap-2064	27	16	number	number	NOUN
ap-2064	27	17	.	.	PUNCT
ap-2064	28	1	a	a	DET
ap-2064	28	2	binet	binet	NOUN
ap-2064	28	3	-	-	PUNCT
ap-2064	28	4	like	like	ADJ
ap-2064	28	5	formula	formula	NOUN
ap-2064	28	6	to	to	ADP
ap-2064	28	7	the	the	DET
ap-2064	28	8	biperiodic	biperiodic	ADJ
ap-2064	28	9	fibonacci	fibonacci	NOUN
ap-2064	28	10	sequence	sequence	NOUN
ap-2064	28	11	is	be	AUX
ap-2064	28	12	qn	qn	NOUN
ap-2064	28	13	=	=	X
ap-2064	28	14	(	(	PUNCT
ap-2064	28	15	a1−ξ(n	a1−ξ(n	PROPN
ap-2064	28	16	)	)	PUNCT
ap-2064	28	17	(	(	PUNCT
ap-2064	28	18	ab)bn2	ab)bn2	NOUN
ap-2064	28	19	c	c	PROPN
ap-2064	28	20	)	)	PUNCT
ap-2064	28	21	αn	αn	NOUN
ap-2064	29	1	−	−	NOUN
ap-2064	29	2	βn	βn	NOUN
ap-2064	29	3	α−	α−	ADP
ap-2064	29	4	β	β	X
ap-2064	29	5	,	,	PUNCT
ap-2064	29	6	(	(	PUNCT
ap-2064	29	7	1	1	X
ap-2064	29	8	)	)	PUNCT
ap-2064	29	9	where	where	SCONJ
ap-2064	29	10	α	α	NOUN
ap-2064	29	11	=	=	SYM
ap-2064	29	12	ab+	ab+	NOUN
ap-2064	29	13	√	√	PROPN
ap-2064	29	14	(	(	PUNCT
ap-2064	29	15	ab)2	ab)2	PROPN
ap-2064	29	16	+	+	CCONJ
ap-2064	29	17	4ab	4ab	ADJ
ap-2064	29	18	2	2	NUM
ap-2064	29	19	,	,	PUNCT
ap-2064	30	1	β	β	X
ap-2064	30	2	=	=	SYM
ap-2064	30	3	ab−	ab−	NUM
ap-2064	30	4	√	√	PROPN
ap-2064	30	5	(	(	PUNCT
ap-2064	30	6	ab)2	ab)2	PROPN
ap-2064	30	7	+	+	CCONJ
ap-2064	30	8	4ab	4ab	ADJ
ap-2064	30	9	2	2	NUM
ap-2064	30	10	,	,	PUNCT
ap-2064	30	11	and	and	CCONJ
ap-2064	30	12	ξ(n	ξ(n	NUM
ap-2064	30	13	)	)	PUNCT
ap-2064	30	14	:	:	PUNCT
ap-2064	31	1	=	=	PUNCT
ap-2064	31	2	n−	n−	NOUN
ap-2064	31	3	2	2	NUM
ap-2064	31	4	⌊n	⌊n	SYM
ap-2064	31	5	2	2	NUM
ap-2064	31	6	⌋	⌋	NOUN
ap-2064	31	7	.	.	PUNCT
ap-2064	32	1	on	on	ADP
ap-2064	32	2	the	the	DET
ap-2064	32	3	other	other	ADJ
ap-2064	32	4	hand	hand	NOUN
ap-2064	32	5	,	,	PUNCT
ap-2064	32	6	there	there	PRON
ap-2064	32	7	exists	exist	VERB
ap-2064	32	8	a	a	DET
ap-2064	32	9	well	well	ADV
ap-2064	32	10	-	-	PUNCT
ap-2064	32	11	known	know	VERB
ap-2064	32	12	wordcombinatorial	wordcombinatorial	ADJ
ap-2064	32	13	interpretation	interpretation	NOUN
ap-2064	32	14	of	of	ADP
ap-2064	32	15	the	the	DET
ap-2064	32	16	fibonacci	fibonacci	NOUN
ap-2064	32	17	sequence	sequence	NOUN
ap-2064	32	18	.	.	PUNCT
ap-2064	33	1	let	let	VERB
ap-2064	33	2	fn	fn	PRON
ap-2064	33	3	be	be	AUX
ap-2064	33	4	a	a	DET
ap-2064	33	5	binary	binary	ADJ
ap-2064	33	6	word	word	NOUN
ap-2064	33	7	defined	define	VERB
ap-2064	33	8	inductively	inductively	ADV
ap-2064	33	9	as	as	SCONJ
ap-2064	33	10	follows	follow	VERB
ap-2064	33	11	f0	f0	PROPN
ap-2064	33	12	=	=	SYM
ap-2064	33	13	1	1	NUM
ap-2064	33	14	,	,	PUNCT
ap-2064	33	15	f1	f1	NOUN
ap-2064	33	16	=	=	SYM
ap-2064	33	17	0	0	NUM
ap-2064	33	18	,	,	PUNCT
ap-2064	33	19	fn	fn	NOUN
ap-2064	33	20	=	=	SYM
ap-2064	33	21	fn−1fn−2	fn−1fn−2	PROPN
ap-2064	33	22	,	,	PUNCT
ap-2064	33	23	for	for	ADP
ap-2064	33	24	n	n	PRON
ap-2064	33	25	≥	≥	NOUN
ap-2064	33	26	2	2	NUM
ap-2064	33	27	.	.	PUNCT
ap-2064	34	1	it	it	PRON
ap-2064	34	2	is	be	AUX
ap-2064	34	3	clear	clear	ADJ
ap-2064	34	4	that	that	SCONJ
ap-2064	34	5	|fn|	|fn|	PROPN
ap-2064	34	6	=	=	SYM
ap-2064	34	7	fn	fn	NOUN
ap-2064	34	8	,	,	PUNCT
ap-2064	34	9	i.e.	i.e.	X
ap-2064	34	10	,	,	PUNCT
ap-2064	34	11	the	the	DET
ap-2064	34	12	length	length	NOUN
ap-2064	34	13	of	of	ADP
ap-2064	34	14	the	the	DET
ap-2064	34	15	word	word	NOUN
ap-2064	34	16	fn	fn	NOUN
ap-2064	34	17	is	be	AUX
ap-2064	34	18	the	the	DET
ap-2064	34	19	nth	nth	PROPN
ap-2064	34	20	fibonacci	fibonacci	NOUN
ap-2064	34	21	number	number	NOUN
ap-2064	34	22	.	.	PUNCT
ap-2064	35	1	the	the	DET
ap-2064	35	2	words	word	NOUN
ap-2064	35	3	fn	fn	NOUN
ap-2064	35	4	are	be	AUX
ap-2064	35	5	called	call	VERB
ap-2064	35	6	finite	finite	ADJ
ap-2064	35	7	fibonacci	fibonacci	NOUN
ap-2064	35	8	words	word	NOUN
ap-2064	35	9	.	.	PUNCT
ap-2064	36	1	the	the	DET
ap-2064	36	2	infinite	infinite	ADJ
ap-2064	36	3	fibonacci	fibonacci	NOUN
ap-2064	36	4	word	word	NOUN
ap-2064	36	5	,	,	PUNCT
ap-2064	36	6	f	f	PROPN
ap-2064	36	7	=	=	SYM
ap-2064	36	8	0100101001001010010100100101	0100101001001010010100100101	NUM
ap-2064	36	9	·	·	PUNCT
ap-2064	36	10	·	·	PUNCT
ap-2064	36	11	·	·	PUNCT
ap-2064	36	12	is	be	AUX
ap-2064	36	13	defined	define	VERB
ap-2064	36	14	by	by	ADP
ap-2064	36	15	the	the	DET
ap-2064	36	16	limit	limit	NOUN
ap-2064	36	17	sequence	sequence	NOUN
ap-2064	36	18	of	of	ADP
ap-2064	36	19	the	the	DET
ap-2064	36	20	infinite	infinite	ADJ
ap-2064	36	21	sequence	sequence	NOUN
ap-2064	36	22	{	{	PUNCT
ap-2064	36	23	fn}n∈n	fn}n∈n	INTJ
ap-2064	36	24	.	.	PUNCT
ap-2064	37	1	it	it	PRON
ap-2064	37	2	is	be	AUX
ap-2064	37	3	the	the	DET
ap-2064	37	4	archetype	archetype	NOUN
ap-2064	37	5	of	of	ADP
ap-2064	37	6	a	a	DET
ap-2064	37	7	sturmian	sturmian	ADJ
ap-2064	37	8	word	word	NOUN
ap-2064	37	9	[	[	X
ap-2064	37	10	2	2	NUM
ap-2064	37	11	,	,	PUNCT
ap-2064	37	12	14	14	NUM
ap-2064	37	13	]	]	PUNCT
ap-2064	37	14	,	,	PUNCT
ap-2064	37	15	and	and	CCONJ
ap-2064	37	16	one	one	NUM
ap-2064	37	17	of	of	ADP
ap-2064	37	18	the	the	DET
ap-2064	37	19	most	most	ADV
ap-2064	37	20	studied	studied	ADJ
ap-2064	37	21	examples	example	NOUN
ap-2064	37	22	in	in	ADP
ap-2064	37	23	the	the	DET
ap-2064	37	24	combinatorial	combinatorial	ADJ
ap-2064	37	25	theory	theory	NOUN
ap-2064	37	26	of	of	ADP
ap-2064	37	27	infinite	infinite	ADJ
ap-2064	37	28	words	word	NOUN
ap-2064	37	29	;	;	PUNCT
ap-2064	37	30	see	see	VERB
ap-2064	37	31	,	,	PUNCT
ap-2064	37	32	e.g.	e.g.	ADV
ap-2064	37	33	,	,	PUNCT
ap-2064	37	34	[	[	X
ap-2064	37	35	3	3	NUM
ap-2064	37	36	,	,	PUNCT
ap-2064	37	37	5	5	NUM
ap-2064	37	38	,	,	PUNCT
ap-2064	37	39	6	6	NUM
ap-2064	37	40	,	,	PUNCT
ap-2064	37	41	8	8	NUM
ap-2064	37	42	,	,	PUNCT
ap-2064	37	43	10	10	NUM
ap-2064	37	44	,	,	PUNCT
ap-2064	37	45	15	15	NUM
ap-2064	37	46	,	,	PUNCT
ap-2064	37	47	17	17	NUM
ap-2064	37	48	]	]	PUNCT
ap-2064	37	49	.	.	PUNCT
ap-2064	38	1	the	the	DET
ap-2064	38	2	word	word	NOUN
ap-2064	38	3	f	f	PROPN
ap-2064	38	4	can	can	AUX
ap-2064	38	5	be	be	AUX
ap-2064	38	6	associated	associate	VERB
ap-2064	38	7	with	with	ADP
ap-2064	38	8	a	a	DET
ap-2064	38	9	curve	curve	NOUN
ap-2064	38	10	from	from	ADP
ap-2064	38	11	a	a	DET
ap-2064	38	12	drawing	drawing	NOUN
ap-2064	38	13	rule	rule	NOUN
ap-2064	38	14	,	,	PUNCT
ap-2064	38	15	which	which	PRON
ap-2064	38	16	has	have	VERB
ap-2064	38	17	geometric	geometric	ADJ
ap-2064	38	18	properties	property	NOUN
ap-2064	38	19	obtained	obtain	VERB
ap-2064	38	20	from	from	ADP
ap-2064	38	21	the	the	DET
ap-2064	38	22	combinatorial	combinatorial	ADJ
ap-2064	38	23	properties	property	NOUN
ap-2064	38	24	of	of	ADP
ap-2064	38	25	f	f	PROPN
ap-2064	39	1	[	[	X
ap-2064	39	2	4	4	NUM
ap-2064	39	3	,	,	PUNCT
ap-2064	39	4	16	16	NUM
ap-2064	39	5	]	]	PUNCT
ap-2064	39	6	.	.	PUNCT
ap-2064	40	1	the	the	DET
ap-2064	40	2	curve	curve	NOUN
ap-2064	40	3	produced	produce	VERB
ap-2064	40	4	depends	depend	VERB
ap-2064	40	5	on	on	ADP
ap-2064	40	6	the	the	DET
ap-2064	40	7	rules	rule	NOUN
ap-2064	40	8	given	give	VERB
ap-2064	40	9	.	.	PUNCT
ap-2064	41	1	we	we	PRON
ap-2064	41	2	read	read	VERB
ap-2064	41	3	the	the	DET
ap-2064	41	4	symbols	symbol	NOUN
ap-2064	41	5	of	of	ADP
ap-2064	41	6	the	the	DET
ap-2064	41	7	word	word	NOUN
ap-2064	41	8	in	in	ADP
ap-2064	41	9	order	order	NOUN
ap-2064	41	10	and	and	CCONJ
ap-2064	41	11	depending	depend	VERB
ap-2064	41	12	on	on	ADP
ap-2064	41	13	what	what	PRON
ap-2064	41	14	we	we	PRON
ap-2064	41	15	read	read	VERB
ap-2064	41	16	,	,	PUNCT
ap-2064	41	17	we	we	PRON
ap-2064	41	18	draw	draw	VERB
ap-2064	41	19	a	a	DET
ap-2064	41	20	line	line	NOUN
ap-2064	41	21	segment	segment	NOUN
ap-2064	41	22	in	in	ADP
ap-2064	41	23	a	a	DET
ap-2064	41	24	certain	certain	ADJ
ap-2064	41	25	direction	direction	NOUN
ap-2064	41	26	;	;	PUNCT
ap-2064	41	27	this	this	DET
ap-2064	41	28	idea	idea	NOUN
ap-2064	41	29	is	be	AUX
ap-2064	41	30	the	the	DET
ap-2064	41	31	same	same	ADJ
ap-2064	41	32	as	as	ADP
ap-2064	41	33	that	that	PRON
ap-2064	41	34	used	use	VERB
ap-2064	41	35	in	in	ADP
ap-2064	41	36	lsystems	lsystem	NOUN
ap-2064	41	37	[	[	X
ap-2064	41	38	18	18	NUM
ap-2064	41	39	]	]	PUNCT
ap-2064	41	40	.	.	PUNCT
ap-2064	42	1	in	in	ADP
ap-2064	42	2	this	this	DET
ap-2064	42	3	case	case	NOUN
ap-2064	42	4	,	,	PUNCT
ap-2064	42	5	the	the	DET
ap-2064	42	6	drawing	drawing	NOUN
ap-2064	42	7	rule	rule	NOUN
ap-2064	42	8	is	be	AUX
ap-2064	42	9	called	call	VERB
ap-2064	42	10	the	the	DET
ap-2064	42	11	“	"	PUNCT
ap-2064	42	12	odd	odd	ADJ
ap-2064	42	13	-	-	PUNCT
ap-2064	42	14	even	even	ADV
ap-2064	42	15	drawing	draw	VERB
ap-2064	42	16	rule	rule	NOUN
ap-2064	42	17	”	"	PUNCT
ap-2064	42	18	[	[	X
ap-2064	42	19	16	16	NUM
ap-2064	42	20	]	]	PUNCT
ap-2064	42	21	.	.	PUNCT
ap-2064	43	1	this	this	PRON
ap-2064	43	2	is	be	AUX
ap-2064	43	3	defined	define	VERB
ap-2064	43	4	as	as	SCONJ
ap-2064	43	5	shown	show	VERB
ap-2064	43	6	in	in	ADP
ap-2064	43	7	table	table	NOUN
ap-2064	43	8	1	1	NUM
ap-2064	43	9	.	.	PUNCT
ap-2064	44	1	the	the	DET
ap-2064	44	2	nth	nth	NOUN
ap-2064	44	3	-	-	PUNCT
ap-2064	44	4	curve	curve	NOUN
ap-2064	44	5	of	of	ADP
ap-2064	44	6	fibonacci	fibonacci	NOUN
ap-2064	44	7	,	,	PUNCT
ap-2064	44	8	denoted	denote	VERB
ap-2064	44	9	by	by	ADP
ap-2064	44	10	fn	fn	NOUN
ap-2064	44	11	,	,	PUNCT
ap-2064	44	12	is	be	AUX
ap-2064	44	13	obtained	obtain	VERB
ap-2064	44	14	by	by	ADP
ap-2064	44	15	applying	apply	VERB
ap-2064	44	16	the	the	DET
ap-2064	44	17	odd	odd	ADV
ap-2064	44	18	-	-	PUNCT
ap-2064	44	19	even	even	ADV
ap-2064	44	20	drawing	draw	VERB
ap-2064	44	21	rule	rule	NOUN
ap-2064	44	22	to	to	ADP
ap-2064	44	23	the	the	DET
ap-2064	44	24	word	word	NOUN
ap-2064	44	25	fn	fn	NOUN
ap-2064	44	26	.	.	PUNCT
ap-2064	45	1	the	the	DET
ap-2064	45	2	fibonacci	fibonacci	NOUN
ap-2064	45	3	word	word	PROPN
ap-2064	45	4	fractal	fractal	PROPN
ap-2064	45	5	f	f	PROPN
ap-2064	45	6	,	,	PUNCT
ap-2064	45	7	is	be	AUX
ap-2064	45	8	defined	define	VERB
ap-2064	45	9	as	as	ADP
ap-2064	45	10	f	f	PROPN
ap-2064	45	11	=	=	PUNCT
ap-2064	45	12	limn→∞	limn→∞	PROPN
ap-2064	46	1	fn	fn	NOUN
ap-2064	46	2	.	.	PUNCT
ap-2064	47	1	for	for	ADP
ap-2064	47	2	example	example	NOUN
ap-2064	47	3	,	,	PUNCT
ap-2064	47	4	in	in	ADP
ap-2064	47	5	figure	figure	NOUN
ap-2064	47	6	1	1	NUM
ap-2064	47	7	,	,	PUNCT
ap-2064	47	8	we	we	PRON
ap-2064	47	9	show	show	VERB
ap-2064	47	10	the	the	DET
ap-2064	47	11	curve	curve	NOUN
ap-2064	47	12	f10	f10	NOUN
ap-2064	47	13	and	and	CCONJ
ap-2064	47	14	f17	f17	NOUN
ap-2064	47	15	.	.	PUNCT
ap-2064	48	1	the	the	DET
ap-2064	48	2	graphics	graphic	NOUN
ap-2064	48	3	in	in	ADP
ap-2064	48	4	the	the	DET
ap-2064	48	5	present	present	ADJ
ap-2064	48	6	article	article	NOUN
ap-2064	48	7	were	be	AUX
ap-2064	48	8	generated	generate	VERB
ap-2064	48	9	using	use	VERB
ap-2064	48	10	the	the	DET
ap-2064	48	11	mathematica	mathematica	PROPN
ap-2064	48	12	9.0	9.0	NUM
ap-2064	48	13	software	software	NOUN
ap-2064	48	14	,	,	PUNCT
ap-2064	48	15	[	[	X
ap-2064	48	16	19	19	NUM
ap-2064	48	17	,	,	PUNCT
ap-2064	48	18	21	21	NUM
ap-2064	48	19	]	]	PUNCT
ap-2064	48	20	.	.	PUNCT
ap-2064	49	1	f10	f10	PROPN
ap-2064	49	2	=	=	SYM
ap-2064	49	3	0100101001001010010100100101001	0100101001001010010100100101001	NUM
ap-2064	49	4	0010100101001001010010100100101	0010100101001001010010100100101	NUM
ap-2064	49	5	001001010010100100101001001	001001010010100100101001001	NUM
ap-2064	49	6	.	.	PUNCT
ap-2064	50	1	50	50	NUM
ap-2064	50	2	http://dx.doi.org/10.14311/ap.2015.55.0050	http://dx.doi.org/10.14311/ap.2015.55.0050	NOUN
ap-2064	50	3	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-2064	50	4	vol	vol	NOUN
ap-2064	50	5	.	.	PUNCT
ap-2064	51	1	55	55	NUM
ap-2064	51	2	no	no	NOUN
ap-2064	51	3	.	.	PUNCT
ap-2064	52	1	1/2015	1/2015	NUM
ap-2064	52	2	biperiodic	biperiodic	PROPN
ap-2064	52	3	fibonacci	fibonacci	NOUN
ap-2064	52	4	word	word	NOUN
ap-2064	52	5	and	and	CCONJ
ap-2064	52	6	its	its	PRON
ap-2064	52	7	fractal	fractal	ADJ
ap-2064	52	8	curve	curve	NOUN
ap-2064	52	9	figure	figure	NOUN
ap-2064	52	10	1	1	NUM
ap-2064	52	11	.	.	PUNCT
ap-2064	53	1	fibonacci	fibonacci	NOUN
ap-2064	53	2	curves	curve	VERB
ap-2064	53	3	f10	f10	NOUN
ap-2064	53	4	and	and	CCONJ
ap-2064	53	5	f17	f17	NOUN
ap-2064	53	6	corresponding	correspond	VERB
ap-2064	53	7	to	to	ADP
ap-2064	53	8	the	the	DET
ap-2064	53	9	words	word	NOUN
ap-2064	53	10	f10	f10	NOUN
ap-2064	53	11	and	and	CCONJ
ap-2064	53	12	f17	f17	ADJ
ap-2064	53	13	.	.	PUNCT
ap-2064	54	1	symbol	symbol	NOUN
ap-2064	54	2	action	action	NOUN
ap-2064	54	3	1	1	NUM
ap-2064	54	4	draw	draw	VERB
ap-2064	54	5	a	a	DET
ap-2064	54	6	line	line	NOUN
ap-2064	54	7	forward	forward	ADV
ap-2064	54	8	.	.	PUNCT
ap-2064	55	1	0	0	PUNCT
ap-2064	55	2	draw	draw	VERB
ap-2064	55	3	a	a	DET
ap-2064	55	4	line	line	NOUN
ap-2064	55	5	forward	forward	ADV
ap-2064	55	6	and	and	CCONJ
ap-2064	55	7	if	if	SCONJ
ap-2064	55	8	the	the	DET
ap-2064	55	9	symbol	symbol	NOUN
ap-2064	55	10	0	0	PUNCT
ap-2064	55	11	is	be	AUX
ap-2064	55	12	in	in	ADP
ap-2064	55	13	an	an	DET
ap-2064	55	14	even	even	ADJ
ap-2064	55	15	position	position	NOUN
ap-2064	55	16	,	,	PUNCT
ap-2064	55	17	then	then	ADV
ap-2064	55	18	turn	turn	VERB
ap-2064	55	19	left	left	ADJ
ap-2064	55	20	and	and	CCONJ
ap-2064	55	21	if	if	SCONJ
ap-2064	55	22	0	0	NUM
ap-2064	55	23	is	be	AUX
ap-2064	55	24	in	in	ADP
ap-2064	55	25	an	an	DET
ap-2064	55	26	odd	odd	ADJ
ap-2064	55	27	position	position	NOUN
ap-2064	55	28	,	,	PUNCT
ap-2064	55	29	then	then	ADV
ap-2064	55	30	turn	turn	VERB
ap-2064	55	31	right	right	ADV
ap-2064	55	32	.	.	PUNCT
ap-2064	56	1	table	table	NOUN
ap-2064	56	2	1	1	NUM
ap-2064	56	3	.	.	PUNCT
ap-2064	57	1	odd	odd	ADV
ap-2064	57	2	-	-	PUNCT
ap-2064	57	3	even	even	ADV
ap-2064	57	4	drawing	draw	VERB
ap-2064	57	5	rule	rule	NOUN
ap-2064	57	6	.	.	PUNCT
ap-2064	58	1	ramírez	ramírez	PROPN
ap-2064	58	2	et	et	PROPN
ap-2064	58	3	al	al	PROPN
ap-2064	58	4	.	.	PUNCT
ap-2064	59	1	[	[	X
ap-2064	59	2	22	22	NUM
ap-2064	59	3	]	]	PUNCT
ap-2064	59	4	introduced	introduce	VERB
ap-2064	59	5	a	a	DET
ap-2064	59	6	generalization	generalization	NOUN
ap-2064	59	7	of	of	ADP
ap-2064	59	8	the	the	DET
ap-2064	59	9	fibonacci	fibonacci	NOUN
ap-2064	59	10	word	word	NOUN
ap-2064	59	11	,	,	PUNCT
ap-2064	59	12	the	the	DET
ap-2064	59	13	i	i	PROPN
ap-2064	59	14	-	-	PUNCT
ap-2064	59	15	fibonacci	fibonacci	NOUN
ap-2064	59	16	word	word	NOUN
ap-2064	59	17	.	.	PUNCT
ap-2064	60	1	specifically	specifically	ADV
ap-2064	60	2	,	,	PUNCT
ap-2064	60	3	the	the	PRON
ap-2064	60	4	(	(	PUNCT
ap-2064	60	5	n	n	CCONJ
ap-2064	60	6	,	,	PUNCT
ap-2064	60	7	i)-fibonacci	i)-fibonacci	NOUN
ap-2064	60	8	words	word	NOUN
ap-2064	60	9	are	be	AUX
ap-2064	60	10	words	word	NOUN
ap-2064	60	11	over	over	ADP
ap-2064	60	12	{	{	PUNCT
ap-2064	60	13	0,1	0,1	NOUN
ap-2064	60	14	}	}	PUNCT
ap-2064	60	15	defined	define	VERB
ap-2064	60	16	inductively	inductively	ADV
ap-2064	60	17	as	as	SCONJ
ap-2064	60	18	follows	follow	VERB
ap-2064	60	19	:	:	PUNCT
ap-2064	60	20	f	f	PROPN
ap-2064	61	1	[	[	X
ap-2064	61	2	i	i	X
ap-2064	61	3	]	]	PUNCT
ap-2064	61	4	0	0	PUNCT
ap-2064	62	1	=	=	SYM
ap-2064	62	2	0	0	PROPN
ap-2064	62	3	,	,	PUNCT
ap-2064	62	4	f	f	PROPN
ap-2064	63	1	[	[	X
ap-2064	63	2	i	i	X
ap-2064	63	3	]	]	X
ap-2064	63	4	1	1	NUM
ap-2064	63	5	=	=	SYM
ap-2064	63	6	0i−11	0i−11	PROPN
ap-2064	63	7	,	,	PUNCT
ap-2064	63	8	f	f	PROPN
ap-2064	64	1	[	[	X
ap-2064	64	2	i	i	X
ap-2064	64	3	]	]	X
ap-2064	64	4	n	n	PROPN
ap-2064	64	5	=	=	SYM
ap-2064	64	6	f	f	X
ap-2064	65	1	[	[	X
ap-2064	65	2	i	i	X
ap-2064	65	3	]	]	PUNCT
ap-2064	65	4	n−1f	n−1f	PROPN
ap-2064	66	1	[	[	X
ap-2064	66	2	i	i	X
ap-2064	66	3	]	]	X
ap-2064	66	4	n−2	n−2	PROPN
ap-2064	66	5	,	,	PUNCT
ap-2064	66	6	for	for	ADP
ap-2064	66	7	n	n	PRON
ap-2064	66	8	≥	≥	NUM
ap-2064	66	9	2	2	NUM
ap-2064	66	10	and	and	CCONJ
ap-2064	66	11	≥	≥	NUM
ap-2064	66	12	1	1	NUM
ap-2064	66	13	.	.	PUNCT
ap-2064	67	1	the	the	DET
ap-2064	67	2	infinite	infinite	ADJ
ap-2064	67	3	word	word	NOUN
ap-2064	67	4	f	f	PROPN
ap-2064	68	1	[	[	X
ap-2064	68	2	i	i	X
ap-2064	68	3	]	]	X
ap-2064	68	4	:	:	PUNCT
ap-2064	68	5	=	=	SYM
ap-2064	68	6	limn→∞	limn→∞	SYM
ap-2064	68	7	f	f	X
ap-2064	69	1	[	[	X
ap-2064	69	2	i	i	X
ap-2064	69	3	]	]	PUNCT
ap-2064	69	4	n	n	CCONJ
ap-2064	69	5	is	be	AUX
ap-2064	69	6	called	call	VERB
ap-2064	69	7	the	the	DET
ap-2064	69	8	i	i	PROPN
ap-2064	69	9	-	-	PUNCT
ap-2064	69	10	fibonacci	fibonacci	NOUN
ap-2064	69	11	word	word	NOUN
ap-2064	69	12	.	.	PUNCT
ap-2064	70	1	for	for	ADP
ap-2064	70	2	i	i	PRON
ap-2064	70	3	=	=	NOUN
ap-2064	70	4	2	2	NUM
ap-2064	70	5	,	,	PUNCT
ap-2064	70	6	we	we	PRON
ap-2064	70	7	have	have	VERB
ap-2064	70	8	the	the	DET
ap-2064	70	9	classical	classical	ADJ
ap-2064	70	10	fibonacci	fibonacci	NOUN
ap-2064	70	11	word	word	NOUN
ap-2064	70	12	.	.	PUNCT
ap-2064	71	1	note	note	VERB
ap-2064	71	2	that	that	SCONJ
ap-2064	71	3	|f	|f	PROPN
ap-2064	72	1	[	[	X
ap-2064	72	2	i	i	X
ap-2064	72	3	]	]	PUNCT
ap-2064	72	4	n	n	CCONJ
ap-2064	72	5	|	|	ADV
ap-2064	72	6	=	=	SYM
ap-2064	72	7	f	f	X
ap-2064	73	1	[	[	X
ap-2064	73	2	i	i	X
ap-2064	73	3	]	]	PUNCT
ap-2064	73	4	n	n	CCONJ
ap-2064	73	5	,	,	PUNCT
ap-2064	73	6	where	where	SCONJ
ap-2064	73	7	f	f	PROPN
ap-2064	73	8	[	[	X
ap-2064	73	9	i	i	X
ap-2064	73	10	]	]	PUNCT
ap-2064	73	11	n	n	PRON
ap-2064	73	12	is	be	AUX
ap-2064	73	13	the	the	DET
ap-2064	73	14	integer	integer	NOUN
ap-2064	73	15	sequence	sequence	NOUN
ap-2064	73	16	defined	define	VERB
ap-2064	73	17	recursively	recursively	ADV
ap-2064	73	18	by	by	ADP
ap-2064	73	19	f	f	PROPN
ap-2064	74	1	[	[	X
ap-2064	74	2	i	i	X
ap-2064	74	3	]	]	PUNCT
ap-2064	74	4	0	0	PUNCT
ap-2064	74	5	=	=	SYM
ap-2064	74	6	1	1	NUM
ap-2064	74	7	,	,	PUNCT
ap-2064	74	8	f	f	PROPN
ap-2064	75	1	[	[	X
ap-2064	75	2	i	i	X
ap-2064	75	3	]	]	X
ap-2064	75	4	1	1	X
ap-2064	75	5	=	=	SYM
ap-2064	75	6	i	i	PROPN
ap-2064	75	7	,	,	PUNCT
ap-2064	75	8	f	f	PROPN
ap-2064	76	1	[	[	X
ap-2064	76	2	i	i	X
ap-2064	76	3	]	]	X
ap-2064	76	4	n	n	PROPN
ap-2064	76	5	=	=	SYM
ap-2064	76	6	f	f	X
ap-2064	77	1	[	[	X
ap-2064	77	2	i	i	X
ap-2064	77	3	]	]	X
ap-2064	77	4	n−1	n−1	PROPN
ap-2064	78	1	+	+	NUM
ap-2064	78	2	f	f	X
ap-2064	79	1	[	[	X
ap-2064	79	2	i	i	X
ap-2064	79	3	]	]	X
ap-2064	79	4	n−2	n−2	PROPN
ap-2064	79	5	,	,	PUNCT
ap-2064	79	6	for	for	ADP
ap-2064	79	7	n	n	PRON
ap-2064	79	8	≥	≥	NUM
ap-2064	79	9	2	2	NUM
ap-2064	79	10	,	,	PUNCT
ap-2064	79	11	and	and	CCONJ
ap-2064	79	12	i	i	PRON
ap-2064	79	13	≥	≥	VERB
ap-2064	79	14	1	1	NUM
ap-2064	79	15	.	.	PUNCT
ap-2064	80	1	for	for	ADP
ap-2064	80	2	i	i	PRON
ap-2064	80	3	=	=	SYM
ap-2064	80	4	1	1	NUM
ap-2064	80	5	,	,	PUNCT
ap-2064	80	6	2	2	NUM
ap-2064	80	7	we	we	PRON
ap-2064	80	8	have	have	VERB
ap-2064	80	9	the	the	DET
ap-2064	80	10	fibonacci	fibonacci	NOUN
ap-2064	80	11	numbers	number	NOUN
ap-2064	80	12	.	.	PUNCT
ap-2064	81	1	ramírez	ramírez	NOUN
ap-2064	81	2	and	and	CCONJ
ap-2064	81	3	rubiano	rubiano	VERB
ap-2064	82	1	[	[	X
ap-2064	82	2	20	20	NUM
ap-2064	82	3	]	]	PUNCT
ap-2064	82	4	studied	study	VERB
ap-2064	82	5	a	a	DET
ap-2064	82	6	similar	similar	ADJ
ap-2064	82	7	binary	binary	ADJ
ap-2064	82	8	word	word	NOUN
ap-2064	82	9	to	to	ADP
ap-2064	82	10	f	f	PROPN
ap-2064	83	1	[	[	X
ap-2064	83	2	i	i	X
ap-2064	83	3	]	]	X
ap-2064	83	4	n	n	CCONJ
ap-2064	83	5	,	,	PUNCT
ap-2064	83	6	which	which	PRON
ap-2064	83	7	is	be	AUX
ap-2064	83	8	denoted	denote	VERB
ap-2064	83	9	by	by	ADP
ap-2064	83	10	fk	fk	INTJ
ap-2064	83	11	,	,	PUNCT
ap-2064	83	12	n.	n.	VERB
ap-2064	83	13	the	the	DET
ap-2064	83	14	initial	initial	ADJ
ap-2064	83	15	values	value	NOUN
ap-2064	83	16	are	be	AUX
ap-2064	83	17	the	the	DET
ap-2064	83	18	same	same	ADJ
ap-2064	83	19	,	,	PUNCT
ap-2064	83	20	i.e.	i.e.	X
ap-2064	83	21	,	,	PUNCT
ap-2064	83	22	fk,0	fk,0	NOUN
ap-2064	83	23	=	=	SYM
ap-2064	83	24	0	0	NUM
ap-2064	83	25	,	,	PUNCT
ap-2064	83	26	fk,1	fk,1	NOUN
ap-2064	83	27	=	=	SYM
ap-2064	83	28	0k−11	0k−11	NOUN
ap-2064	83	29	.	.	PUNCT
ap-2064	84	1	however	however	ADV
ap-2064	84	2	,	,	PUNCT
ap-2064	84	3	the	the	DET
ap-2064	84	4	recurrence	recurrence	NOUN
ap-2064	84	5	relation	relation	NOUN
ap-2064	84	6	is	be	AUX
ap-2064	84	7	defined	define	VERB
ap-2064	84	8	by	by	ADP
ap-2064	84	9	fk	fk	INTJ
ap-2064	84	10	,	,	PUNCT
ap-2064	84	11	n	n	NOUN
ap-2064	84	12	=	=	SYM
ap-2064	84	13	fkk	fkk	NOUN
ap-2064	84	14	,	,	PUNCT
ap-2064	84	15	n−1fk	n−1fk	NOUN
ap-2064	84	16	,	,	PUNCT
ap-2064	84	17	n−2	n−2	PROPN
ap-2064	84	18	,	,	PUNCT
ap-2064	84	19	for	for	ADP
ap-2064	84	20	n	n	PRON
ap-2064	84	21	≥	≥	NUM
ap-2064	84	22	2	2	NUM
ap-2064	84	23	,	,	PUNCT
ap-2064	84	24	and	and	CCONJ
ap-2064	84	25	k	k	PROPN
ap-2064	84	26	≥	≥	NUM
ap-2064	84	27	1	1	NUM
ap-2064	84	28	.	.	PUNCT
ap-2064	85	1	the	the	DET
ap-2064	85	2	infinite	infinite	ADJ
ap-2064	85	3	word	word	NOUN
ap-2064	85	4	fk	fk	INTJ
ap-2064	85	5	is	be	AUX
ap-2064	85	6	the	the	DET
ap-2064	85	7	limit	limit	NOUN
ap-2064	85	8	sequence	sequence	NOUN
ap-2064	85	9	of	of	ADP
ap-2064	85	10	the	the	DET
ap-2064	85	11	infinite	infinite	ADJ
ap-2064	85	12	sequence	sequence	NOUN
ap-2064	85	13	{	{	PUNCT
ap-2064	85	14	fk	fk	INTJ
ap-2064	85	15	,	,	PUNCT
ap-2064	85	16	n}n∈n	n}n∈n	PROPN
ap-2064	85	17	.	.	PROPN
ap-2064	86	1	for	for	ADP
ap-2064	86	2	k	k	PROPN
ap-2064	86	3	=	=	SYM
ap-2064	86	4	1	1	NUM
ap-2064	86	5	,	,	PUNCT
ap-2064	86	6	we	we	PRON
ap-2064	86	7	have	have	VERB
ap-2064	86	8	the	the	DET
ap-2064	86	9	word	word	NOUN
ap-2064	86	10	f	f	PROPN
ap-2064	86	11	=	=	SYM
ap-2064	86	12	1011010110110	1011010110110	NUM
ap-2064	86	13	.	.	PUNCT
ap-2064	86	14	.	.	PUNCT
ap-2064	87	1	..	..	PUNCT
ap-2064	88	1	here	here	ADV
ap-2064	88	2	the	the	DET
ap-2064	88	3	overline	overline	NOUN
ap-2064	88	4	is	be	AUX
ap-2064	88	5	shorthand	shorthand	NOUN
ap-2064	88	6	for	for	ADP
ap-2064	88	7	the	the	DET
ap-2064	88	8	morphism	morphism	NOUN
ap-2064	88	9	that	that	PRON
ap-2064	88	10	maps	map	VERB
ap-2064	88	11	0	0	NUM
ap-2064	88	12	to	to	ADP
ap-2064	88	13	1	1	NUM
ap-2064	88	14	and	and	CCONJ
ap-2064	88	15	1	1	NUM
ap-2064	88	16	to	to	ADP
ap-2064	88	17	0	0	NUM
ap-2064	88	18	.	.	PUNCT
ap-2064	89	1	it	it	PRON
ap-2064	89	2	is	be	AUX
ap-2064	89	3	clear	clear	ADJ
ap-2064	89	4	that	that	SCONJ
ap-2064	89	5	|fk	|fk	X
ap-2064	89	6	,	,	PUNCT
ap-2064	89	7	n|	n|	NOUN
ap-2064	89	8	=	=	SYM
ap-2064	89	9	fk	fk	PROPN
ap-2064	89	10	,	,	PUNCT
ap-2064	89	11	n+1	n+1	PROPN
ap-2064	89	12	,	,	PUNCT
ap-2064	89	13	where	where	SCONJ
ap-2064	89	14	fk	fk	INTJ
ap-2064	89	15	,	,	PUNCT
ap-2064	89	16	n+1	n+1	PROPN
ap-2064	89	17	is	be	AUX
ap-2064	89	18	the	the	DET
ap-2064	89	19	integer	integer	NOUN
ap-2064	89	20	sequence	sequence	NOUN
ap-2064	89	21	defined	define	VERB
ap-2064	89	22	recursively	recursively	ADV
ap-2064	89	23	by	by	ADP
ap-2064	89	24	fk,0	fk,0	PROPN
ap-2064	89	25	=	=	SYM
ap-2064	89	26	0	0	NUM
ap-2064	89	27	,	,	PUNCT
ap-2064	89	28	fk,1	fk,1	NOUN
ap-2064	89	29	=	=	SYM
ap-2064	89	30	1	1	NUM
ap-2064	89	31	,	,	PUNCT
ap-2064	89	32	fk	fk	INTJ
ap-2064	89	33	,	,	PUNCT
ap-2064	89	34	n+1	n+1	PROPN
ap-2064	89	35	=	=	PUNCT
ap-2064	89	36	kfk	kfk	NOUN
ap-2064	89	37	,	,	PUNCT
ap-2064	89	38	n	n	PROPN
ap-2064	89	39	+	+	X
ap-2064	89	40	fk	fk	INTJ
ap-2064	89	41	,	,	PUNCT
ap-2064	89	42	n−1	n−1	PROPN
ap-2064	89	43	,	,	PUNCT
ap-2064	89	44	for	for	ADP
ap-2064	89	45	n	n	PRON
ap-2064	89	46	≥	≥	NUM
ap-2064	89	47	1	1	NUM
ap-2064	89	48	.	.	PUNCT
ap-2064	90	1	by	by	ADP
ap-2064	90	2	analogy	analogy	NOUN
ap-2064	90	3	with	with	ADP
ap-2064	90	4	the	the	DET
ap-2064	90	5	fibonacci	fibonacci	NOUN
ap-2064	90	6	word	word	NOUN
ap-2064	90	7	fractal	fractal	PROPN
ap-2064	90	8	,	,	PUNCT
ap-2064	90	9	when	when	SCONJ
ap-2064	90	10	we	we	PRON
ap-2064	90	11	apply	apply	VERB
ap-2064	90	12	the	the	DET
ap-2064	90	13	odd	odd	ADV
ap-2064	90	14	-	-	PUNCT
ap-2064	90	15	even	even	ADV
ap-2064	90	16	drawing	draw	VERB
ap-2064	90	17	rule	rule	NOUN
ap-2064	90	18	to	to	ADP
ap-2064	90	19	the	the	DET
ap-2064	90	20	words	word	NOUN
ap-2064	91	1	f	f	PROPN
ap-2064	92	1	[	[	X
ap-2064	92	2	i	i	X
ap-2064	92	3	]	]	X
ap-2064	92	4	n	n	PROPN
ap-2064	92	5	and	and	CCONJ
ap-2064	92	6	fk	fk	INTJ
ap-2064	92	7	,	,	PUNCT
ap-2064	92	8	n	n	CCONJ
ap-2064	92	9	,	,	PUNCT
ap-2064	92	10	we	we	PRON
ap-2064	92	11	obtain	obtain	VERB
ap-2064	92	12	the	the	DET
ap-2064	92	13	nth	nth	NOUN
ap-2064	92	14	word	word	NOUN
ap-2064	92	15	fractal	fractal	PROPN
ap-2064	92	16	f	f	PROPN
ap-2064	93	1	[	[	X
ap-2064	93	2	i	i	X
ap-2064	93	3	]	]	X
ap-2064	93	4	n	n	PROPN
ap-2064	93	5	and	and	CCONJ
ap-2064	93	6	fk	fk	INTJ
ap-2064	93	7	,	,	PUNCT
ap-2064	93	8	n	n	CCONJ
ap-2064	93	9	,	,	PUNCT
ap-2064	93	10	respectively	respectively	ADV
ap-2064	93	11	.	.	PUNCT
ap-2064	94	1	moreover	moreover	ADV
ap-2064	94	2	,	,	PUNCT
ap-2064	94	3	we	we	PRON
ap-2064	94	4	have	have	VERB
ap-2064	94	5	the	the	DET
ap-2064	94	6	curves	curve	NOUN
ap-2064	94	7	f	f	NOUN
ap-2064	95	1	[	[	X
ap-2064	95	2	i	i	X
ap-2064	95	3	]	]	PUNCT
ap-2064	95	4	and	and	CCONJ
ap-2064	95	5	fk	fk	INTJ
ap-2064	95	6	,	,	PUNCT
ap-2064	95	7	which	which	PRON
ap-2064	95	8	are	be	AUX
ap-2064	95	9	defined	define	VERB
ap-2064	95	10	as	as	ADP
ap-2064	95	11	f	f	PROPN
ap-2064	96	1	[	[	X
ap-2064	96	2	i	i	X
ap-2064	96	3	]	]	X
ap-2064	96	4	=	=	SYM
ap-2064	96	5	limn→∞	limn→∞	X
ap-2064	96	6	f	f	X
ap-2064	97	1	[	[	X
ap-2064	97	2	i	i	X
ap-2064	97	3	]	]	PUNCT
ap-2064	97	4	n	n	CCONJ
ap-2064	97	5	and	and	CCONJ
ap-2064	97	6	fk	fk	INTJ
ap-2064	97	7	=	=	PUNCT
ap-2064	97	8	limn→∞	limn→∞	X
ap-2064	97	9	fk	fk	INTJ
ap-2064	97	10	,	,	PUNCT
ap-2064	97	11	n.	n.	NOUN
ap-2064	97	12	in	in	ADP
ap-2064	97	13	table	table	NOUN
ap-2064	97	14	2	2	NUM
ap-2064	97	15	,	,	PUNCT
ap-2064	97	16	we	we	PRON
ap-2064	97	17	show	show	VERB
ap-2064	97	18	some	some	DET
ap-2064	97	19	curves	curve	NOUN
ap-2064	97	20	f	f	NOUN
ap-2064	98	1	[	[	X
ap-2064	98	2	i	i	X
ap-2064	98	3	]	]	PUNCT
ap-2064	98	4	16	16	NUM
ap-2064	98	5	and	and	CCONJ
ap-2064	98	6	fk	fk	INTJ
ap-2064	98	7	,	,	PUNCT
ap-2064	98	8	n	n	CCONJ
ap-2064	98	9	,	,	PUNCT
ap-2064	98	10	and	and	CCONJ
ap-2064	98	11	their	their	PRON
ap-2064	98	12	associated	associated	ADJ
ap-2064	98	13	words	word	NOUN
ap-2064	98	14	.	.	PUNCT
ap-2064	99	1	in	in	ADP
ap-2064	99	2	this	this	DET
ap-2064	99	3	paper	paper	NOUN
ap-2064	99	4	,	,	PUNCT
ap-2064	99	5	we	we	PRON
ap-2064	99	6	study	study	VERB
ap-2064	99	7	a	a	DET
ap-2064	99	8	word	word	NOUN
ap-2064	99	9	-	-	PUNCT
ap-2064	99	10	combinatorial	combinatorial	ADJ
ap-2064	99	11	interpretation	interpretation	NOUN
ap-2064	99	12	of	of	ADP
ap-2064	99	13	the	the	DET
ap-2064	99	14	biperiodic	biperiodic	ADJ
ap-2064	99	15	fibonacci	fibonacci	NOUN
ap-2064	99	16	sequence	sequence	NOUN
ap-2064	99	17	[	[	X
ap-2064	99	18	12	12	NUM
ap-2064	99	19	]	]	PUNCT
ap-2064	99	20	.	.	PUNCT
ap-2064	100	1	this	this	DET
ap-2064	100	2	problem	problem	NOUN
ap-2064	100	3	was	be	AUX
ap-2064	100	4	recently	recently	ADV
ap-2064	100	5	proposed	propose	VERB
ap-2064	100	6	by	by	ADP
ap-2064	100	7	ramírez	ramírez	PROPN
ap-2064	100	8	et	et	PROPN
ap-2064	100	9	al	al	PROPN
ap-2064	100	10	.	.	PUNCT
ap-2064	101	1	[	[	X
ap-2064	101	2	22	22	NUM
ap-2064	101	3	]	]	PUNCT
ap-2064	101	4	.	.	PUNCT
ap-2064	102	1	we	we	PRON
ap-2064	102	2	study	study	VERB
ap-2064	102	3	a	a	DET
ap-2064	102	4	family	family	NOUN
ap-2064	102	5	of	of	ADP
ap-2064	102	6	infinite	infinite	ADJ
ap-2064	102	7	words	word	NOUN
ap-2064	102	8	f(a	f(a	PROPN
ap-2064	102	9	,	,	PUNCT
ap-2064	102	10	b	b	NOUN
ap-2064	102	11	)	)	PUNCT
ap-2064	102	12	that	that	PRON
ap-2064	102	13	generalize	generalize	VERB
ap-2064	102	14	the	the	DET
ap-2064	102	15	fibonacci	fibonacci	NOUN
ap-2064	102	16	word	word	NOUN
ap-2064	102	17	and	and	CCONJ
ap-2064	102	18	the	the	DET
ap-2064	102	19	word	word	NOUN
ap-2064	102	20	fk	fk	INTJ
ap-2064	102	21	.	.	PUNCT
ap-2064	103	1	specifically	specifically	ADV
ap-2064	103	2	,	,	PUNCT
ap-2064	103	3	the	the	DET
ap-2064	103	4	nth	nth	PROPN
ap-2064	103	5	biperiodic	biperiodic	PROPN
ap-2064	103	6	fibonacci	fibonacci	PROPN
ap-2064	103	7	words	word	NOUN
ap-2064	103	8	are	be	AUX
ap-2064	103	9	words	word	NOUN
ap-2064	103	10	over	over	ADP
ap-2064	103	11	{	{	PUNCT
ap-2064	103	12	0,1	0,1	NOUN
ap-2064	103	13	}	}	PUNCT
ap-2064	103	14	defined	define	VERB
ap-2064	103	15	inductively	inductively	ADV
ap-2064	103	16	as	as	SCONJ
ap-2064	103	17	follows	follow	VERB
ap-2064	103	18	f(a	f(a	PROPN
ap-2064	103	19	,	,	PUNCT
ap-2064	103	20	b,0	b,0	ADJ
ap-2064	103	21	)	)	PUNCT
ap-2064	103	22	=	=	SYM
ap-2064	103	23	ε	ε	PROPN
ap-2064	103	24	,	,	PUNCT
ap-2064	103	25	f(a	f(a	NOUN
ap-2064	103	26	,	,	PUNCT
ap-2064	103	27	b,1	b,1	X
ap-2064	103	28	)	)	PUNCT
ap-2064	104	1	=	=	SYM
ap-2064	104	2	0	0	NUM
ap-2064	104	3	,	,	PUNCT
ap-2064	104	4	f(a	f(a	NOUN
ap-2064	104	5	,	,	PUNCT
ap-2064	104	6	b,2	b,2	VERB
ap-2064	104	7	)	)	PUNCT
ap-2064	104	8	=	=	SYM
ap-2064	104	9	0a−11	0a−11	NOUN
ap-2064	104	10	,	,	PUNCT
ap-2064	104	11	f(a	f(a	NOUN
ap-2064	104	12	,	,	PUNCT
ap-2064	104	13	b	b	NOUN
ap-2064	104	14	,	,	PUNCT
ap-2064	104	15	n	n	CCONJ
ap-2064	104	16	)	)	PUNCT
ap-2064	104	17	=	=	PRON
ap-2064	104	18	{	{	PUNCT
ap-2064	104	19	fa(a	fa(a	PROPN
ap-2064	104	20	,	,	PUNCT
ap-2064	104	21	b	b	NOUN
ap-2064	104	22	,	,	PUNCT
ap-2064	104	23	n−1)f(a	n−1)f(a	ADJ
ap-2064	104	24	,	,	PUNCT
ap-2064	104	25	b	b	NOUN
ap-2064	104	26	,	,	PUNCT
ap-2064	104	27	n−2	n−2	PROPN
ap-2064	104	28	)	)	PUNCT
ap-2064	104	29	,	,	PUNCT
ap-2064	104	30	if	if	SCONJ
ap-2064	104	31	n	n	PRON
ap-2064	104	32	≥	≥	NOUN
ap-2064	104	33	3	3	NUM
ap-2064	104	34	is	be	AUX
ap-2064	104	35	even	even	ADV
ap-2064	104	36	,	,	PUNCT
ap-2064	104	37	f	f	PROPN
ap-2064	104	38	b(a	b(a	PROPN
ap-2064	104	39	,	,	PUNCT
ap-2064	104	40	b	b	NOUN
ap-2064	104	41	,	,	PUNCT
ap-2064	104	42	n−1)f(a	n−1)f(a	ADJ
ap-2064	104	43	,	,	PUNCT
ap-2064	104	44	b	b	NOUN
ap-2064	104	45	,	,	PUNCT
ap-2064	104	46	n−2	n−2	PROPN
ap-2064	104	47	)	)	PUNCT
ap-2064	104	48	,	,	PUNCT
ap-2064	104	49	if	if	SCONJ
ap-2064	104	50	n	n	PRON
ap-2064	104	51	≥	≥	NOUN
ap-2064	104	52	3	3	NUM
ap-2064	104	53	is	be	AUX
ap-2064	104	54	odd	odd	ADJ
ap-2064	104	55	,	,	PUNCT
ap-2064	104	56	for	for	ADP
ap-2064	104	57	all	all	DET
ap-2064	104	58	a	a	PRON
ap-2064	104	59	,	,	PUNCT
ap-2064	104	60	b	b	NOUN
ap-2064	104	61	≥	≥	NUM
ap-2064	104	62	1	1	NUM
ap-2064	104	63	.	.	PUNCT
ap-2064	105	1	it	it	PRON
ap-2064	105	2	is	be	AUX
ap-2064	105	3	clear	clear	ADJ
ap-2064	105	4	that	that	SCONJ
ap-2064	105	5	|f(a	|f(a	NOUN
ap-2064	105	6	,	,	PUNCT
ap-2064	105	7	b	b	PROPN
ap-2064	105	8	,	,	PUNCT
ap-2064	105	9	n)|	n)|	NOUN
ap-2064	105	10	=	=	SYM
ap-2064	105	11	f	f	PROPN
ap-2064	105	12	(	(	PUNCT
ap-2064	105	13	a	a	DET
ap-2064	105	14	,	,	PUNCT
ap-2064	105	15	b	b	NOUN
ap-2064	105	16	)	)	PUNCT
ap-2064	105	17	n	n	NOUN
ap-2064	105	18	=	=	SYM
ap-2064	105	19	qn	qn	PROPN
ap-2064	105	20	.	.	PUNCT
ap-2064	106	1	the	the	DET
ap-2064	106	2	infinite	infinite	ADJ
ap-2064	106	3	word	word	NOUN
ap-2064	106	4	f(a	f(a	PROPN
ap-2064	106	5	,	,	PUNCT
ap-2064	106	6	b	b	NOUN
ap-2064	106	7	)	)	PUNCT
ap-2064	106	8	:	:	PUNCT
ap-2064	106	9	=	=	PUNCT
ap-2064	106	10	lim	lim	PROPN
ap-2064	106	11	n→∞	n→∞	NUM
ap-2064	106	12	f(a	f(a	PROPN
ap-2064	106	13	,	,	PUNCT
ap-2064	106	14	b	b	NOUN
ap-2064	106	15	,	,	PUNCT
ap-2064	106	16	n	n	CCONJ
ap-2064	106	17	)	)	PUNCT
ap-2064	106	18	,	,	PUNCT
ap-2064	106	19	is	be	AUX
ap-2064	106	20	called	call	VERB
ap-2064	106	21	the	the	DET
ap-2064	106	22	biperiodic	biperiodic	ADJ
ap-2064	106	23	fibonacci	fibonacci	NOUN
ap-2064	106	24	word	word	NOUN
ap-2064	106	25	.	.	PUNCT
ap-2064	107	1	in	in	ADP
ap-2064	107	2	addition	addition	NOUN
ap-2064	107	3	to	to	ADP
ap-2064	107	4	this	this	DET
ap-2064	107	5	definition	definition	NOUN
ap-2064	107	6	,	,	PUNCT
ap-2064	107	7	we	we	PRON
ap-2064	107	8	investigate	investigate	VERB
ap-2064	107	9	some	some	DET
ap-2064	107	10	new	new	ADJ
ap-2064	107	11	combinatorial	combinatorial	ADJ
ap-2064	107	12	properties	property	NOUN
ap-2064	107	13	and	and	CCONJ
ap-2064	107	14	we	we	PRON
ap-2064	107	15	associate	associate	VERB
ap-2064	107	16	a	a	DET
ap-2064	107	17	family	family	NOUN
ap-2064	107	18	of	of	ADP
ap-2064	107	19	curves	curve	NOUN
ap-2064	107	20	with	with	ADP
ap-2064	107	21	interesting	interesting	ADJ
ap-2064	107	22	geometric	geometric	ADJ
ap-2064	107	23	properties	property	NOUN
ap-2064	107	24	.	.	PUNCT
ap-2064	108	1	these	these	DET
ap-2064	108	2	properties	property	NOUN
ap-2064	108	3	are	be	AUX
ap-2064	108	4	obtained	obtain	VERB
ap-2064	108	5	from	from	ADP
ap-2064	108	6	the	the	DET
ap-2064	108	7	combinatorial	combinatorial	ADJ
ap-2064	108	8	properties	property	NOUN
ap-2064	108	9	of	of	ADP
ap-2064	108	10	the	the	DET
ap-2064	108	11	word	word	NOUN
ap-2064	108	12	f(a	f(a	PROPN
ap-2064	108	13	,	,	PUNCT
ap-2064	108	14	b	b	NOUN
ap-2064	108	15	)	)	PUNCT
ap-2064	108	16	.	.	PUNCT
ap-2064	109	1	2	2	X
ap-2064	109	2	.	.	X
ap-2064	109	3	definitions	definition	NOUN
ap-2064	109	4	and	and	CCONJ
ap-2064	109	5	notation	notation	VERB
ap-2064	109	6	the	the	DET
ap-2064	109	7	terminology	terminology	NOUN
ap-2064	109	8	and	and	CCONJ
ap-2064	109	9	notations	notation	NOUN
ap-2064	109	10	are	be	AUX
ap-2064	109	11	mainly	mainly	ADV
ap-2064	109	12	those	those	PRON
ap-2064	109	13	of	of	ADP
ap-2064	109	14	lothaire	lothaire	NOUN
ap-2064	109	15	[	[	X
ap-2064	109	16	14	14	NUM
ap-2064	109	17	]	]	PUNCT
ap-2064	109	18	and	and	CCONJ
ap-2064	109	19	allouche	allouche	NOUN
ap-2064	109	20	and	and	CCONJ
ap-2064	110	1	shallit	shallit	ADJ
ap-2064	110	2	[	[	X
ap-2064	110	3	1	1	NUM
ap-2064	110	4	]	]	PUNCT
ap-2064	110	5	.	.	PUNCT
ap-2064	111	1	let	let	VERB
ap-2064	111	2	σ	σ	NOUN
ap-2064	111	3	be	be	AUX
ap-2064	111	4	a	a	DET
ap-2064	111	5	finite	finite	ADJ
ap-2064	111	6	alphabet	alphabet	NOUN
ap-2064	111	7	,	,	PUNCT
ap-2064	111	8	whose	whose	DET
ap-2064	111	9	elements	element	NOUN
ap-2064	111	10	are	be	AUX
ap-2064	111	11	called	call	VERB
ap-2064	111	12	symbols	symbol	NOUN
ap-2064	111	13	.	.	PUNCT
ap-2064	112	1	a	a	DET
ap-2064	112	2	word	word	NOUN
ap-2064	112	3	over	over	ADP
ap-2064	112	4	σ	σ	PROPN
ap-2064	112	5	is	be	AUX
ap-2064	112	6	a	a	DET
ap-2064	112	7	finite	finite	ADJ
ap-2064	112	8	sequence	sequence	NOUN
ap-2064	112	9	of	of	ADP
ap-2064	112	10	symbols	symbol	NOUN
ap-2064	112	11	from	from	ADP
ap-2064	112	12	σ	σ	PROPN
ap-2064	112	13	.	.	PUNCT
ap-2064	113	1	the	the	DET
ap-2064	113	2	set	set	NOUN
ap-2064	113	3	of	of	ADP
ap-2064	113	4	all	all	DET
ap-2064	113	5	words	word	NOUN
ap-2064	113	6	over	over	ADP
ap-2064	113	7	σ	σ	PROPN
ap-2064	113	8	,	,	PUNCT
ap-2064	113	9	i.e.	i.e.	X
ap-2064	113	10	,	,	PUNCT
ap-2064	113	11	the	the	DET
ap-2064	113	12	free	free	PROPN
ap-2064	113	13	monoid	monoid	NOUN
ap-2064	113	14	generated	generate	VERB
ap-2064	113	15	by	by	ADP
ap-2064	113	16	σ	σ	PROPN
ap-2064	113	17	,	,	PUNCT
ap-2064	113	18	is	be	AUX
ap-2064	113	19	denoted	denote	VERB
ap-2064	113	20	by	by	ADP
ap-2064	113	21	σ∗.	σ∗.	NOUN
ap-2064	113	22	the	the	DET
ap-2064	113	23	identity	identity	NOUN
ap-2064	113	24	element	element	NOUN
ap-2064	113	25	ε	ε	PROPN
ap-2064	113	26	of	of	ADP
ap-2064	113	27	σ∗	σ∗	PROPN
ap-2064	113	28	is	be	AUX
ap-2064	113	29	called	call	VERB
ap-2064	113	30	the	the	DET
ap-2064	113	31	empty	empty	ADJ
ap-2064	113	32	word	word	NOUN
ap-2064	113	33	.	.	PUNCT
ap-2064	114	1	for	for	ADP
ap-2064	114	2	any	any	DET
ap-2064	114	3	word	word	NOUN
ap-2064	114	4	w	w	PROPN
ap-2064	114	5	∈	∈	PROPN
ap-2064	114	6	σ∗	σ∗	NOUN
ap-2064	114	7	,	,	PUNCT
ap-2064	114	8	|w|	|w|	ADJ
ap-2064	114	9	denotes	denote	NOUN
ap-2064	114	10	its	its	PRON
ap-2064	114	11	length	length	NOUN
ap-2064	114	12	,	,	PUNCT
ap-2064	114	13	i.e.	i.e.	X
ap-2064	114	14	,	,	PUNCT
ap-2064	114	15	the	the	DET
ap-2064	114	16	number	number	NOUN
ap-2064	114	17	of	of	ADP
ap-2064	114	18	symbols	symbol	NOUN
ap-2064	114	19	occurring	occur	VERB
ap-2064	114	20	in	in	ADP
ap-2064	114	21	w.	w.	PROPN
ap-2064	114	22	the	the	DET
ap-2064	114	23	length	length	NOUN
ap-2064	114	24	of	of	ADP
ap-2064	114	25	ε	ε	PROPN
ap-2064	114	26	is	be	AUX
ap-2064	114	27	taken	take	VERB
ap-2064	114	28	to	to	PART
ap-2064	114	29	be	be	AUX
ap-2064	114	30	equal	equal	ADJ
ap-2064	114	31	to	to	ADP
ap-2064	114	32	0	0	NUM
ap-2064	114	33	.	.	PUNCT
ap-2064	115	1	if	if	SCONJ
ap-2064	115	2	a	a	DET
ap-2064	115	3	∈	∈	PROPN
ap-2064	115	4	σ	σ	NOUN
ap-2064	115	5	and	and	CCONJ
ap-2064	115	6	w	w	PROPN
ap-2064	115	7	∈	∈	PROPN
ap-2064	115	8	σ∗	σ∗	NOUN
ap-2064	115	9	,	,	PUNCT
ap-2064	115	10	then	then	ADV
ap-2064	115	11	|w|a	|w|a	VERB
ap-2064	115	12	denotes	denote	VERB
ap-2064	115	13	the	the	DET
ap-2064	115	14	number	number	NOUN
ap-2064	115	15	of	of	ADP
ap-2064	115	16	occurrences	occurrence	NOUN
ap-2064	115	17	of	of	ADP
ap-2064	115	18	a	a	PRON
ap-2064	115	19	in	in	ADP
ap-2064	115	20	w.	w.	NOUN
ap-2064	115	21	for	for	ADP
ap-2064	115	22	two	two	NUM
ap-2064	115	23	words	word	NOUN
ap-2064	115	24	u	u	NOUN
ap-2064	115	25	=	=	PUNCT
ap-2064	115	26	a1a2	a1a2	PROPN
ap-2064	115	27	·	·	PUNCT
ap-2064	115	28	·	·	PUNCT
ap-2064	115	29	·	·	PUNCT
ap-2064	115	30	ak	ak	PROPN
ap-2064	115	31	and	and	CCONJ
ap-2064	115	32	v	v	NOUN
ap-2064	115	33	=	=	SYM
ap-2064	115	34	b1b2	b1b2	PROPN
ap-2064	115	35	·	·	PUNCT
ap-2064	115	36	·	·	PUNCT
ap-2064	115	37	·	·	PUNCT
ap-2064	115	38	bs	b	NOUN
ap-2064	115	39	in	in	ADP
ap-2064	115	40	σ∗	σ∗	NOUN
ap-2064	115	41	we	we	PRON
ap-2064	115	42	denote	denote	VERB
ap-2064	115	43	by	by	ADP
ap-2064	115	44	uv	uv	NOUN
ap-2064	115	45	the	the	DET
ap-2064	115	46	concatenation	concatenation	NOUN
ap-2064	115	47	of	of	ADP
ap-2064	115	48	the	the	DET
ap-2064	115	49	two	two	NUM
ap-2064	115	50	words	word	NOUN
ap-2064	115	51	,	,	PUNCT
ap-2064	115	52	that	that	ADV
ap-2064	115	53	is	is	ADV
ap-2064	115	54	,	,	PUNCT
ap-2064	115	55	uv	uv	NOUN
ap-2064	115	56	=	=	PUNCT
ap-2064	115	57	a1a2	a1a2	PROPN
ap-2064	115	58	·	·	PUNCT
ap-2064	115	59	·	·	PUNCT
ap-2064	115	60	·	·	PUNCT
ap-2064	116	1	akb1b2	akb1b2	VERB
ap-2064	116	2	·	·	PUNCT
ap-2064	116	3	·	·	PUNCT
ap-2064	116	4	·	·	PUNCT
ap-2064	116	5	bs	bs	X
ap-2064	116	6	.	.	PUNCT
ap-2064	117	1	if	if	SCONJ
ap-2064	117	2	v	v	NOUN
ap-2064	117	3	=	=	SYM
ap-2064	117	4	ε	ε	PROPN
ap-2064	117	5	,	,	PUNCT
ap-2064	117	6	then	then	ADV
ap-2064	117	7	uε	uε	X
ap-2064	117	8	=	=	NOUN
ap-2064	117	9	εu	εu	PROPN
ap-2064	117	10	=	=	SYM
ap-2064	117	11	u.	u.	VERB
ap-2064	117	12	moreover	moreover	ADV
ap-2064	117	13	,	,	PUNCT
ap-2064	117	14	by	by	ADP
ap-2064	117	15	un	un	PROPN
ap-2064	117	16	we	we	PRON
ap-2064	117	17	denote	denote	VERB
ap-2064	117	18	the	the	DET
ap-2064	117	19	word	word	NOUN
ap-2064	117	20	uu	uu	X
ap-2064	117	21	·	·	PUNCT
ap-2064	117	22	·	·	PUNCT
ap-2064	117	23	·	·	PUNCT
ap-2064	117	24	u	u	SYM
ap-2064	117	25	(	(	PUNCT
ap-2064	117	26	n	n	NUM
ap-2064	117	27	times	time	NOUN
ap-2064	117	28	)	)	PUNCT
ap-2064	117	29	.	.	PUNCT
ap-2064	118	1	a	a	DET
ap-2064	118	2	word	word	NOUN
ap-2064	118	3	v	v	NOUN
ap-2064	118	4	is	be	AUX
ap-2064	118	5	a	a	DET
ap-2064	118	6	factor	factor	NOUN
ap-2064	118	7	or	or	CCONJ
ap-2064	118	8	subword	subword	NOUN
ap-2064	118	9	of	of	ADP
ap-2064	118	10	u	u	PRON
ap-2064	118	11	if	if	SCONJ
ap-2064	118	12	there	there	PRON
ap-2064	118	13	exist	exist	VERB
ap-2064	118	14	x	x	NOUN
ap-2064	118	15	,	,	PUNCT
ap-2064	118	16	y	y	PROPN
ap-2064	118	17	∈	∈	PROPN
ap-2064	118	18	σ∗	σ∗	VERB
ap-2064	118	19	such	such	ADJ
ap-2064	118	20	that	that	DET
ap-2064	118	21	u	u	NOUN
ap-2064	118	22	=	=	SYM
ap-2064	118	23	xvy	xvy	VERB
ap-2064	118	24	.	.	PUNCT
ap-2064	119	1	if	if	SCONJ
ap-2064	119	2	x	x	NOUN
ap-2064	119	3	=	=	SYM
ap-2064	119	4	ε	ε	PROPN
ap-2064	119	5	(	(	PUNCT
ap-2064	119	6	y	y	PROPN
ap-2064	119	7	=	=	PUNCT
ap-2064	119	8	ε	ε	PROPN
ap-2064	119	9	)	)	PUNCT
ap-2064	119	10	,	,	PUNCT
ap-2064	119	11	then	then	ADV
ap-2064	119	12	v	v	NOUN
ap-2064	119	13	is	be	AUX
ap-2064	119	14	called	call	VERB
ap-2064	119	15	a	a	DET
ap-2064	119	16	prefix	prefix	NOUN
ap-2064	119	17	(	(	PUNCT
ap-2064	119	18	suffix	suffix	NOUN
ap-2064	119	19	)	)	PUNCT
ap-2064	119	20	of	of	ADP
ap-2064	119	21	u.	u.	NOUN
ap-2064	119	22	the	the	DET
ap-2064	119	23	reversal	reversal	NOUN
ap-2064	119	24	of	of	ADP
ap-2064	119	25	a	a	DET
ap-2064	119	26	word	word	NOUN
ap-2064	119	27	u	u	NOUN
ap-2064	119	28	=	=	PROPN
ap-2064	119	29	a1a2	a1a2	X
ap-2064	119	30	·	·	PUNCT
ap-2064	119	31	·	·	PUNCT
ap-2064	119	32	·	·	PUNCT
ap-2064	120	1	an	an	PRON
ap-2064	120	2	is	be	AUX
ap-2064	120	3	the	the	DET
ap-2064	120	4	word	word	NOUN
ap-2064	120	5	ur	ur	INTJ
ap-2064	120	6	=	=	NOUN
ap-2064	120	7	an	an	PRON
ap-2064	120	8	·	·	PUNCT
ap-2064	120	9	·	·	PUNCT
ap-2064	120	10	·	·	PUNCT
ap-2064	120	11	a2a1	a2a1	PUNCT
ap-2064	120	12	and	and	CCONJ
ap-2064	120	13	εr	εr	X
ap-2064	120	14	=	=	SYM
ap-2064	120	15	ε	ε	PROPN
ap-2064	120	16	.	.	PUNCT
ap-2064	121	1	a	a	DET
ap-2064	121	2	word	word	NOUN
ap-2064	121	3	u	u	NOUN
ap-2064	121	4	is	be	AUX
ap-2064	121	5	a	a	DET
ap-2064	121	6	palindrome	palindrome	NOUN
ap-2064	121	7	if	if	SCONJ
ap-2064	121	8	ur	ur	PRON
ap-2064	121	9	=	=	PUNCT
ap-2064	121	10	u.	u.	VERB
ap-2064	121	11	an	an	DET
ap-2064	121	12	infinite	infinite	ADJ
ap-2064	121	13	word	word	NOUN
ap-2064	121	14	over	over	ADP
ap-2064	121	15	σ	σ	PROPN
ap-2064	121	16	is	be	AUX
ap-2064	121	17	a	a	DET
ap-2064	121	18	map	map	NOUN
ap-2064	121	19	u	u	NOUN
ap-2064	121	20	:	:	PUNCT
ap-2064	121	21	n→	n→	PROPN
ap-2064	121	22	σ	σ	PROPN
ap-2064	121	23	.	.	PUNCT
ap-2064	122	1	it	it	PRON
ap-2064	122	2	is	be	AUX
ap-2064	122	3	written	write	VERB
ap-2064	122	4	u	u	NOUN
ap-2064	122	5	=	=	NOUN
ap-2064	122	6	a1a2a3	a1a2a3	VERB
ap-2064	122	7	.	.	PUNCT
ap-2064	122	8	.	.	PUNCT
ap-2064	123	1	..	..	PUNCT
ap-2064	124	1	the	the	DET
ap-2064	124	2	set	set	NOUN
ap-2064	124	3	of	of	ADP
ap-2064	124	4	all	all	DET
ap-2064	124	5	infinite	infinite	ADJ
ap-2064	124	6	words	word	NOUN
ap-2064	124	7	over	over	ADP
ap-2064	124	8	σ	σ	PROPN
ap-2064	124	9	is	be	AUX
ap-2064	124	10	denoted	denote	VERB
ap-2064	124	11	by	by	ADP
ap-2064	124	12	σω	σω	PROPN
ap-2064	124	13	.	.	PROPN
ap-2064	124	14	51	51	NUM
ap-2064	124	15	josé	josé	PROPN
ap-2064	124	16	l.	l.	PROPN
ap-2064	124	17	ramírez	ramírez	PROPN
ap-2064	124	18	,	,	PUNCT
ap-2064	125	1	gustavo	gustavo	PROPN
ap-2064	125	2	n.	n.	PROPN
ap-2064	125	3	rubiano	rubiano	PROPN
ap-2064	125	4	acta	acta	PROPN
ap-2064	125	5	polytechnica	polytechnica	PROPN
ap-2064	125	6	f	f	PROPN
ap-2064	126	1	[	[	X
ap-2064	126	2	1	1	NUM
ap-2064	126	3	]	]	X
ap-2064	126	4	=	=	PUNCT
ap-2064	126	5	1011010110110	1011010110110	NUM
ap-2064	126	6	·	·	PUNCT
ap-2064	126	7	·	·	PUNCT
ap-2064	126	8	·	·	PUNCT
ap-2064	127	1	=	=	PUNCT
ap-2064	127	2	f	f	X
ap-2064	127	3	f	f	X
ap-2064	128	1	[	[	X
ap-2064	128	2	2	2	NUM
ap-2064	128	3	]	]	X
ap-2064	128	4	=	=	SYM
ap-2064	128	5	0100101001001	0100101001001	NUM
ap-2064	128	6	·	·	PUNCT
ap-2064	128	7	·	·	PUNCT
ap-2064	128	8	·	·	PUNCT
ap-2064	129	1	=	=	PUNCT
ap-2064	129	2	f	f	X
ap-2064	129	3	f	f	X
ap-2064	130	1	[	[	X
ap-2064	130	2	3	3	X
ap-2064	130	3	]	]	X
ap-2064	130	4	=	=	SYM
ap-2064	130	5	0010001001000	0010001001000	NUM
ap-2064	130	6	·	·	PUNCT
ap-2064	130	7	·	·	PUNCT
ap-2064	130	8	·	·	PUNCT
ap-2064	130	9	f	f	X
ap-2064	131	1	[	[	X
ap-2064	131	2	1	1	NUM
ap-2064	131	3	]	]	PUNCT
ap-2064	131	4	16	16	NUM
ap-2064	131	5	f	f	NOUN
ap-2064	132	1	[	[	X
ap-2064	132	2	2	2	NUM
ap-2064	132	3	]	]	PUNCT
ap-2064	132	4	16	16	NUM
ap-2064	132	5	f	f	NOUN
ap-2064	132	6	[	[	X
ap-2064	132	7	3	3	NUM
ap-2064	132	8	]	]	PUNCT
ap-2064	132	9	16	16	NUM
ap-2064	132	10	f1	f1	NOUN
ap-2064	132	11	=	=	SYM
ap-2064	132	12	1011010110110	1011010110110	NUM
ap-2064	132	13	·	·	PUNCT
ap-2064	132	14	·	·	PUNCT
ap-2064	132	15	·	·	PUNCT
ap-2064	133	1	=	=	PUNCT
ap-2064	133	2	f	f	X
ap-2064	133	3	f5	f5	PROPN
ap-2064	133	4	=	=	SYM
ap-2064	133	5	0000100001000	0000100001000	NUM
ap-2064	133	6	·	·	PUNCT
ap-2064	133	7	·	·	PUNCT
ap-2064	133	8	·	·	PUNCT
ap-2064	133	9	f6	f6	X
ap-2064	134	1	=	=	PUNCT
ap-2064	134	2	0000010000010	0000010000010	NUM
ap-2064	134	3	·	·	PUNCT
ap-2064	134	4	·	·	PUNCT
ap-2064	134	5	·	·	PUNCT
ap-2064	134	6	f1,18	f1,18	NUM
ap-2064	135	1	f5,6	f5,6	NUM
ap-2064	135	2	f6,6	f6,6	PROPN
ap-2064	135	3	table	table	NOUN
ap-2064	135	4	2	2	NUM
ap-2064	135	5	.	.	PUNCT
ap-2064	136	1	some	some	DET
ap-2064	136	2	curves	curve	NOUN
ap-2064	136	3	f	f	NOUN
ap-2064	137	1	[	[	X
ap-2064	137	2	i	i	X
ap-2064	137	3	]	]	PUNCT
ap-2064	137	4	16	16	NUM
ap-2064	137	5	and	and	CCONJ
ap-2064	137	6	fk	fk	INTJ
ap-2064	137	7	,	,	PUNCT
ap-2064	137	8	n.	n.	ADJ
ap-2064	137	9	example	example	NOUN
ap-2064	137	10	1	1	X
ap-2064	137	11	.	.	PUNCT
ap-2064	138	1	let	let	VERB
ap-2064	138	2	p	p	NOUN
ap-2064	138	3	=	=	NOUN
ap-2064	138	4	(	(	PUNCT
ap-2064	138	5	pn)n≥1	pn)n≥1	NOUN
ap-2064	138	6	=	=	SYM
ap-2064	138	7	0110101000101	0110101000101	NUM
ap-2064	138	8	·	·	PUNCT
ap-2064	138	9	·	·	PUNCT
ap-2064	138	10	·	·	PUNCT
ap-2064	138	11	be	be	AUX
ap-2064	138	12	an	an	DET
ap-2064	138	13	infinite	infinite	ADJ
ap-2064	138	14	word	word	NOUN
ap-2064	138	15	,	,	PUNCT
ap-2064	138	16	where	where	SCONJ
ap-2064	138	17	pn	pn	PROPN
ap-2064	138	18	=	=	NOUN
ap-2064	138	19	1	1	NUM
ap-2064	138	20	if	if	SCONJ
ap-2064	138	21	n	n	PRON
ap-2064	138	22	is	be	AUX
ap-2064	138	23	a	a	DET
ap-2064	138	24	prime	prime	ADJ
ap-2064	138	25	number	number	NOUN
ap-2064	138	26	and	and	CCONJ
ap-2064	138	27	pn	pn	NOUN
ap-2064	138	28	=	=	NOUN
ap-2064	138	29	0	0	PUNCT
ap-2064	138	30	otherwise	otherwise	ADV
ap-2064	138	31	.	.	PUNCT
ap-2064	139	1	the	the	DET
ap-2064	139	2	word	word	NOUN
ap-2064	139	3	p	p	NOUN
ap-2064	139	4	is	be	AUX
ap-2064	139	5	called	call	VERB
ap-2064	139	6	the	the	DET
ap-2064	139	7	characteristic	characteristic	ADJ
ap-2064	139	8	sequence	sequence	NOUN
ap-2064	139	9	of	of	ADP
ap-2064	139	10	the	the	DET
ap-2064	139	11	prime	prime	ADJ
ap-2064	139	12	numbers	number	NOUN
ap-2064	139	13	.	.	PUNCT
ap-2064	140	1	let	let	VERB
ap-2064	140	2	σ	σ	NOUN
ap-2064	140	3	and	and	CCONJ
ap-2064	140	4	∆	∆	PROPN
ap-2064	140	5	be	be	VERB
ap-2064	140	6	alphabets	alphabet	NOUN
ap-2064	140	7	.	.	PUNCT
ap-2064	141	1	a	a	DET
ap-2064	141	2	morphism	morphism	NOUN
ap-2064	141	3	is	be	AUX
ap-2064	141	4	a	a	DET
ap-2064	141	5	map	map	NOUN
ap-2064	141	6	h	h	NOUN
ap-2064	141	7	:	:	PUNCT
ap-2064	141	8	σ∗	σ∗	PROPN
ap-2064	141	9	→	→	PUNCT
ap-2064	141	10	∆∗	∆∗	VERB
ap-2064	141	11	such	such	ADJ
ap-2064	141	12	that	that	DET
ap-2064	141	13	h(xy	h(xy	NOUN
ap-2064	141	14	)	)	PUNCT
ap-2064	141	15	=	=	PUNCT
ap-2064	141	16	h(x)h(y	h(x)h(y	X
ap-2064	141	17	)	)	PUNCT
ap-2064	141	18	for	for	ADP
ap-2064	141	19	all	all	DET
ap-2064	141	20	x	x	NOUN
ap-2064	141	21	,	,	PUNCT
ap-2064	141	22	y	y	PROPN
ap-2064	141	23	∈	∈	PROPN
ap-2064	141	24	σ∗.	σ∗.	NOUN
ap-2064	141	25	it	it	PRON
ap-2064	141	26	is	be	AUX
ap-2064	141	27	clear	clear	ADJ
ap-2064	142	1	that	that	SCONJ
ap-2064	142	2	h(ε	h(ε	PROPN
ap-2064	142	3	)	)	PUNCT
ap-2064	142	4	=	=	SYM
ap-2064	142	5	ε	ε	PROPN
ap-2064	142	6	.	.	PUNCT
ap-2064	143	1	furthermore	furthermore	ADV
ap-2064	143	2	,	,	PUNCT
ap-2064	143	3	a	a	DET
ap-2064	143	4	morphism	morphism	NOUN
ap-2064	143	5	is	be	AUX
ap-2064	143	6	completely	completely	ADV
ap-2064	143	7	determined	determine	VERB
ap-2064	143	8	by	by	ADP
ap-2064	143	9	its	its	PRON
ap-2064	143	10	action	action	NOUN
ap-2064	143	11	on	on	ADP
ap-2064	143	12	single	single	ADJ
ap-2064	143	13	symbols	symbol	NOUN
ap-2064	143	14	.	.	PUNCT
ap-2064	144	1	for	for	ADP
ap-2064	144	2	instance	instance	NOUN
ap-2064	144	3	,	,	PUNCT
ap-2064	144	4	the	the	DET
ap-2064	144	5	fibonacci	fibonacci	NOUN
ap-2064	144	6	word	word	NOUN
ap-2064	144	7	f	f	PROPN
ap-2064	144	8	satisfies	satisfy	VERB
ap-2064	144	9	limn→∞	limn→∞	PROPN
ap-2064	144	10	σn(1	σn(1	NOUN
ap-2064	144	11	)	)	PUNCT
ap-2064	144	12	=	=	SYM
ap-2064	144	13	f	f	PROPN
ap-2064	144	14	,	,	PUNCT
ap-2064	144	15	where	where	SCONJ
ap-2064	144	16	σ	σ	NOUN
ap-2064	144	17	:	:	PUNCT
ap-2064	144	18	{	{	PUNCT
ap-2064	144	19	0,1	0,1	NOUN
ap-2064	144	20	}	}	PUNCT
ap-2064	144	21	→	→	SYM
ap-2064	144	22	{	{	PUNCT
ap-2064	144	23	0,1}∗	0,1}∗	NOUN
ap-2064	144	24	is	be	AUX
ap-2064	144	25	the	the	DET
ap-2064	144	26	morphism	morphism	NOUN
ap-2064	144	27	defined	define	VERB
ap-2064	144	28	by	by	ADP
ap-2064	144	29	σ(0	σ(0	PROPN
ap-2064	144	30	)	)	PUNCT
ap-2064	144	31	=	=	SYM
ap-2064	144	32	01	01	NUM
ap-2064	144	33	and	and	CCONJ
ap-2064	144	34	σ(1	σ(1	PROPN
ap-2064	144	35	)	)	PUNCT
ap-2064	145	1	=	=	SYM
ap-2064	145	2	0	0	X
ap-2064	145	3	.	.	PUNCT
ap-2064	146	1	this	this	DET
ap-2064	146	2	morphism	morphism	NOUN
ap-2064	146	3	is	be	AUX
ap-2064	146	4	called	call	VERB
ap-2064	146	5	the	the	DET
ap-2064	146	6	fibonacci	fibonacci	NOUN
ap-2064	146	7	morphism	morphism	NOUN
ap-2064	146	8	.	.	PUNCT
ap-2064	147	1	the	the	DET
ap-2064	147	2	fibonacci	fibonacci	NOUN
ap-2064	147	3	word	word	NOUN
ap-2064	147	4	f	f	PROPN
ap-2064	147	5	can	can	AUX
ap-2064	147	6	be	be	AUX
ap-2064	147	7	defined	define	VERB
ap-2064	147	8	in	in	ADP
ap-2064	147	9	several	several	ADJ
ap-2064	147	10	different	different	ADJ
ap-2064	147	11	ways	way	NOUN
ap-2064	147	12	,	,	PUNCT
ap-2064	147	13	see	see	VERB
ap-2064	147	14	,	,	PUNCT
ap-2064	147	15	e.g.	e.g.	ADV
ap-2064	147	16	,	,	PUNCT
ap-2064	147	17	[	[	X
ap-2064	147	18	3	3	NUM
ap-2064	147	19	]	]	PUNCT
ap-2064	147	20	.	.	PUNCT
ap-2064	148	1	there	there	PRON
ap-2064	148	2	is	be	VERB
ap-2064	148	3	a	a	DET
ap-2064	148	4	special	special	ADJ
ap-2064	148	5	class	class	NOUN
ap-2064	148	6	of	of	ADP
ap-2064	148	7	infinite	infinite	ADJ
ap-2064	148	8	words	word	NOUN
ap-2064	148	9	,	,	PUNCT
ap-2064	148	10	with	with	ADP
ap-2064	148	11	many	many	ADJ
ap-2064	148	12	remarkable	remarkable	ADJ
ap-2064	148	13	properties	property	NOUN
ap-2064	148	14	,	,	PUNCT
ap-2064	148	15	the	the	DET
ap-2064	148	16	so	so	ADV
ap-2064	148	17	-	-	PUNCT
ap-2064	148	18	called	call	VERB
ap-2064	148	19	sturmian	sturmian	NOUN
ap-2064	148	20	words	word	NOUN
ap-2064	148	21	.	.	PUNCT
ap-2064	149	1	these	these	DET
ap-2064	149	2	words	word	NOUN
ap-2064	149	3	admit	admit	VERB
ap-2064	149	4	several	several	ADJ
ap-2064	149	5	equivalent	equivalent	ADJ
ap-2064	149	6	definitions	definition	NOUN
ap-2064	149	7	(	(	PUNCT
ap-2064	149	8	see	see	VERB
ap-2064	149	9	,	,	PUNCT
ap-2064	149	10	e.g.	e.g.	ADV
ap-2064	149	11	[	[	X
ap-2064	149	12	1	1	NUM
ap-2064	149	13	,	,	PUNCT
ap-2064	149	14	2	2	NUM
ap-2064	149	15	,	,	PUNCT
ap-2064	149	16	14	14	NUM
ap-2064	149	17	]	]	PUNCT
ap-2064	149	18	)	)	PUNCT
ap-2064	149	19	.	.	PUNCT
ap-2064	150	1	let	let	VERB
ap-2064	150	2	w	w	PROPN
ap-2064	150	3	∈	∈	PROPN
ap-2064	150	4	σω	σω	VERB
ap-2064	150	5	.	.	PUNCT
ap-2064	151	1	we	we	PRON
ap-2064	151	2	define	define	VERB
ap-2064	151	3	p	p	X
ap-2064	151	4	(	(	PUNCT
ap-2064	151	5	w	w	PROPN
ap-2064	151	6	,	,	PUNCT
ap-2064	151	7	n	n	CCONJ
ap-2064	151	8	)	)	PUNCT
ap-2064	151	9	,	,	PUNCT
ap-2064	151	10	the	the	DET
ap-2064	151	11	complexity	complexity	NOUN
ap-2064	151	12	function	function	NOUN
ap-2064	151	13	of	of	ADP
ap-2064	151	14	w	w	PROPN
ap-2064	151	15	,	,	PUNCT
ap-2064	151	16	to	to	PART
ap-2064	151	17	be	be	AUX
ap-2064	151	18	the	the	DET
ap-2064	151	19	map	map	NOUN
ap-2064	151	20	that	that	PRON
ap-2064	151	21	counts	count	VERB
ap-2064	151	22	,	,	PUNCT
ap-2064	151	23	for	for	ADP
ap-2064	151	24	all	all	DET
ap-2064	151	25	integer	integer	NOUN
ap-2064	151	26	n	n	PRON
ap-2064	151	27	≥	≥	NOUN
ap-2064	151	28	0	0	NUM
ap-2064	151	29	,	,	PUNCT
ap-2064	151	30	the	the	DET
ap-2064	151	31	number	number	NOUN
ap-2064	151	32	of	of	ADP
ap-2064	151	33	subwords	subword	NOUN
ap-2064	151	34	of	of	ADP
ap-2064	151	35	length	length	NOUN
ap-2064	151	36	n	n	PROPN
ap-2064	151	37	in	in	ADP
ap-2064	151	38	w.	w.	PROPN
ap-2064	151	39	an	an	DET
ap-2064	151	40	infinite	infinite	ADJ
ap-2064	151	41	word	word	NOUN
ap-2064	151	42	w	w	NOUN
ap-2064	151	43	is	be	AUX
ap-2064	151	44	a	a	DET
ap-2064	151	45	sturmian	sturmian	ADJ
ap-2064	151	46	word	word	NOUN
ap-2064	151	47	if	if	SCONJ
ap-2064	151	48	p	p	PROPN
ap-2064	151	49	(	(	PUNCT
ap-2064	151	50	w	w	PROPN
ap-2064	151	51	,	,	PUNCT
ap-2064	151	52	n	n	CCONJ
ap-2064	151	53	)	)	PUNCT
ap-2064	151	54	=	=	SYM
ap-2064	151	55	n+	n+	ADP
ap-2064	151	56	1	1	NUM
ap-2064	151	57	for	for	ADP
ap-2064	151	58	all	all	DET
ap-2064	151	59	integers	integer	NOUN
ap-2064	151	60	n	n	PRON
ap-2064	151	61	≥	≥	NOUN
ap-2064	151	62	0	0	NUM
ap-2064	151	63	.	.	PUNCT
ap-2064	152	1	since	since	SCONJ
ap-2064	152	2	for	for	ADP
ap-2064	152	3	any	any	DET
ap-2064	152	4	sturmian	sturmian	ADJ
ap-2064	152	5	word	word	NOUN
ap-2064	152	6	p	p	X
ap-2064	152	7	(	(	PUNCT
ap-2064	152	8	w	w	PROPN
ap-2064	152	9	,	,	PUNCT
ap-2064	152	10	1	1	NUM
ap-2064	152	11	)	)	PUNCT
ap-2064	152	12	=	=	SYM
ap-2064	152	13	2	2	NUM
ap-2064	152	14	,	,	PUNCT
ap-2064	152	15	sturmian	sturmian	NOUN
ap-2064	152	16	words	word	NOUN
ap-2064	152	17	are	be	AUX
ap-2064	152	18	over	over	ADP
ap-2064	152	19	two	two	NUM
ap-2064	152	20	symbols	symbol	NOUN
ap-2064	152	21	.	.	PUNCT
ap-2064	153	1	the	the	DET
ap-2064	153	2	word	word	NOUN
ap-2064	153	3	p	p	X
ap-2064	153	4	,	,	PUNCT
ap-2064	153	5	in	in	ADP
ap-2064	153	6	example	example	NOUN
ap-2064	153	7	1	1	NUM
ap-2064	153	8	,	,	PUNCT
ap-2064	153	9	is	be	AUX
ap-2064	153	10	not	not	PART
ap-2064	153	11	a	a	DET
ap-2064	153	12	sturmian	sturmian	NOUN
ap-2064	153	13	word	word	NOUN
ap-2064	153	14	because	because	SCONJ
ap-2064	153	15	p	p	PROPN
ap-2064	153	16	(	(	PUNCT
ap-2064	153	17	p	p	X
ap-2064	153	18	,	,	PUNCT
ap-2064	153	19	2	2	NUM
ap-2064	153	20	)	)	PUNCT
ap-2064	153	21	=	=	SYM
ap-2064	153	22	4	4	X
ap-2064	153	23	.	.	PUNCT
ap-2064	153	24	given	give	VERB
ap-2064	153	25	two	two	NUM
ap-2064	153	26	real	real	ADJ
ap-2064	153	27	numbers	number	NOUN
ap-2064	153	28	θ	θ	PROPN
ap-2064	153	29	,	,	PUNCT
ap-2064	153	30	ρ	ρ	PROPN
ap-2064	153	31	∈	∈	PROPN
ap-2064	153	32	r	r	NOUN
ap-2064	153	33	with	with	ADP
ap-2064	153	34	θ	θ	PROPN
ap-2064	153	35	irrational	irrational	ADJ
ap-2064	153	36	and	and	CCONJ
ap-2064	153	37	0	0	NUM
ap-2064	153	38	<	<	X
ap-2064	153	39	θ	θ	X
ap-2064	153	40	<	<	X
ap-2064	153	41	1	1	NUM
ap-2064	153	42	,	,	PUNCT
ap-2064	153	43	0	0	NUM
ap-2064	153	44	≤	≤	NUM
ap-2064	153	45	ρ	ρ	NOUN
ap-2064	153	46	<	<	X
ap-2064	153	47	1	1	NUM
ap-2064	153	48	,	,	PUNCT
ap-2064	153	49	we	we	PRON
ap-2064	153	50	define	define	VERB
ap-2064	153	51	the	the	DET
ap-2064	153	52	infinite	infinite	ADJ
ap-2064	153	53	word	word	NOUN
ap-2064	153	54	w	w	NOUN
ap-2064	153	55	=	=	SYM
ap-2064	153	56	w1w2w3	w1w2w3	NOUN
ap-2064	153	57	·	·	PUNCT
ap-2064	153	58	·	·	PUNCT
ap-2064	153	59	·	·	PUNCT
ap-2064	153	60	by	by	ADP
ap-2064	153	61	wn	wn	PROPN
ap-2064	153	62	=	=	SYM
ap-2064	153	63	b(n+	b(n+	PROPN
ap-2064	153	64	1)θ	1)θ	NUM
ap-2064	153	65	+	+	CCONJ
ap-2064	153	66	ρc	ρc	VERB
ap-2064	153	67	−	−	PROPN
ap-2064	153	68	bnθ	bnθ	NOUN
ap-2064	153	69	+	+	CCONJ
ap-2064	153	70	ρc	ρc	VERB
ap-2064	153	71	.	.	PUNCT
ap-2064	154	1	the	the	DET
ap-2064	154	2	numbers	number	NOUN
ap-2064	154	3	θ	θ	PROPN
ap-2064	154	4	and	and	CCONJ
ap-2064	154	5	ρ	ρ	PROPN
ap-2064	154	6	are	be	AUX
ap-2064	154	7	called	call	VERB
ap-2064	154	8	the	the	DET
ap-2064	154	9	slope	slope	NOUN
ap-2064	154	10	and	and	CCONJ
ap-2064	154	11	the	the	DET
ap-2064	154	12	intercept	intercept	NOUN
ap-2064	154	13	,	,	PUNCT
ap-2064	154	14	respectively	respectively	ADV
ap-2064	154	15	.	.	PUNCT
ap-2064	155	1	words	word	NOUN
ap-2064	155	2	of	of	ADP
ap-2064	155	3	this	this	DET
ap-2064	155	4	form	form	NOUN
ap-2064	155	5	are	be	AUX
ap-2064	155	6	called	call	VERB
ap-2064	155	7	lower	low	ADJ
ap-2064	155	8	mechanical	mechanical	ADJ
ap-2064	155	9	words	word	NOUN
ap-2064	155	10	and	and	CCONJ
ap-2064	155	11	are	be	AUX
ap-2064	155	12	known	know	VERB
ap-2064	155	13	to	to	PART
ap-2064	155	14	be	be	AUX
ap-2064	155	15	equivalent	equivalent	ADJ
ap-2064	155	16	to	to	ADP
ap-2064	155	17	sturmian	sturmian	NOUN
ap-2064	155	18	words	word	NOUN
ap-2064	155	19	[	[	X
ap-2064	155	20	14	14	NUM
ap-2064	155	21	]	]	PUNCT
ap-2064	155	22	.	.	PUNCT
ap-2064	156	1	as	as	ADP
ap-2064	156	2	a	a	DET
ap-2064	156	3	special	special	ADJ
ap-2064	156	4	case	case	NOUN
ap-2064	156	5	,	,	PUNCT
ap-2064	156	6	when	when	SCONJ
ap-2064	156	7	ρ	ρ	PROPN
ap-2064	156	8	=	=	SYM
ap-2064	156	9	0	0	NUM
ap-2064	156	10	,	,	PUNCT
ap-2064	156	11	we	we	PRON
ap-2064	156	12	obtain	obtain	VERB
ap-2064	156	13	characteristic	characteristic	ADJ
ap-2064	156	14	words	word	NOUN
ap-2064	156	15	.	.	PUNCT
ap-2064	157	1	definition	definition	NOUN
ap-2064	157	2	2	2	NUM
ap-2064	157	3	.	.	PUNCT
ap-2064	158	1	let	let	VERB
ap-2064	158	2	θ	θ	NOUN
ap-2064	158	3	be	be	AUX
ap-2064	158	4	an	an	DET
ap-2064	158	5	irrational	irrational	ADJ
ap-2064	158	6	number	number	NOUN
ap-2064	158	7	with	with	ADP
ap-2064	158	8	0	0	NUM
ap-2064	158	9	<	<	X
ap-2064	158	10	θ	θ	X
ap-2064	158	11	<	<	X
ap-2064	158	12	1	1	NUM
ap-2064	158	13	.	.	PUNCT
ap-2064	158	14	for	for	ADP
ap-2064	158	15	n	n	PRON
ap-2064	158	16	≥	≥	NUM
ap-2064	158	17	1	1	NUM
ap-2064	158	18	,	,	PUNCT
ap-2064	158	19	define	define	VERB
ap-2064	158	20	wθ(n	wθ(n	PRON
ap-2064	158	21	)	)	PUNCT
ap-2064	158	22	:	:	PUNCT
ap-2064	159	1	=	=	PUNCT
ap-2064	159	2	b(n	b(n	NOUN
ap-2064	159	3	+	+	CCONJ
ap-2064	159	4	1)θc	1)θc	PROPN
ap-2064	159	5	−	−	PROPN
ap-2064	159	6	bnθc	bnθc	NOUN
ap-2064	159	7	,	,	PUNCT
ap-2064	159	8	and	and	CCONJ
ap-2064	159	9	w(θ	w(θ	PROPN
ap-2064	159	10	)	)	PUNCT
ap-2064	159	11	:	:	PUNCT
ap-2064	159	12	=	=	SYM
ap-2064	159	13	wθ(1)wθ(2)wθ(3	wθ(1)wθ(2)wθ(3	PROPN
ap-2064	159	14	)	)	PUNCT
ap-2064	159	15	·	·	PUNCT
ap-2064	159	16	·	·	PUNCT
ap-2064	159	17	·	·	PUNCT
ap-2064	159	18	.	.	PUNCT
ap-2064	160	1	then	then	ADV
ap-2064	160	2	w(θ	w(θ	PROPN
ap-2064	160	3	)	)	PUNCT
ap-2064	160	4	is	be	AUX
ap-2064	160	5	called	call	VERB
ap-2064	160	6	the	the	DET
ap-2064	160	7	characteristic	characteristic	ADJ
ap-2064	160	8	word	word	NOUN
ap-2064	160	9	with	with	ADP
ap-2064	160	10	slope	slope	NOUN
ap-2064	160	11	θ	θ	PROPN
ap-2064	160	12	.	.	PUNCT
ap-2064	160	13	note	note	VERB
ap-2064	160	14	that	that	SCONJ
ap-2064	160	15	every	every	DET
ap-2064	160	16	irrational	irrational	ADJ
ap-2064	160	17	θ	θ	PROPN
ap-2064	160	18	∈	∈	PROPN
ap-2064	160	19	(	(	PUNCT
ap-2064	160	20	0	0	NUM
ap-2064	160	21	,	,	PUNCT
ap-2064	160	22	1	1	NUM
ap-2064	160	23	)	)	PUNCT
ap-2064	160	24	has	have	VERB
ap-2064	160	25	a	a	DET
ap-2064	160	26	unique	unique	ADJ
ap-2064	160	27	continued	continue	VERB
ap-2064	160	28	fraction	fraction	NOUN
ap-2064	160	29	expansion	expansion	NOUN
ap-2064	160	30	θ	θ	NOUN
ap-2064	161	1	=	=	PUNCT
ap-2064	162	1	[	[	X
ap-2064	162	2	0	0	NUM
ap-2064	162	3	,	,	PUNCT
ap-2064	162	4	a1	a1	NOUN
ap-2064	162	5	,	,	PUNCT
ap-2064	162	6	a2	a2	PROPN
ap-2064	162	7	,	,	PUNCT
ap-2064	162	8	a3	a3	NOUN
ap-2064	162	9	,	,	PUNCT
ap-2064	162	10	.	.	PUNCT
ap-2064	162	11	.	.	PUNCT
ap-2064	162	12	.	.	PUNCT
ap-2064	162	13	]	]	PUNCT
ap-2064	163	1	=	=	SYM
ap-2064	163	2	1	1	NUM
ap-2064	163	3	a1	a1	NOUN
ap-2064	163	4	+	+	CCONJ
ap-2064	163	5	1	1	NUM
ap-2064	163	6	a2	a2	NOUN
ap-2064	163	7	+	+	CCONJ
ap-2064	163	8	1	1	NUM
ap-2064	163	9	a3	a3	NOUN
ap-2064	163	10	+	+	X
ap-2064	163	11	·	·	PUNCT
ap-2064	163	12	·	·	PUNCT
ap-2064	163	13	·	·	PUNCT
ap-2064	163	14	,	,	PUNCT
ap-2064	163	15	where	where	SCONJ
ap-2064	163	16	each	each	PRON
ap-2064	163	17	ai	ai	VERB
ap-2064	163	18	is	be	AUX
ap-2064	163	19	a	a	DET
ap-2064	163	20	positive	positive	ADJ
ap-2064	163	21	integer	integer	NOUN
ap-2064	163	22	.	.	PUNCT
ap-2064	164	1	let	let	VERB
ap-2064	164	2	θ	θ	NOUN
ap-2064	164	3	=	=	PUNCT
ap-2064	165	1	[	[	X
ap-2064	165	2	0	0	NUM
ap-2064	165	3	,	,	PUNCT
ap-2064	165	4	1	1	NUM
ap-2064	165	5	+	+	CCONJ
ap-2064	165	6	d1	d1	PROPN
ap-2064	165	7	,	,	PUNCT
ap-2064	165	8	d2	d2	PROPN
ap-2064	165	9	,	,	PUNCT
ap-2064	165	10	.	.	PUNCT
ap-2064	165	11	.	.	PUNCT
ap-2064	165	12	.	.	PUNCT
ap-2064	166	1	]	]	PUNCT
ap-2064	166	2	be	be	AUX
ap-2064	166	3	an	an	DET
ap-2064	166	4	irrational	irrational	ADJ
ap-2064	166	5	number	number	NOUN
ap-2064	166	6	with	with	ADP
ap-2064	166	7	d1	d1	PROPN
ap-2064	166	8	≥	≥	NOUN
ap-2064	166	9	0	0	NUM
ap-2064	166	10	and	and	CCONJ
ap-2064	166	11	dn	dn	ADP
ap-2064	166	12	>	>	X
ap-2064	166	13	0	0	PUNCT
ap-2064	167	1	for	for	ADP
ap-2064	167	2	n	n	X
ap-2064	167	3	>	>	X
ap-2064	167	4	1	1	X
ap-2064	167	5	.	.	PUNCT
ap-2064	168	1	we	we	PRON
ap-2064	168	2	use	use	VERB
ap-2064	168	3	the	the	DET
ap-2064	168	4	continued	continue	VERB
ap-2064	168	5	fraction	fraction	NOUN
ap-2064	168	6	expansion	expansion	NOUN
ap-2064	168	7	of	of	ADP
ap-2064	168	8	θ	θ	PROPN
ap-2064	168	9	as	as	ADP
ap-2064	168	10	a	a	DET
ap-2064	168	11	directive	directive	NOUN
ap-2064	168	12	sequence	sequence	NOUN
ap-2064	168	13	in	in	ADP
ap-2064	168	14	the	the	DET
ap-2064	168	15	following	following	ADJ
ap-2064	168	16	way	way	NOUN
ap-2064	168	17	.	.	PUNCT
ap-2064	169	1	we	we	PRON
ap-2064	169	2	associate	associate	VERB
ap-2064	169	3	a	a	DET
ap-2064	169	4	sequence	sequence	NOUN
ap-2064	169	5	(	(	PUNCT
ap-2064	169	6	sn)n≥−1	sn)n≥−1	NOUN
ap-2064	169	7	of	of	ADP
ap-2064	169	8	words	word	NOUN
ap-2064	169	9	defined	define	VERB
ap-2064	169	10	by	by	ADP
ap-2064	169	11	s−1	s−1	PROPN
ap-2064	169	12	=	=	SYM
ap-2064	169	13	1	1	NUM
ap-2064	169	14	,	,	PUNCT
ap-2064	169	15	s0	s0	PROPN
ap-2064	169	16	=	=	SYM
ap-2064	169	17	0	0	NUM
ap-2064	169	18	,	,	PUNCT
ap-2064	169	19	sn	sn	NOUN
ap-2064	169	20	=	=	SYM
ap-2064	169	21	sdnn−1sn−2	sdnn−1sn−2	PROPN
ap-2064	169	22	,	,	PUNCT
ap-2064	169	23	for	for	ADP
ap-2064	169	24	n	n	PRON
ap-2064	169	25	≥	≥	NUM
ap-2064	169	26	1	1	NUM
ap-2064	169	27	.	.	PUNCT
ap-2064	170	1	such	such	DET
ap-2064	170	2	a	a	DET
ap-2064	170	3	sequence	sequence	NOUN
ap-2064	170	4	of	of	ADP
ap-2064	170	5	words	word	NOUN
ap-2064	170	6	is	be	AUX
ap-2064	170	7	called	call	VERB
ap-2064	170	8	a	a	DET
ap-2064	170	9	standard	standard	ADJ
ap-2064	170	10	sequence	sequence	NOUN
ap-2064	170	11	.	.	PUNCT
ap-2064	171	1	this	this	DET
ap-2064	171	2	sequence	sequence	NOUN
ap-2064	171	3	is	be	AUX
ap-2064	171	4	related	relate	VERB
ap-2064	171	5	to	to	ADP
ap-2064	171	6	characteristic	characteristic	ADJ
ap-2064	171	7	words	word	NOUN
ap-2064	171	8	in	in	ADP
ap-2064	171	9	the	the	DET
ap-2064	171	10	following	following	ADJ
ap-2064	171	11	way	way	NOUN
ap-2064	171	12	.	.	PUNCT
ap-2064	172	1	observe	observe	VERB
ap-2064	172	2	that	that	SCONJ
ap-2064	172	3	,	,	PUNCT
ap-2064	172	4	for	for	ADP
ap-2064	172	5	any	any	DET
ap-2064	172	6	n	n	PRON
ap-2064	172	7	≥	≥	NOUN
ap-2064	172	8	0	0	NUM
ap-2064	172	9	,	,	PUNCT
ap-2064	172	10	sn	sn	PROPN
ap-2064	172	11	is	be	AUX
ap-2064	172	12	a	a	DET
ap-2064	172	13	prefix	prefix	NOUN
ap-2064	172	14	of	of	ADP
ap-2064	172	15	sn+1	sn+1	NOUN
ap-2064	172	16	,	,	PUNCT
ap-2064	172	17	which	which	PRON
ap-2064	172	18	gives	give	VERB
ap-2064	172	19	meaning	meaning	NOUN
ap-2064	172	20	to	to	ADP
ap-2064	172	21	limn→∞	limn→∞	PROPN
ap-2064	172	22	sn	sn	PROPN
ap-2064	172	23	as	as	ADP
ap-2064	172	24	52	52	NUM
ap-2064	172	25	vol	vol	NOUN
ap-2064	172	26	.	.	PUNCT
ap-2064	173	1	55	55	NUM
ap-2064	173	2	no	no	NOUN
ap-2064	173	3	.	.	PUNCT
ap-2064	174	1	1/2015	1/2015	NUM
ap-2064	174	2	biperiodic	biperiodic	PROPN
ap-2064	174	3	fibonacci	fibonacci	NOUN
ap-2064	174	4	word	word	NOUN
ap-2064	174	5	and	and	CCONJ
ap-2064	174	6	its	its	PRON
ap-2064	174	7	fractal	fractal	ADJ
ap-2064	174	8	curve	curve	NOUN
ap-2064	174	9	a	a	DET
ap-2064	174	10	b	b	PROPN
ap-2064	174	11	f(a	f(a	PROPN
ap-2064	174	12	,	,	PUNCT
ap-2064	174	13	b	b	NOUN
ap-2064	174	14	)	)	PUNCT
ap-2064	174	15	1	1	NUM
ap-2064	174	16	2	2	NUM
ap-2064	174	17	1101110111011011101110111011011	1101110111011011101110111011011	NUM
ap-2064	174	18	·	·	PUNCT
ap-2064	174	19	·	·	PUNCT
ap-2064	175	1	·	·	PUNCT
ap-2064	175	2	2	2	NUM
ap-2064	175	3	1	1	NUM
ap-2064	175	4	0100100101001001001010010010010	0100100101001001001010010010010	NUM
ap-2064	175	5	·	·	PUNCT
ap-2064	175	6	·	·	PUNCT
ap-2064	175	7	·	·	PUNCT
ap-2064	176	1	2	2	NUM
ap-2064	176	2	3	3	NUM
ap-2064	176	3	0101010010101001010101001010100	0101010010101001010101001010100	NUM
ap-2064	176	4	·	·	PUNCT
ap-2064	176	5	·	·	PUNCT
ap-2064	176	6	·	·	PUNCT
ap-2064	176	7	3	3	NUM
ap-2064	176	8	2	2	NUM
ap-2064	176	9	0010010001001000100100010010010	0010010001001000100100010010010	NUM
ap-2064	176	10	·	·	PUNCT
ap-2064	176	11	·	·	PUNCT
ap-2064	176	12	·	·	PUNCT
ap-2064	176	13	3	3	NUM
ap-2064	176	14	5	5	NUM
ap-2064	176	15	0010010010010010001001001001001	0010010010010010001001001001001	NUM
ap-2064	176	16	·	·	PUNCT
ap-2064	176	17	·	·	PUNCT
ap-2064	176	18	·	·	PUNCT
ap-2064	176	19	table	table	NOUN
ap-2064	176	20	3	3	X
ap-2064	176	21	.	.	PUNCT
ap-2064	177	1	some	some	DET
ap-2064	177	2	biperiodic	biperiodic	ADJ
ap-2064	177	3	fibonacci	fibonacci	NOUN
ap-2064	177	4	words	word	NOUN
ap-2064	177	5	.	.	PUNCT
ap-2064	178	1	an	an	DET
ap-2064	178	2	infinite	infinite	ADJ
ap-2064	178	3	word	word	NOUN
ap-2064	178	4	.	.	PUNCT
ap-2064	179	1	in	in	ADP
ap-2064	179	2	fact	fact	NOUN
ap-2064	179	3	,	,	PUNCT
ap-2064	179	4	one	one	PRON
ap-2064	179	5	can	can	AUX
ap-2064	179	6	prove	prove	VERB
ap-2064	179	7	[	[	X
ap-2064	179	8	14	14	NUM
ap-2064	179	9	]	]	PUNCT
ap-2064	179	10	that	that	SCONJ
ap-2064	179	11	each	each	DET
ap-2064	179	12	sn	sn	PROPN
ap-2064	179	13	is	be	AUX
ap-2064	179	14	a	a	DET
ap-2064	179	15	prefix	prefix	NOUN
ap-2064	179	16	of	of	ADP
ap-2064	179	17	w(θ	w(θ	NOUN
ap-2064	179	18	)	)	PUNCT
ap-2064	179	19	for	for	ADP
ap-2064	179	20	all	all	DET
ap-2064	179	21	n	n	PRON
ap-2064	179	22	≥	≥	NOUN
ap-2064	179	23	0	0	NUM
ap-2064	179	24	and	and	CCONJ
ap-2064	179	25	w(θ	w(θ	ADJ
ap-2064	179	26	)	)	PUNCT
ap-2064	179	27	=	=	VERB
ap-2064	179	28	lim	lim	PROPN
ap-2064	179	29	n→∞	n→∞	NUM
ap-2064	179	30	sn	sn	PROPN
ap-2064	179	31	.	.	PUNCT
ap-2064	180	1	(	(	PUNCT
ap-2064	180	2	2	2	NUM
ap-2064	180	3	)	)	PUNCT
ap-2064	180	4	3	3	NUM
ap-2064	180	5	.	.	PUNCT
ap-2064	180	6	biperiodic	biperiodic	ADJ
ap-2064	180	7	fibonacci	fibonacci	PROPN
ap-2064	180	8	words	word	NOUN
ap-2064	180	9	let	let	VERB
ap-2064	180	10	f(a	f(a	NOUN
ap-2064	180	11	,	,	PUNCT
ap-2064	180	12	b	b	NOUN
ap-2064	180	13	,	,	PUNCT
ap-2064	180	14	n	n	CCONJ
ap-2064	180	15	)	)	PUNCT
ap-2064	180	16	and	and	CCONJ
ap-2064	180	17	f(a	f(a	PROPN
ap-2064	180	18	,	,	PUNCT
ap-2064	180	19	b	b	X
ap-2064	180	20	)	)	PUNCT
ap-2064	180	21	be	be	AUX
ap-2064	180	22	the	the	DET
ap-2064	180	23	nth	nth	PROPN
ap-2064	180	24	biperiodic	biperiodic	PROPN
ap-2064	180	25	fibonacci	fibonacci	NOUN
ap-2064	180	26	word	word	NOUN
ap-2064	180	27	and	and	CCONJ
ap-2064	180	28	the	the	DET
ap-2064	180	29	infinite	infinite	ADJ
ap-2064	180	30	biperiodic	biperiodic	ADJ
ap-2064	180	31	fibonacci	fibonacci	NOUN
ap-2064	180	32	word	word	NOUN
ap-2064	180	33	,	,	PUNCT
ap-2064	180	34	respectively	respectively	ADV
ap-2064	180	35	.	.	PUNCT
ap-2064	181	1	note	note	VERB
ap-2064	181	2	that	that	SCONJ
ap-2064	181	3	,	,	PUNCT
ap-2064	181	4	for	for	ADP
ap-2064	181	5	a	a	DET
ap-2064	181	6	=	=	SYM
ap-2064	181	7	b	b	NOUN
ap-2064	181	8	=	=	SYM
ap-2064	181	9	1	1	NUM
ap-2064	181	10	we	we	PRON
ap-2064	181	11	have	have	VERB
ap-2064	181	12	the	the	DET
ap-2064	181	13	word	word	NOUN
ap-2064	181	14	f	f	PROPN
ap-2064	181	15	=	=	SYM
ap-2064	181	16	1011010110110	1011010110110	NUM
ap-2064	181	17	.	.	PUNCT
ap-2064	181	18	.	.	PUNCT
ap-2064	182	1	..	..	PUNCT
ap-2064	183	1	if	if	SCONJ
ap-2064	183	2	a	a	DET
ap-2064	183	3	=	=	SYM
ap-2064	183	4	b	b	NOUN
ap-2064	183	5	=	=	SYM
ap-2064	183	6	k	k	PROPN
ap-2064	183	7	,	,	PUNCT
ap-2064	183	8	we	we	PRON
ap-2064	183	9	obtain	obtain	VERB
ap-2064	183	10	k	k	ADJ
ap-2064	183	11	-	-	PUNCT
ap-2064	183	12	fibonacci	fibonacci	NOUN
ap-2064	183	13	words	word	NOUN
ap-2064	183	14	,	,	PUNCT
ap-2064	183	15	i.e.	i.e.	X
ap-2064	183	16	,	,	PUNCT
ap-2064	183	17	f(k	f(k	ADJ
ap-2064	183	18	,	,	PUNCT
ap-2064	183	19	k	k	NOUN
ap-2064	183	20	)	)	PUNCT
ap-2064	183	21	=	=	SYM
ap-2064	183	22	fk	fk	INTJ
ap-2064	183	23	.	.	PUNCT
ap-2064	183	24	definition	definition	NOUN
ap-2064	183	25	3	3	NUM
ap-2064	183	26	.	.	PUNCT
ap-2064	184	1	the	the	DET
ap-2064	184	2	(	(	PUNCT
ap-2064	184	3	a	a	PROPN
ap-2064	184	4	,	,	PUNCT
ap-2064	184	5	b)-fibonacci	b)-fibonacci	PUNCT
ap-2064	184	6	morphism	morphism	PROPN
ap-2064	184	7	σ(a	σ(a	PROPN
ap-2064	184	8	,	,	PUNCT
ap-2064	184	9	b	b	NOUN
ap-2064	184	10	)	)	PUNCT
ap-2064	184	11	:	:	PUNCT
ap-2064	184	12	{	{	PUNCT
ap-2064	184	13	0,1	0,1	NOUN
ap-2064	184	14	}	}	PUNCT
ap-2064	184	15	→	→	SYM
ap-2064	184	16	{	{	PUNCT
ap-2064	184	17	0,1}∗	0,1}∗	NOUN
ap-2064	184	18	is	be	AUX
ap-2064	184	19	defined	define	VERB
ap-2064	184	20	by	by	ADP
ap-2064	184	21	σ(a	σ(a	PROPN
ap-2064	184	22	,	,	PUNCT
ap-2064	184	23	b)(0	b)(0	NUM
ap-2064	184	24	)	)	PUNCT
ap-2064	184	25	=	=	SYM
ap-2064	184	26	(	(	PUNCT
ap-2064	184	27	0a−11)b0	0a−11)b0	X
ap-2064	184	28	and	and	CCONJ
ap-2064	184	29	σ(a	σ(a	PROPN
ap-2064	184	30	,	,	PUNCT
ap-2064	184	31	b)(1	b)(1	NOUN
ap-2064	184	32	)	)	PUNCT
ap-2064	184	33	=	=	SYM
ap-2064	184	34	(	(	PUNCT
ap-2064	184	35	0a−11)b0a1	0a−11)b0a1	NOUN
ap-2064	184	36	.	.	PUNCT
ap-2064	184	37	theorem	theorem	VERB
ap-2064	184	38	4	4	NUM
ap-2064	184	39	.	.	PUNCT
ap-2064	185	1	for	for	ADP
ap-2064	185	2	all	all	DET
ap-2064	185	3	n	n	PRON
ap-2064	185	4	≥	≥	NOUN
ap-2064	185	5	0	0	NUM
ap-2064	185	6	,	,	PUNCT
ap-2064	185	7	σn(a	σn(a	X
ap-2064	185	8	,	,	PUNCT
ap-2064	185	9	b)(0	b)(0	NUM
ap-2064	185	10	)	)	PUNCT
ap-2064	185	11	=	=	SYM
ap-2064	185	12	f(a	f(a	NOUN
ap-2064	185	13	,	,	PUNCT
ap-2064	185	14	b,2n+1	b,2n+1	NOUN
ap-2064	185	15	)	)	PUNCT
ap-2064	185	16	and	and	CCONJ
ap-2064	185	17	σn(a	σn(a	NUM
ap-2064	185	18	,	,	PUNCT
ap-2064	185	19	b)(0a−11	b)(0a−11	NOUN
ap-2064	185	20	)	)	PUNCT
ap-2064	185	21	=	=	SYM
ap-2064	185	22	f(a	f(a	NOUN
ap-2064	185	23	,	,	PUNCT
ap-2064	185	24	b,2n+2	b,2n+2	NOUN
ap-2064	185	25	)	)	PUNCT
ap-2064	185	26	.	.	PUNCT
ap-2064	186	1	hence	hence	ADV
ap-2064	186	2	,	,	PUNCT
ap-2064	186	3	the	the	DET
ap-2064	186	4	biperiodic	biperiodic	ADJ
ap-2064	186	5	fibonacci	fibonacci	PROPN
ap-2064	186	6	word	word	PROPN
ap-2064	186	7	f(a	f(a	PROPN
ap-2064	186	8	,	,	PUNCT
ap-2064	186	9	b	b	NOUN
ap-2064	186	10	)	)	PUNCT
ap-2064	186	11	satisfies	satisfie	NOUN
ap-2064	186	12	that	that	PRON
ap-2064	186	13	lim	lim	PROPN
ap-2064	186	14	n→∞	n→∞	NUM
ap-2064	186	15	σn(a	σn(a	PUNCT
ap-2064	186	16	,	,	PUNCT
ap-2064	186	17	b)(0	b)(0	NUM
ap-2064	186	18	)	)	PUNCT
ap-2064	186	19	=	=	SYM
ap-2064	186	20	f(a	f(a	PROPN
ap-2064	186	21	,	,	PUNCT
ap-2064	186	22	b	b	NOUN
ap-2064	186	23	)	)	PUNCT
ap-2064	186	24	.	.	PUNCT
ap-2064	187	1	proof	proof	NOUN
ap-2064	187	2	.	.	PUNCT
ap-2064	188	1	we	we	PRON
ap-2064	188	2	prove	prove	VERB
ap-2064	188	3	the	the	DET
ap-2064	188	4	two	two	NUM
ap-2064	188	5	assertions	assertion	NOUN
ap-2064	188	6	about	about	ADP
ap-2064	188	7	σn(a	σn(a	NOUN
ap-2064	188	8	,	,	PUNCT
ap-2064	188	9	b	b	NOUN
ap-2064	188	10	)	)	PUNCT
ap-2064	188	11	by	by	ADP
ap-2064	188	12	induction	induction	NOUN
ap-2064	188	13	on	on	ADP
ap-2064	188	14	n.	n.	NOUN
ap-2064	188	15	they	they	PRON
ap-2064	188	16	are	be	AUX
ap-2064	188	17	clearly	clearly	ADV
ap-2064	188	18	true	true	ADJ
ap-2064	188	19	for	for	ADP
ap-2064	188	20	n	n	NOUN
ap-2064	188	21	=	=	SYM
ap-2064	188	22	0	0	NUM
ap-2064	188	23	,	,	PUNCT
ap-2064	188	24	1	1	NUM
ap-2064	188	25	.	.	PUNCT
ap-2064	189	1	now	now	ADV
ap-2064	189	2	assume	assume	VERB
ap-2064	189	3	the	the	DET
ap-2064	189	4	result	result	NOUN
ap-2064	189	5	holds	hold	VERB
ap-2064	189	6	for	for	ADP
ap-2064	189	7	n	n	CCONJ
ap-2064	189	8	,	,	PUNCT
ap-2064	189	9	and	and	CCONJ
ap-2064	189	10	let	let	VERB
ap-2064	189	11	us	we	PRON
ap-2064	189	12	prove	prove	VERB
ap-2064	189	13	it	it	PRON
ap-2064	189	14	for	for	ADP
ap-2064	189	15	n+	n+	ADP
ap-2064	189	16	1	1	NUM
ap-2064	189	17	:	:	SYM
ap-2064	189	18	σn+1	σn+1	PROPN
ap-2064	189	19	(	(	PUNCT
ap-2064	189	20	a	a	PRON
ap-2064	189	21	,	,	PUNCT
ap-2064	189	22	b)(0	b)(0	NUM
ap-2064	189	23	)	)	PUNCT
ap-2064	189	24	=	=	PUNCT
ap-2064	189	25	σn(a	σn(a	NOUN
ap-2064	189	26	,	,	PUNCT
ap-2064	189	27	b)((0a−11)b0	b)((0a−11)b0	NOUN
ap-2064	189	28	)	)	PUNCT
ap-2064	190	1	=	=	SYM
ap-2064	190	2	(	(	PUNCT
ap-2064	190	3	σn(a	σn(a	PROPN
ap-2064	190	4	,	,	PUNCT
ap-2064	190	5	b)(0a−11))bσn(a	b)(0a−11))bσn(a	PROPN
ap-2064	190	6	,	,	PUNCT
ap-2064	190	7	b)(0	b)(0	NUM
ap-2064	190	8	)	)	PUNCT
ap-2064	190	9	=	=	SYM
ap-2064	191	1	f	f	PROPN
ap-2064	191	2	b(a	b(a	NOUN
ap-2064	191	3	,	,	PUNCT
ap-2064	191	4	b,2n+2)f(a	b,2n+2)f(a	ADJ
ap-2064	191	5	,	,	PUNCT
ap-2064	191	6	b,2n+1	b,2n+1	NOUN
ap-2064	191	7	)	)	PUNCT
ap-2064	191	8	=	=	SYM
ap-2064	191	9	f(a	f(a	PROPN
ap-2064	191	10	,	,	PUNCT
ap-2064	191	11	b,2n+3	b,2n+3	NOUN
ap-2064	191	12	)	)	PUNCT
ap-2064	191	13	,	,	PUNCT
ap-2064	191	14	and	and	CCONJ
ap-2064	191	15	σn+1	σn+1	PROPN
ap-2064	191	16	(	(	PUNCT
ap-2064	191	17	a	a	PRON
ap-2064	191	18	,	,	PUNCT
ap-2064	191	19	b)(0	b)(0	NUM
ap-2064	191	20	a−11	a−11	PUNCT
ap-2064	191	21	)	)	PUNCT
ap-2064	191	22	=	=	SYM
ap-2064	191	23	σn(a	σn(a	NOUN
ap-2064	191	24	,	,	PUNCT
ap-2064	191	25	b)(((0a−11)b0)a−1(0a−11)b0a1	b)(((0a−11)b0)a−1(0a−11)b0a1	NOUN
ap-2064	191	26	)	)	PUNCT
ap-2064	191	27	=	=	PUNCT
ap-2064	191	28	σn(a	σn(a	NOUN
ap-2064	191	29	,	,	PUNCT
ap-2064	191	30	b)(((0a−11)b0)a0a−11	b)(((0a−11)b0)a0a−11	ADJ
ap-2064	191	31	)	)	PUNCT
ap-2064	191	32	=	=	SYM
ap-2064	191	33	(	(	PUNCT
ap-2064	191	34	(	(	PUNCT
ap-2064	191	35	σn(a	σn(a	NOUN
ap-2064	191	36	,	,	PUNCT
ap-2064	191	37	b)(0a−11))bσn(a	b)(0a−11))bσn(a	PROPN
ap-2064	191	38	,	,	PUNCT
ap-2064	191	39	b)(0))aσn(a	b)(0))aσn(a	PROPN
ap-2064	191	40	,	,	PUNCT
ap-2064	191	41	b)(0a−11	b)(0a−11	NOUN
ap-2064	191	42	)	)	PUNCT
ap-2064	191	43	=	=	PUNCT
ap-2064	192	1	(	(	PUNCT
ap-2064	192	2	f	f	PROPN
ap-2064	192	3	b(a	b(a	NOUN
ap-2064	192	4	,	,	PUNCT
ap-2064	192	5	b,2n+2)f(a	b,2n+2)f(a	PROPN
ap-2064	192	6	,	,	PUNCT
ap-2064	192	7	b,2n+1))af(a	b,2n+1))af(a	NOUN
ap-2064	192	8	,	,	PUNCT
ap-2064	192	9	b,2n+2	b,2n+2	NOUN
ap-2064	192	10	)	)	PUNCT
ap-2064	192	11	=	=	SYM
ap-2064	193	1	fa(a	fa(a	NOUN
ap-2064	193	2	,	,	PUNCT
ap-2064	193	3	b,2n+3)f(a	b,2n+3)f(a	NOUN
ap-2064	193	4	,	,	PUNCT
ap-2064	193	5	b,2n+2	b,2n+2	NOUN
ap-2064	193	6	)	)	PUNCT
ap-2064	193	7	=	=	SYM
ap-2064	193	8	f(a	f(a	PROPN
ap-2064	193	9	,	,	PUNCT
ap-2064	193	10	b,2n+4	b,2n+4	PROPN
ap-2064	193	11	)	)	PUNCT
ap-2064	193	12	.	.	PUNCT
ap-2064	194	1	example	example	NOUN
ap-2064	195	1	5	5	NUM
ap-2064	195	2	.	.	X
ap-2064	196	1	in	in	ADP
ap-2064	196	2	table	table	NOUN
ap-2064	196	3	3	3	NUM
ap-2064	196	4	we	we	PRON
ap-2064	196	5	show	show	VERB
ap-2064	196	6	some	some	DET
ap-2064	196	7	biperiodic	biperiodic	ADJ
ap-2064	196	8	fibonacci	fibonacci	NOUN
ap-2064	196	9	words	word	NOUN
ap-2064	196	10	for	for	ADP
ap-2064	196	11	specific	specific	ADJ
ap-2064	196	12	values	value	NOUN
ap-2064	196	13	of	of	ADP
ap-2064	196	14	a	a	DET
ap-2064	196	15	and	and	CCONJ
ap-2064	196	16	b.	b.	NOUN
ap-2064	196	17	proposition	proposition	NOUN
ap-2064	196	18	6	6	NUM
ap-2064	196	19	.	.	PUNCT
ap-2064	197	1	the	the	DET
ap-2064	197	2	number	number	NOUN
ap-2064	197	3	of	of	ADP
ap-2064	197	4	occurrences	occurrence	NOUN
ap-2064	197	5	of	of	ADP
ap-2064	197	6	1	1	NUM
ap-2064	197	7	and	and	CCONJ
ap-2064	197	8	0	0	NUM
ap-2064	197	9	in	in	ADP
ap-2064	197	10	the	the	DET
ap-2064	197	11	finite	finite	PROPN
ap-2064	197	12	biperiodic	biperiodic	PROPN
ap-2064	197	13	fibonacci	fibonacci	PROPN
ap-2064	197	14	words	word	NOUN
ap-2064	197	15	are	be	AUX
ap-2064	197	16	given	give	VERB
ap-2064	197	17	by	by	ADP
ap-2064	197	18	pn	pn	PROPN
ap-2064	197	19	and	and	CCONJ
ap-2064	197	20	hn	hn	NOUN
ap-2064	197	21	,	,	PUNCT
ap-2064	197	22	where	where	SCONJ
ap-2064	197	23	pn	pn	PROPN
ap-2064	197	24	and	and	CCONJ
ap-2064	197	25	hn	hn	PROPN
ap-2064	197	26	satisfy	satisfy	VERB
ap-2064	197	27	the	the	DET
ap-2064	197	28	following	follow	VERB
ap-2064	197	29	recurrences	recurrence	NOUN
ap-2064	197	30	:	:	PUNCT
ap-2064	197	31	p0	p0	NOUN
ap-2064	197	32	=	=	SYM
ap-2064	197	33	0	0	NUM
ap-2064	197	34	,	,	PUNCT
ap-2064	197	35	p1	p1	NOUN
ap-2064	197	36	=	=	SYM
ap-2064	197	37	0	0	NUM
ap-2064	197	38	,	,	PUNCT
ap-2064	197	39	p2	p2	X
ap-2064	197	40	=	=	SYM
ap-2064	197	41	1	1	NUM
ap-2064	197	42	,	,	PUNCT
ap-2064	197	43	pn	pn	NOUN
ap-2064	197	44	=	=	SYM
ap-2064	197	45	{	{	PUNCT
ap-2064	197	46	apn−1	apn−1	PROPN
ap-2064	197	47	+	+	NUM
ap-2064	197	48	pn−2	pn−2	PROPN
ap-2064	197	49	,	,	PUNCT
ap-2064	197	50	if	if	SCONJ
ap-2064	197	51	n	n	PRON
ap-2064	197	52	≥	≥	NOUN
ap-2064	197	53	3	3	NUM
ap-2064	197	54	is	be	AUX
ap-2064	197	55	even	even	ADV
ap-2064	197	56	,	,	PUNCT
ap-2064	197	57	bpn−1	bpn−1	PROPN
ap-2064	197	58	+	+	SYM
ap-2064	197	59	pn−2	pn−2	PROPN
ap-2064	197	60	,	,	PUNCT
ap-2064	197	61	if	if	SCONJ
ap-2064	197	62	n	n	PRON
ap-2064	197	63	≥	≥	NOUN
ap-2064	197	64	3	3	NUM
ap-2064	197	65	is	be	AUX
ap-2064	197	66	odd	odd	ADJ
ap-2064	197	67	.	.	PUNCT
ap-2064	198	1	(	(	PUNCT
ap-2064	198	2	3	3	X
ap-2064	198	3	)	)	PUNCT
ap-2064	198	4	moreover	moreover	ADV
ap-2064	198	5	,	,	PUNCT
ap-2064	198	6	pn	pn	PROPN
ap-2064	198	7	=	=	SYM
ap-2064	198	8	(	(	PUNCT
ap-2064	198	9	b1−ξ(n−1	b1−ξ(n−1	PROPN
ap-2064	198	10	)	)	PUNCT
ap-2064	198	11	(	(	PUNCT
ap-2064	198	12	ab)bn−1	ab)bn−1	PROPN
ap-2064	198	13	2	2	NUM
ap-2064	198	14	c	c	NOUN
ap-2064	198	15	)	)	PUNCT
ap-2064	199	1	αn−1	αn−1	ADV
ap-2064	199	2	−	−	NOUN
ap-2064	200	1	βn−1	βn−1	ADJ
ap-2064	200	2	α−	α−	ADP
ap-2064	200	3	β	β	X
ap-2064	200	4	,	,	PUNCT
ap-2064	200	5	for	for	ADP
ap-2064	200	6	n	n	PRON
ap-2064	200	7	≥	≥	NUM
ap-2064	200	8	1	1	NUM
ap-2064	200	9	,	,	PUNCT
ap-2064	200	10	(	(	PUNCT
ap-2064	200	11	4	4	NUM
ap-2064	200	12	)	)	PUNCT
ap-2064	200	13	and	and	CCONJ
ap-2064	200	14	h0	h0	NOUN
ap-2064	200	15	=	=	PROPN
ap-2064	200	16	0	0	NUM
ap-2064	200	17	,	,	PUNCT
ap-2064	200	18	h1	h1	NOUN
ap-2064	200	19	=	=	SYM
ap-2064	200	20	1	1	NUM
ap-2064	200	21	,	,	PUNCT
ap-2064	200	22	h2	h2	NOUN
ap-2064	200	23	=	=	SYM
ap-2064	200	24	a−	a−	PROPN
ap-2064	200	25	1	1	NUM
ap-2064	200	26	,	,	PUNCT
ap-2064	201	1	hn	hn	NOUN
ap-2064	201	2	=	=	PUNCT
ap-2064	201	3	{	{	PUNCT
ap-2064	201	4	ahn−1	ahn−1	PROPN
ap-2064	201	5	+	+	CCONJ
ap-2064	201	6	hn−2	hn−2	PROPN
ap-2064	201	7	,	,	PUNCT
ap-2064	201	8	if	if	SCONJ
ap-2064	201	9	n	n	NUM
ap-2064	201	10	≥	≥	NOUN
ap-2064	201	11	3	3	NUM
ap-2064	201	12	is	be	AUX
ap-2064	201	13	even	even	ADV
ap-2064	201	14	,	,	PUNCT
ap-2064	201	15	bhn−1	bhn−1	PROPN
ap-2064	201	16	+	+	CCONJ
ap-2064	201	17	hn−2	hn−2	PROPN
ap-2064	201	18	,	,	PUNCT
ap-2064	201	19	if	if	SCONJ
ap-2064	201	20	n	n	NUM
ap-2064	201	21	≥	≥	NOUN
ap-2064	201	22	3	3	NUM
ap-2064	201	23	is	be	AUX
ap-2064	201	24	odd	odd	ADJ
ap-2064	201	25	.	.	PUNCT
ap-2064	202	1	(	(	PUNCT
ap-2064	202	2	5	5	NUM
ap-2064	202	3	)	)	PUNCT
ap-2064	202	4	moreover	moreover	ADV
ap-2064	202	5	,	,	PUNCT
ap-2064	202	6	for	for	ADP
ap-2064	202	7	n	n	PRON
ap-2064	202	8	≥	≥	NUM
ap-2064	202	9	1	1	NUM
ap-2064	202	10	,	,	PUNCT
ap-2064	203	1	hn	hn	PROPN
ap-2064	203	2	=	=	SYM
ap-2064	203	3	(	(	PUNCT
ap-2064	203	4	a−	a−	PROPN
ap-2064	203	5	1)f	1)f	NUM
ap-2064	203	6	(	(	PUNCT
ap-2064	203	7	b	b	NOUN
ap-2064	203	8	,	,	PUNCT
ap-2064	203	9	a	a	PRON
ap-2064	203	10	)	)	PUNCT
ap-2064	203	11	n−1	n−1	PROPN
ap-2064	203	12	+	+	CCONJ
ap-2064	203	13	(	(	PUNCT
ap-2064	203	14	a	a	DET
ap-2064	203	15	b	b	NOUN
ap-2064	203	16	)	)	PUNCT
ap-2064	203	17	ξ(n−1	ξ(n−1	PROPN
ap-2064	203	18	)	)	PUNCT
ap-2064	203	19	f	f	NOUN
ap-2064	203	20	(	(	PUNCT
ap-2064	203	21	b	b	NOUN
ap-2064	203	22	,	,	PUNCT
ap-2064	203	23	a	a	PRON
ap-2064	203	24	)	)	PUNCT
ap-2064	203	25	n−2	n−2	PROPN
ap-2064	203	26	.	.	PUNCT
ap-2064	204	1	(	(	PUNCT
ap-2064	204	2	6	6	X
ap-2064	204	3	)	)	PUNCT
ap-2064	204	4	proof	proof	NOUN
ap-2064	204	5	.	.	PUNCT
ap-2064	205	1	recurrences	recurrence	NOUN
ap-2064	205	2	(	(	PUNCT
ap-2064	205	3	3	3	NUM
ap-2064	205	4	)	)	PUNCT
ap-2064	205	5	and	and	CCONJ
ap-2064	205	6	(	(	PUNCT
ap-2064	205	7	5	5	X
ap-2064	205	8	)	)	PUNCT
ap-2064	205	9	are	be	AUX
ap-2064	205	10	clear	clear	ADJ
ap-2064	205	11	from	from	ADP
ap-2064	205	12	definition	definition	NOUN
ap-2064	205	13	of	of	ADP
ap-2064	205	14	f(a	f(a	PROPN
ap-2064	205	15	,	,	PUNCT
ap-2064	205	16	b	b	NOUN
ap-2064	205	17	,	,	PUNCT
ap-2064	205	18	n	n	CCONJ
ap-2064	205	19	)	)	PUNCT
ap-2064	205	20	.	.	PUNCT
ap-2064	206	1	equation	equation	NOUN
ap-2064	206	2	(	(	PUNCT
ap-2064	206	3	4	4	X
ap-2064	206	4	)	)	PUNCT
ap-2064	206	5	is	be	AUX
ap-2064	206	6	clear	clear	ADJ
ap-2064	206	7	from	from	ADP
ap-2064	206	8	the	the	DET
ap-2064	206	9	binetlike	binetlike	ADJ
ap-2064	206	10	formula	formula	NOUN
ap-2064	206	11	;	;	PUNCT
ap-2064	206	12	see	see	VERB
ap-2064	206	13	equation	equation	NOUN
ap-2064	206	14	(	(	PUNCT
ap-2064	206	15	1	1	NUM
ap-2064	206	16	)	)	PUNCT
ap-2064	206	17	.	.	PUNCT
ap-2064	207	1	we	we	PRON
ap-2064	207	2	obtain	obtain	VERB
ap-2064	207	3	equation	equation	NOUN
ap-2064	207	4	(	(	PUNCT
ap-2064	207	5	6	6	NUM
ap-2064	207	6	)	)	PUNCT
ap-2064	207	7	from	from	ADP
ap-2064	207	8	[	[	X
ap-2064	207	9	12	12	NUM
ap-2064	207	10	,	,	PUNCT
ap-2064	207	11	theorem	theorem	VERB
ap-2064	207	12	8	8	NUM
ap-2064	207	13	]	]	PUNCT
ap-2064	207	14	.	.	PUNCT
ap-2064	208	1	proposition	proposition	NOUN
ap-2064	208	2	7	7	NUM
ap-2064	208	3	.	.	X
ap-2064	209	1	one	one	PRON
ap-2064	209	2	has	have	VERB
ap-2064	209	3	(	(	PUNCT
ap-2064	209	4	1	1	NUM
ap-2064	209	5	.	.	PUNCT
ap-2064	209	6	)	)	PUNCT
ap-2064	210	1	limn→∞	limn→∞	PROPN
ap-2064	210	2	|f(a	|f(a	NOUN
ap-2064	210	3	,	,	PUNCT
ap-2064	210	4	b	b	PROPN
ap-2064	210	5	,	,	PUNCT
ap-2064	210	6	n)|	n)|	PROPN
ap-2064	210	7	|f(a	|f(a	PROPN
ap-2064	210	8	,	,	PUNCT
ap-2064	210	9	b	b	NOUN
ap-2064	210	10	,	,	PUNCT
ap-2064	210	11	n)|1	n)|1	NOUN
ap-2064	210	12	=	=	PUNCT
ap-2064	210	13	α	α	X
ap-2064	210	14	b	b	X
ap-2064	210	15	=	=	PUNCT
ap-2064	210	16	ab+	ab+	NOUN
ap-2064	210	17	√	√	PROPN
ap-2064	210	18	(	(	PUNCT
ap-2064	210	19	ab)2	ab)2	PROPN
ap-2064	210	20	+	+	NOUN
ap-2064	210	21	4ab	4ab	ADJ
ap-2064	210	22	2b	2b	NUM
ap-2064	210	23	.	.	PUNCT
ap-2064	211	1	(	(	PUNCT
ap-2064	211	2	2	2	NUM
ap-2064	211	3	.	.	PUNCT
ap-2064	211	4	)	)	PUNCT
ap-2064	212	1	limn→∞	limn→∞	PROPN
ap-2064	212	2	|f(a	|f(a	NOUN
ap-2064	212	3	,	,	PUNCT
ap-2064	212	4	b	b	PROPN
ap-2064	212	5	,	,	PUNCT
ap-2064	212	6	n)|	n)|	PROPN
ap-2064	212	7	|f(a	|f(a	PROPN
ap-2064	212	8	,	,	PUNCT
ap-2064	212	9	b	b	PROPN
ap-2064	212	10	,	,	PUNCT
ap-2064	212	11	n)|0	n)|0	NOUN
ap-2064	212	12	=	=	PUNCT
ap-2064	212	13	a(α+1	a(α+1	NOUN
ap-2064	212	14	)	)	PUNCT
ap-2064	212	15	α(a−1)+a	α(a−1)+a	PROPN
ap-2064	212	16	.	.	PUNCT
ap-2064	213	1	(	(	PUNCT
ap-2064	213	2	3	3	NUM
ap-2064	213	3	.	.	PUNCT
ap-2064	213	4	)	)	PUNCT
ap-2064	214	1	limn→∞	limn→∞	PROPN
ap-2064	214	2	|f(a	|f(a	NOUN
ap-2064	214	3	,	,	PUNCT
ap-2064	214	4	b	b	PROPN
ap-2064	214	5	,	,	PUNCT
ap-2064	214	6	n)|0	n)|0	NOUN
ap-2064	214	7	|f(a	|f(a	PROPN
ap-2064	214	8	,	,	PUNCT
ap-2064	214	9	b	b	NOUN
ap-2064	214	10	,	,	PUNCT
ap-2064	214	11	n)|1	n)|1	NOUN
ap-2064	214	12	=	=	PUNCT
ap-2064	214	13	a	a	PRON
ap-2064	214	14	(	(	PUNCT
ap-2064	214	15	1	1	NUM
ap-2064	214	16	+	+	SYM
ap-2064	214	17	1	1	NUM
ap-2064	214	18	α	α	NOUN
ap-2064	214	19	)	)	PUNCT
ap-2064	214	20	−	−	PROPN
ap-2064	215	1	1	1	X
ap-2064	215	2	.	.	PUNCT
ap-2064	215	3	proof	proof	NOUN
ap-2064	215	4	.	.	PUNCT
ap-2064	216	1	(	(	PUNCT
ap-2064	216	2	1	1	NUM
ap-2064	216	3	.	.	PUNCT
ap-2064	216	4	)	)	PUNCT
ap-2064	217	1	from	from	ADP
ap-2064	217	2	equations	equation	NOUN
ap-2064	217	3	(	(	PUNCT
ap-2064	217	4	1	1	NUM
ap-2064	217	5	)	)	PUNCT
ap-2064	217	6	and	and	CCONJ
ap-2064	217	7	(	(	PUNCT
ap-2064	217	8	4	4	X
ap-2064	217	9	)	)	PUNCT
ap-2064	217	10	we	we	PRON
ap-2064	217	11	obtain	obtain	VERB
ap-2064	217	12	lim	lim	PROPN
ap-2064	217	13	n→∞	n→∞	NUM
ap-2064	217	14	|f(a	|f(a	PROPN
ap-2064	217	15	,	,	PUNCT
ap-2064	217	16	b	b	PROPN
ap-2064	217	17	,	,	PUNCT
ap-2064	217	18	n)|	n)|	PROPN
ap-2064	217	19	|f(a	|f(a	PROPN
ap-2064	217	20	,	,	PUNCT
ap-2064	217	21	b	b	NOUN
ap-2064	217	22	,	,	PUNCT
ap-2064	217	23	n)|1	n)|1	PROPN
ap-2064	217	24	=	=	SYM
ap-2064	217	25	lim	lim	PROPN
ap-2064	217	26	n→∞	n→∞	NUM
ap-2064	218	1	qn	qn	PROPN
ap-2064	218	2	pn	pn	PROPN
ap-2064	218	3	=	=	PROPN
ap-2064	218	4	lim	lim	PROPN
ap-2064	218	5	n→∞	n→∞	X
ap-2064	218	6	(	(	PUNCT
ap-2064	218	7	a1−ξ(n	a1−ξ(n	NOUN
ap-2064	218	8	)	)	PUNCT
ap-2064	218	9	(	(	PUNCT
ap-2064	218	10	ab)b	ab)b	NOUN
ap-2064	218	11	n	n	CCONJ
ap-2064	218	12	2	2	NUM
ap-2064	218	13	c	c	NOUN
ap-2064	218	14	)	)	PUNCT
ap-2064	218	15	αn−βn	αn−βn	NUM
ap-2064	218	16	α−β	α−β	PROPN
ap-2064	218	17	(	(	PUNCT
ap-2064	218	18	b1−ξ(n−1	b1−ξ(n−1	ADJ
ap-2064	218	19	)	)	PUNCT
ap-2064	218	20	(	(	PUNCT
ap-2064	218	21	ab)b	ab)b	PROPN
ap-2064	218	22	n−1	n−1	PROPN
ap-2064	218	23	2	2	NUM
ap-2064	218	24	c	c	NOUN
ap-2064	218	25	)	)	PUNCT
ap-2064	218	26	αn−1−βn−1	αn−1−βn−1	PROPN
ap-2064	218	27	α−β	α−β	PROPN
ap-2064	218	28	=	=	PROPN
ap-2064	218	29	lim	lim	PROPN
ap-2064	218	30	n→∞	n→∞	X
ap-2064	218	31	(	(	PUNCT
ap-2064	218	32	ab)bn−1	ab)bn−1	PROPN
ap-2064	218	33	2	2	NUM
ap-2064	218	34	c(a1−ξ(n))(αn	c(a1−ξ(n))(αn	NOUN
ap-2064	218	35	−	−	NOUN
ap-2064	218	36	βn	βn	NOUN
ap-2064	218	37	)	)	PUNCT
ap-2064	218	38	(	(	PUNCT
ap-2064	218	39	ab)bn2	ab)bn2	NOUN
ap-2064	218	40	c(b1−ξ(n−1))(αn−1	c(b1−ξ(n−1))(αn−1	PROPN
ap-2064	218	41	−	−	PROPN
ap-2064	218	42	βn−1	βn−1	PROPN
ap-2064	218	43	)	)	PUNCT
ap-2064	218	44	.	.	PUNCT
ap-2064	219	1	if	if	SCONJ
ap-2064	219	2	n	n	PRON
ap-2064	219	3	is	be	AUX
ap-2064	219	4	even	even	ADV
ap-2064	219	5	,	,	PUNCT
ap-2064	219	6	then	then	ADV
ap-2064	219	7	lim	lim	PROPN
ap-2064	219	8	n→∞	n→∞	PROPN
ap-2064	219	9	|f(a	|f(a	PROPN
ap-2064	219	10	,	,	PUNCT
ap-2064	219	11	b	b	PROPN
ap-2064	219	12	,	,	PUNCT
ap-2064	219	13	n)|	n)|	PROPN
ap-2064	219	14	|f(a	|f(a	PROPN
ap-2064	219	15	,	,	PUNCT
ap-2064	219	16	b	b	NOUN
ap-2064	219	17	,	,	PUNCT
ap-2064	219	18	n)|1	n)|1	NOUN
ap-2064	219	19	=	=	SYM
ap-2064	219	20	1	1	NUM
ap-2064	219	21	b	b	X
ap-2064	219	22	lim	lim	PROPN
ap-2064	219	23	n→∞	n→∞	NUM
ap-2064	219	24	αn	αn	NOUN
ap-2064	220	1	−	−	PROPN
ap-2064	220	2	βn	βn	ADV
ap-2064	220	3	αn−1	αn−1	ADV
ap-2064	220	4	−	−	PROPN
ap-2064	221	1	βn−1	βn−1	ADJ
ap-2064	222	1	=	=	SYM
ap-2064	223	1	1	1	NUM
ap-2064	223	2	b	b	X
ap-2064	223	3	lim	lim	PROPN
ap-2064	223	4	n→∞	n→∞	NUM
ap-2064	224	1	α	α	NOUN
ap-2064	224	2	1−	1−	NUM
ap-2064	225	1	(	(	PUNCT
ap-2064	225	2	β	β	X
ap-2064	225	3	α	α	NOUN
ap-2064	225	4	)	)	PUNCT
ap-2064	225	5	n	n	PROPN
ap-2064	225	6	1−	1−	NUM
ap-2064	225	7	(	(	PUNCT
ap-2064	225	8	β	β	X
ap-2064	225	9	α	α	NOUN
ap-2064	225	10	)	)	PUNCT
ap-2064	226	1	n−1	n−1	PROPN
ap-2064	226	2	=	=	PUNCT
ap-2064	227	1	α	α	PROPN
ap-2064	227	2	b	b	PROPN
ap-2064	227	3	.	.	PUNCT
ap-2064	228	1	if	if	SCONJ
ap-2064	228	2	n	n	PROPN
ap-2064	228	3	is	be	AUX
ap-2064	228	4	odd	odd	ADJ
ap-2064	228	5	,	,	PUNCT
ap-2064	228	6	the	the	DET
ap-2064	228	7	limit	limit	NOUN
ap-2064	228	8	is	be	AUX
ap-2064	228	9	the	the	DET
ap-2064	228	10	same	same	ADJ
ap-2064	228	11	.	.	PUNCT
ap-2064	229	1	53	53	NUM
ap-2064	229	2	josé	josé	PROPN
ap-2064	229	3	l.	l.	PROPN
ap-2064	229	4	ramírez	ramírez	PROPN
ap-2064	229	5	,	,	PUNCT
ap-2064	229	6	gustavo	gustavo	PROPN
ap-2064	229	7	n.	n.	PROPN
ap-2064	229	8	rubiano	rubiano	PROPN
ap-2064	229	9	acta	acta	PROPN
ap-2064	229	10	polytechnica	polytechnica	PROPN
ap-2064	229	11	(	(	PUNCT
ap-2064	229	12	2	2	NUM
ap-2064	229	13	.	.	PUNCT
ap-2064	229	14	)	)	PUNCT
ap-2064	229	15	from	from	ADP
ap-2064	229	16	equations	equation	NOUN
ap-2064	229	17	(	(	PUNCT
ap-2064	229	18	1	1	NUM
ap-2064	229	19	)	)	PUNCT
ap-2064	229	20	and	and	CCONJ
ap-2064	229	21	(	(	PUNCT
ap-2064	229	22	6	6	X
ap-2064	229	23	)	)	PUNCT
ap-2064	229	24	we	we	PRON
ap-2064	229	25	obtain	obtain	VERB
ap-2064	229	26	lim	lim	PROPN
ap-2064	229	27	n→∞	n→∞	NUM
ap-2064	229	28	|f(a	|f(a	PROPN
ap-2064	229	29	,	,	PUNCT
ap-2064	229	30	b	b	PROPN
ap-2064	229	31	,	,	PUNCT
ap-2064	229	32	n)|	n)|	PROPN
ap-2064	229	33	|f(a	|f(a	PROPN
ap-2064	229	34	,	,	PUNCT
ap-2064	229	35	b	b	NOUN
ap-2064	229	36	,	,	PUNCT
ap-2064	229	37	n)|0	n)|0	NOUN
ap-2064	229	38	=	=	SYM
ap-2064	229	39	lim	lim	PROPN
ap-2064	229	40	n→∞	n→∞	NUM
ap-2064	229	41	qn	qn	PROPN
ap-2064	229	42	pn	pn	PROPN
ap-2064	229	43	=	=	PROPN
ap-2064	229	44	lim	lim	PROPN
ap-2064	229	45	n→∞	n→∞	X
ap-2064	230	1	(	(	PUNCT
ap-2064	230	2	a1−ξ(n	a1−ξ(n	NOUN
ap-2064	230	3	)	)	PUNCT
ap-2064	230	4	(	(	PUNCT
ap-2064	230	5	ab)b	ab)b	NOUN
ap-2064	230	6	n	n	CCONJ
ap-2064	230	7	2	2	NUM
ap-2064	230	8	c	c	NOUN
ap-2064	230	9	)	)	PUNCT
ap-2064	230	10	αn−βn	αn−βn	NUM
ap-2064	230	11	α−β	α−β	PROPN
ap-2064	230	12	(	(	PUNCT
ap-2064	230	13	a−	a−	PROPN
ap-2064	230	14	1)b(n	1)b(n	PROPN
ap-2064	230	15	)	)	PUNCT
ap-2064	231	1	+	+	CCONJ
ap-2064	231	2	(	(	PUNCT
ap-2064	231	3	a	a	DET
ap-2064	231	4	b	b	NOUN
ap-2064	231	5	)	)	PUNCT
ap-2064	231	6	ξ(n−1	ξ(n−1	PROPN
ap-2064	231	7	)	)	PUNCT
ap-2064	231	8	b(n−	b(n−	NUM
ap-2064	231	9	1	1	NUM
ap-2064	231	10	)	)	PUNCT
ap-2064	231	11	,	,	PUNCT
ap-2064	231	12	where	where	SCONJ
ap-2064	231	13	b(n	b(n	NOUN
ap-2064	231	14	)	)	PUNCT
ap-2064	231	15	=	=	PUNCT
ap-2064	231	16	(	(	PUNCT
ap-2064	231	17	b1−ξ(n	b1−ξ(n	PROPN
ap-2064	231	18	)	)	PUNCT
ap-2064	231	19	(	(	PUNCT
ap-2064	231	20	ab)b	ab)b	PROPN
ap-2064	231	21	n−2	n−2	PROPN
ap-2064	231	22	2	2	NUM
ap-2064	231	23	c	c	NOUN
ap-2064	231	24	)	)	PUNCT
ap-2064	232	1	αn−2−βn−2	αn−2−βn−2	PROPN
ap-2064	232	2	α−β	α−β	PROPN
ap-2064	232	3	.	.	PUNCT
ap-2064	233	1	if	if	SCONJ
ap-2064	233	2	n	n	PRON
ap-2064	233	3	is	be	AUX
ap-2064	233	4	even	even	ADV
ap-2064	233	5	,	,	PUNCT
ap-2064	233	6	then	then	ADV
ap-2064	233	7	lim	lim	PROPN
ap-2064	233	8	n→∞	n→∞	PROPN
ap-2064	233	9	|f(a	|f(a	PROPN
ap-2064	233	10	,	,	PUNCT
ap-2064	233	11	b	b	PROPN
ap-2064	233	12	,	,	PUNCT
ap-2064	233	13	n)|	n)|	PROPN
ap-2064	233	14	|f(a	|f(a	PROPN
ap-2064	233	15	,	,	PUNCT
ap-2064	233	16	b	b	NOUN
ap-2064	233	17	,	,	PUNCT
ap-2064	233	18	n)|0	n)|0	NOUN
ap-2064	233	19	=	=	PUNCT
ap-2064	233	20	a	a	DET
ap-2064	233	21	lim	lim	PROPN
ap-2064	233	22	n→∞	n→∞	NUM
ap-2064	233	23	αn	αn	NOUN
ap-2064	233	24	−	−	PROPN
ap-2064	233	25	βn	βn	NOUN
ap-2064	233	26	(	(	PUNCT
ap-2064	233	27	a−1)(ab)(αn−1−βn−1)+a2b(αn−2−βn−2	a−1)(ab)(αn−1−βn−1)+a2b(αn−2−βn−2	ADJ
ap-2064	233	28	)	)	PUNCT
ap-2064	233	29	=	=	PUNCT
ap-2064	233	30	a	a	DET
ap-2064	233	31	1	1	NUM
ap-2064	233	32	1	1	NUM
ap-2064	233	33	α	α	NOUN
ap-2064	233	34	(	(	PUNCT
ap-2064	233	35	a−	a−	PROPN
ap-2064	233	36	1)(ab	1)(ab	NUM
ap-2064	233	37	)	)	PUNCT
ap-2064	233	38	+	+	CCONJ
ap-2064	233	39	a2b	a2b	PROPN
ap-2064	233	40	1	1	NUM
ap-2064	233	41	α2	α2	NOUN
ap-2064	233	42	=	=	SYM
ap-2064	233	43	a(α+	a(α+	NOUN
ap-2064	233	44	1	1	NUM
ap-2064	233	45	)	)	PUNCT
ap-2064	233	46	α(a−	α(a−	NUM
ap-2064	233	47	1	1	NUM
ap-2064	233	48	)	)	PUNCT
ap-2064	233	49	+	+	CCONJ
ap-2064	233	50	a	a	PRON
ap-2064	233	51	.	.	PUNCT
ap-2064	234	1	if	if	SCONJ
ap-2064	234	2	n	n	NOUN
ap-2064	234	3	is	be	AUX
ap-2064	234	4	odd	odd	ADJ
ap-2064	234	5	,	,	PUNCT
ap-2064	234	6	the	the	DET
ap-2064	234	7	limit	limit	NOUN
ap-2064	234	8	is	be	AUX
ap-2064	234	9	the	the	DET
ap-2064	234	10	same	same	ADJ
ap-2064	234	11	.	.	PUNCT
ap-2064	235	1	(	(	PUNCT
ap-2064	235	2	3	3	NUM
ap-2064	235	3	.	.	PUNCT
ap-2064	235	4	)	)	PUNCT
ap-2064	236	1	the	the	DET
ap-2064	236	2	proof	proof	NOUN
ap-2064	236	3	runs	run	VERB
ap-2064	236	4	like	like	ADP
ap-2064	236	5	in	in	ADP
ap-2064	236	6	the	the	DET
ap-2064	236	7	previous	previous	ADJ
ap-2064	236	8	two	two	NUM
ap-2064	236	9	items	item	NOUN
ap-2064	236	10	.	.	PUNCT
ap-2064	237	1	the	the	DET
ap-2064	237	2	previous	previous	ADJ
ap-2064	237	3	proposition	proposition	NOUN
ap-2064	237	4	is	be	AUX
ap-2064	237	5	a	a	DET
ap-2064	237	6	particular	particular	ADJ
ap-2064	237	7	result	result	NOUN
ap-2064	237	8	related	relate	VERB
ap-2064	237	9	to	to	ADP
ap-2064	237	10	the	the	DET
ap-2064	237	11	incidence	incidence	NOUN
ap-2064	237	12	matrix	matrix	NOUN
ap-2064	237	13	of	of	ADP
ap-2064	237	14	a	a	DET
ap-2064	237	15	substitution	substitution	NOUN
ap-2064	237	16	,	,	PUNCT
ap-2064	237	17	see	see	VERB
ap-2064	237	18	,	,	PUNCT
ap-2064	237	19	e.g.	e.g.	ADV
ap-2064	237	20	,	,	PUNCT
ap-2064	237	21	[	[	X
ap-2064	237	22	7	7	NUM
ap-2064	237	23	]	]	PUNCT
ap-2064	237	24	.	.	PUNCT
ap-2064	238	1	proposition	proposition	NOUN
ap-2064	238	2	8	8	NUM
ap-2064	238	3	.	.	PUNCT
ap-2064	239	1	the	the	DET
ap-2064	239	2	biperiodic	biperiodic	ADJ
ap-2064	239	3	fibonacci	fibonacci	NOUN
ap-2064	239	4	word	word	NOUN
ap-2064	239	5	and	and	CCONJ
ap-2064	239	6	the	the	DET
ap-2064	239	7	nth	nth	PROPN
ap-2064	239	8	biperiodic	biperiodic	PROPN
ap-2064	239	9	fibonacci	fibonacci	PROPN
ap-2064	239	10	word	word	NOUN
ap-2064	239	11	satisfy	satisfy	VERB
ap-2064	239	12	the	the	DET
ap-2064	239	13	following	follow	VERB
ap-2064	239	14	properties	property	NOUN
ap-2064	239	15	.	.	PUNCT
ap-2064	240	1	(	(	PUNCT
ap-2064	240	2	1	1	NUM
ap-2064	240	3	.	.	PUNCT
ap-2064	240	4	)	)	PUNCT
ap-2064	241	1	word	word	NOUN
ap-2064	241	2	11	11	NUM
ap-2064	241	3	is	be	AUX
ap-2064	241	4	not	not	PART
ap-2064	241	5	a	a	DET
ap-2064	241	6	subword	subword	NOUN
ap-2064	241	7	of	of	ADP
ap-2064	241	8	the	the	DET
ap-2064	241	9	biperiodic	biperiodic	ADJ
ap-2064	241	10	fibonacci	fibonacci	PROPN
ap-2064	241	11	word	word	NOUN
ap-2064	241	12	,	,	PUNCT
ap-2064	241	13	for	for	ADP
ap-2064	241	14	a	a	DET
ap-2064	241	15	≥	≥	NOUN
ap-2064	241	16	2	2	NUM
ap-2064	241	17	,	,	PUNCT
ap-2064	241	18	and	and	CCONJ
ap-2064	241	19	b	b	NOUN
ap-2064	241	20	≥	≥	NUM
ap-2064	241	21	1	1	NUM
ap-2064	241	22	.	.	PUNCT
ap-2064	242	1	(	(	PUNCT
ap-2064	242	2	2	2	NUM
ap-2064	242	3	.	.	PUNCT
ap-2064	242	4	)	)	PUNCT
ap-2064	242	5	let	let	VERB
ap-2064	242	6	xy	xy	PROPN
ap-2064	242	7	be	be	AUX
ap-2064	242	8	the	the	DET
ap-2064	242	9	last	last	ADJ
ap-2064	242	10	two	two	NUM
ap-2064	242	11	symbols	symbol	NOUN
ap-2064	242	12	of	of	ADP
ap-2064	242	13	f(a	f(a	PROPN
ap-2064	242	14	,	,	PUNCT
ap-2064	242	15	b	b	NOUN
ap-2064	242	16	,	,	PUNCT
ap-2064	242	17	n	n	CCONJ
ap-2064	242	18	)	)	PUNCT
ap-2064	242	19	.	.	PUNCT
ap-2064	243	1	for	for	ADP
ap-2064	243	2	n	n	CCONJ
ap-2064	243	3	,	,	PUNCT
ap-2064	243	4	a	a	DET
ap-2064	243	5	≥	≥	NOUN
ap-2064	243	6	2	2	NUM
ap-2064	243	7	,	,	PUNCT
ap-2064	243	8	and	and	CCONJ
ap-2064	243	9	b	b	X
ap-2064	243	10	≥	≥	NUM
ap-2064	243	11	1	1	NUM
ap-2064	243	12	,	,	PUNCT
ap-2064	243	13	we	we	PRON
ap-2064	243	14	have	have	VERB
ap-2064	243	15	xy	xy	PROPN
ap-2064	243	16	=	=	SYM
ap-2064	243	17	01	01	NUM
ap-2064	243	18	if	if	SCONJ
ap-2064	243	19	n	n	PRON
ap-2064	243	20	is	be	AUX
ap-2064	243	21	even	even	ADV
ap-2064	243	22	,	,	PUNCT
ap-2064	243	23	and	and	CCONJ
ap-2064	243	24	xy	xy	NOUN
ap-2064	243	25	=	=	NOUN
ap-2064	243	26	10	10	NUM
ap-2064	243	27	if	if	SCONJ
ap-2064	243	28	n	n	ADJ
ap-2064	243	29	is	be	AUX
ap-2064	243	30	odd	odd	ADJ
ap-2064	243	31	.	.	PUNCT
ap-2064	244	1	(	(	PUNCT
ap-2064	244	2	3	3	NUM
ap-2064	244	3	.	.	PUNCT
ap-2064	244	4	)	)	PUNCT
ap-2064	245	1	the	the	DET
ap-2064	245	2	concatenation	concatenation	NOUN
ap-2064	245	3	of	of	ADP
ap-2064	245	4	two	two	NUM
ap-2064	245	5	successive	successive	ADJ
ap-2064	245	6	biperiodic	biperiodic	ADJ
ap-2064	245	7	fibonacci	fibonacci	NOUN
ap-2064	245	8	words	word	NOUN
ap-2064	245	9	is	be	AUX
ap-2064	245	10	“	"	PUNCT
ap-2064	245	11	almost	almost	ADV
ap-2064	245	12	commutative	commutative	ADJ
ap-2064	245	13	”	"	PUNCT
ap-2064	245	14	,	,	PUNCT
ap-2064	245	15	i.e.	i.e.	X
ap-2064	245	16	,	,	PUNCT
ap-2064	245	17	f(a	f(a	NOUN
ap-2064	245	18	,	,	PUNCT
ap-2064	245	19	b	b	NOUN
ap-2064	245	20	,	,	PUNCT
ap-2064	245	21	n−1)f(a	n−1)f(a	ADJ
ap-2064	245	22	,	,	PUNCT
ap-2064	245	23	b	b	NOUN
ap-2064	245	24	,	,	PUNCT
ap-2064	245	25	n−2	n−2	PROPN
ap-2064	245	26	)	)	PUNCT
ap-2064	245	27	and	and	CCONJ
ap-2064	245	28	f(a	f(a	PROPN
ap-2064	245	29	,	,	PUNCT
ap-2064	245	30	b	b	NOUN
ap-2064	245	31	,	,	PUNCT
ap-2064	245	32	n−2)f(a	n−2)f(a	NOUN
ap-2064	245	33	,	,	PUNCT
ap-2064	245	34	b	b	NOUN
ap-2064	245	35	,	,	PUNCT
ap-2064	245	36	n−1	n−1	PROPN
ap-2064	245	37	)	)	PUNCT
ap-2064	245	38	have	have	VERB
ap-2064	245	39	a	a	DET
ap-2064	245	40	common	common	ADJ
ap-2064	245	41	prefix	prefix	NOUN
ap-2064	245	42	of	of	ADP
ap-2064	245	43	length	length	NOUN
ap-2064	246	1	qn−1	qn−1	PROPN
ap-2064	246	2	+	+	CCONJ
ap-2064	246	3	qn−2	qn−2	NOUN
ap-2064	246	4	−	−	PROPN
ap-2064	246	5	2	2	NUM
ap-2064	246	6	,	,	PUNCT
ap-2064	246	7	for	for	ADP
ap-2064	246	8	all	all	DET
ap-2064	246	9	n	n	DET
ap-2064	246	10	≥	≥	NOUN
ap-2064	246	11	3	3	NUM
ap-2064	246	12	.	.	PUNCT
ap-2064	247	1	proof	proof	NOUN
ap-2064	247	2	.	.	PUNCT
ap-2064	248	1	(	(	PUNCT
ap-2064	248	2	1	1	NUM
ap-2064	248	3	.	.	PUNCT
ap-2064	248	4	)	)	PUNCT
ap-2064	249	1	it	it	PRON
ap-2064	249	2	suffices	suffice	VERB
ap-2064	249	3	to	to	PART
ap-2064	249	4	prove	prove	VERB
ap-2064	249	5	that	that	SCONJ
ap-2064	249	6	11	11	NUM
ap-2064	249	7	is	be	AUX
ap-2064	249	8	not	not	PART
ap-2064	249	9	a	a	DET
ap-2064	249	10	subword	subword	NOUN
ap-2064	249	11	of	of	ADP
ap-2064	249	12	f(a	f(a	PROPN
ap-2064	249	13	,	,	PUNCT
ap-2064	249	14	b	b	NOUN
ap-2064	249	15	,	,	PUNCT
ap-2064	249	16	n	n	CCONJ
ap-2064	249	17	)	)	PUNCT
ap-2064	249	18	,	,	PUNCT
ap-2064	249	19	for	for	ADP
ap-2064	249	20	all	all	DET
ap-2064	249	21	n	n	PRON
ap-2064	249	22	≥	≥	NOUN
ap-2064	249	23	0	0	NUM
ap-2064	249	24	.	.	PUNCT
ap-2064	250	1	we	we	PRON
ap-2064	250	2	proceed	proceed	VERB
ap-2064	250	3	by	by	ADP
ap-2064	250	4	induction	induction	NOUN
ap-2064	250	5	on	on	ADP
ap-2064	250	6	n.	n.	NOUN
ap-2064	250	7	for	for	ADP
ap-2064	250	8	n	n	NOUN
ap-2064	250	9	=	=	SYM
ap-2064	250	10	0	0	NUM
ap-2064	250	11	,	,	PUNCT
ap-2064	250	12	1	1	NUM
ap-2064	250	13	,	,	PUNCT
ap-2064	250	14	2	2	NUM
ap-2064	250	15	it	it	PRON
ap-2064	250	16	is	be	AUX
ap-2064	250	17	clear	clear	ADJ
ap-2064	250	18	.	.	PUNCT
ap-2064	251	1	now	now	ADV
ap-2064	251	2	,	,	PUNCT
ap-2064	251	3	assume	assume	VERB
ap-2064	251	4	that	that	SCONJ
ap-2064	251	5	it	it	PRON
ap-2064	251	6	is	be	AUX
ap-2064	251	7	true	true	ADJ
ap-2064	251	8	for	for	ADP
ap-2064	251	9	an	an	DET
ap-2064	251	10	arbitrary	arbitrary	ADJ
ap-2064	251	11	integer	integer	NOUN
ap-2064	251	12	n	n	PRON
ap-2064	251	13	≥	≥	NOUN
ap-2064	251	14	2	2	NUM
ap-2064	251	15	.	.	PUNCT
ap-2064	252	1	if	if	SCONJ
ap-2064	252	2	n+1	n+1	PROPN
ap-2064	252	3	is	be	AUX
ap-2064	252	4	even	even	ADV
ap-2064	252	5	,	,	PUNCT
ap-2064	252	6	we	we	PRON
ap-2064	252	7	know	know	VERB
ap-2064	252	8	that	that	SCONJ
ap-2064	252	9	f(a	f(a	NOUN
ap-2064	252	10	,	,	PUNCT
ap-2064	252	11	b	b	NOUN
ap-2064	252	12	,	,	PUNCT
ap-2064	252	13	n+1	n+1	PROPN
ap-2064	252	14	)	)	PUNCT
ap-2064	252	15	=	=	PUNCT
ap-2064	252	16	fa(a	fa(a	PROPN
ap-2064	252	17	,	,	PUNCT
ap-2064	252	18	b	b	NOUN
ap-2064	252	19	,	,	PUNCT
ap-2064	252	20	n)f(a	n)f(a	ADJ
ap-2064	252	21	,	,	PUNCT
ap-2064	252	22	b	b	NOUN
ap-2064	252	23	,	,	PUNCT
ap-2064	252	24	n−1	n−1	PROPN
ap-2064	252	25	)	)	PUNCT
ap-2064	252	26	,	,	PUNCT
ap-2064	252	27	so	so	CCONJ
ap-2064	252	28	by	by	ADP
ap-2064	252	29	the	the	DET
ap-2064	252	30	induction	induction	NOUN
ap-2064	252	31	hypothesis	hypothesis	NOUN
ap-2064	252	32	we	we	PRON
ap-2064	252	33	have	have	VERB
ap-2064	252	34	that	that	SCONJ
ap-2064	252	35	11	11	NUM
ap-2064	252	36	is	be	AUX
ap-2064	252	37	not	not	PART
ap-2064	252	38	a	a	DET
ap-2064	252	39	subword	subword	NOUN
ap-2064	252	40	of	of	ADP
ap-2064	252	41	f(a	f(a	PROPN
ap-2064	252	42	,	,	PUNCT
ap-2064	252	43	b	b	NOUN
ap-2064	252	44	,	,	PUNCT
ap-2064	252	45	n	n	CCONJ
ap-2064	252	46	)	)	PUNCT
ap-2064	252	47	and	and	CCONJ
ap-2064	252	48	f(a	f(a	PROPN
ap-2064	252	49	,	,	PUNCT
ap-2064	252	50	b	b	NOUN
ap-2064	252	51	,	,	PUNCT
ap-2064	252	52	n−1	n−1	PROPN
ap-2064	252	53	)	)	PUNCT
ap-2064	252	54	.	.	PUNCT
ap-2064	253	1	therefore	therefore	ADV
ap-2064	253	2	,	,	PUNCT
ap-2064	253	3	the	the	DET
ap-2064	253	4	only	only	ADJ
ap-2064	253	5	possibility	possibility	NOUN
ap-2064	253	6	is	be	AUX
ap-2064	253	7	that	that	SCONJ
ap-2064	253	8	1	1	NUM
ap-2064	253	9	is	be	AUX
ap-2064	253	10	a	a	DET
ap-2064	253	11	suffix	suffix	NOUN
ap-2064	253	12	and	and	CCONJ
ap-2064	253	13	a	a	DET
ap-2064	253	14	prefix	prefix	NOUN
ap-2064	253	15	of	of	ADP
ap-2064	253	16	f(a	f(a	PROPN
ap-2064	253	17	,	,	PUNCT
ap-2064	253	18	b	b	NOUN
ap-2064	253	19	,	,	PUNCT
ap-2064	253	20	n	n	CCONJ
ap-2064	253	21	)	)	PUNCT
ap-2064	253	22	or	or	CCONJ
ap-2064	253	23	1	1	NUM
ap-2064	253	24	is	be	AUX
ap-2064	253	25	a	a	DET
ap-2064	253	26	suffix	suffix	NOUN
ap-2064	253	27	of	of	ADP
ap-2064	253	28	f(a	f(a	PROPN
ap-2064	253	29	,	,	PUNCT
ap-2064	253	30	b	b	NOUN
ap-2064	253	31	,	,	PUNCT
ap-2064	253	32	n	n	CCONJ
ap-2064	253	33	)	)	PUNCT
ap-2064	253	34	and	and	CCONJ
ap-2064	253	35	a	a	DET
ap-2064	253	36	prefix	prefix	NOUN
ap-2064	253	37	of	of	ADP
ap-2064	253	38	f(a	f(a	PROPN
ap-2064	253	39	,	,	PUNCT
ap-2064	253	40	b	b	NOUN
ap-2064	253	41	,	,	PUNCT
ap-2064	253	42	n−1	n−1	PROPN
ap-2064	253	43	)	)	PUNCT
ap-2064	253	44	,	,	PUNCT
ap-2064	253	45	but	but	CCONJ
ap-2064	253	46	these	these	PRON
ap-2064	253	47	are	be	AUX
ap-2064	253	48	impossible	impossible	ADJ
ap-2064	253	49	by	by	ADP
ap-2064	253	50	the	the	DET
ap-2064	253	51	definition	definition	NOUN
ap-2064	253	52	of	of	ADP
ap-2064	253	53	the	the	DET
ap-2064	253	54	word	word	NOUN
ap-2064	253	55	f(a	f(a	PROPN
ap-2064	253	56	,	,	PUNCT
ap-2064	253	57	b	b	NOUN
ap-2064	253	58	,	,	PUNCT
ap-2064	253	59	n	n	CCONJ
ap-2064	253	60	)	)	PUNCT
ap-2064	253	61	.	.	PUNCT
ap-2064	254	1	if	if	SCONJ
ap-2064	254	2	n	n	PRON
ap-2064	254	3	+	+	SYM
ap-2064	254	4	1	1	NUM
ap-2064	254	5	is	be	AUX
ap-2064	254	6	odd	odd	ADJ
ap-2064	254	7	the	the	DET
ap-2064	254	8	proof	proof	NOUN
ap-2064	254	9	is	be	AUX
ap-2064	254	10	analogous	analogous	ADJ
ap-2064	254	11	.	.	PUNCT
ap-2064	255	1	(	(	PUNCT
ap-2064	255	2	2	2	NUM
ap-2064	255	3	.	.	PUNCT
ap-2064	255	4	)	)	PUNCT
ap-2064	256	1	we	we	PRON
ap-2064	256	2	proceed	proceed	VERB
ap-2064	256	3	by	by	ADP
ap-2064	256	4	induction	induction	NOUN
ap-2064	256	5	on	on	ADP
ap-2064	256	6	n.	n.	NOUN
ap-2064	256	7	for	for	ADP
ap-2064	256	8	n	n	NOUN
ap-2064	256	9	=	=	SYM
ap-2064	256	10	2	2	NUM
ap-2064	256	11	it	it	PRON
ap-2064	256	12	is	be	AUX
ap-2064	256	13	clear	clear	ADJ
ap-2064	256	14	.	.	PUNCT
ap-2064	257	1	now	now	ADV
ap-2064	257	2	,	,	PUNCT
ap-2064	257	3	assume	assume	VERB
ap-2064	257	4	that	that	SCONJ
ap-2064	257	5	it	it	PRON
ap-2064	257	6	is	be	AUX
ap-2064	257	7	true	true	ADJ
ap-2064	257	8	for	for	ADP
ap-2064	257	9	an	an	DET
ap-2064	257	10	arbitrary	arbitrary	ADJ
ap-2064	257	11	integer	integer	NOUN
ap-2064	257	12	n	n	PRON
ap-2064	257	13	≥	≥	NOUN
ap-2064	257	14	2	2	NUM
ap-2064	257	15	.	.	PUNCT
ap-2064	258	1	if	if	SCONJ
ap-2064	258	2	n	n	PRON
ap-2064	258	3	+	+	SYM
ap-2064	258	4	1	1	NUM
ap-2064	258	5	is	be	AUX
ap-2064	258	6	even	even	ADV
ap-2064	258	7	,	,	PUNCT
ap-2064	258	8	we	we	PRON
ap-2064	258	9	know	know	VERB
ap-2064	258	10	that	that	SCONJ
ap-2064	258	11	f(a	f(a	NOUN
ap-2064	258	12	,	,	PUNCT
ap-2064	258	13	b	b	NOUN
ap-2064	258	14	,	,	PUNCT
ap-2064	258	15	n+1	n+1	PROPN
ap-2064	258	16	)	)	PUNCT
ap-2064	258	17	=	=	PUNCT
ap-2064	258	18	fa(a	fa(a	PROPN
ap-2064	258	19	,	,	PUNCT
ap-2064	258	20	b	b	NOUN
ap-2064	258	21	,	,	PUNCT
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ap-2064	258	23	,	,	PUNCT
ap-2064	258	24	b	b	NOUN
ap-2064	258	25	,	,	PUNCT
ap-2064	258	26	n−1	n−1	PROPN
ap-2064	258	27	)	)	PUNCT
ap-2064	258	28	,	,	PUNCT
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ap-2064	258	30	by	by	ADP
ap-2064	258	31	the	the	DET
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ap-2064	258	33	hypothesis	hypothesis	NOUN
ap-2064	258	34	the	the	DET
ap-2064	258	35	last	last	ADJ
ap-2064	258	36	two	two	NUM
ap-2064	258	37	symbols	symbol	NOUN
ap-2064	258	38	of	of	ADP
ap-2064	258	39	f(a	f(a	PROPN
ap-2064	258	40	,	,	PUNCT
ap-2064	258	41	b	b	NOUN
ap-2064	258	42	,	,	PUNCT
ap-2064	258	43	n−1	n−1	PROPN
ap-2064	258	44	)	)	PUNCT
ap-2064	258	45	are	be	AUX
ap-2064	258	46	01	01	NUM
ap-2064	258	47	,	,	PUNCT
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ap-2064	258	49	the	the	DET
ap-2064	258	50	last	last	ADJ
ap-2064	258	51	two	two	NUM
ap-2064	258	52	symbols	symbol	NOUN
ap-2064	258	53	of	of	ADP
ap-2064	258	54	f(a	f(a	PROPN
ap-2064	258	55	,	,	PUNCT
ap-2064	258	56	b	b	NOUN
ap-2064	258	57	,	,	PUNCT
ap-2064	258	58	n+1	n+1	NOUN
ap-2064	258	59	)	)	PUNCT
ap-2064	258	60	are	be	AUX
ap-2064	258	61	01	01	NUM
ap-2064	258	62	.	.	PUNCT
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ap-2064	259	2	,	,	PUNCT
ap-2064	259	3	if	if	SCONJ
ap-2064	259	4	n+	n+	ADP
ap-2064	259	5	1	1	NUM
ap-2064	259	6	is	be	AUX
ap-2064	259	7	odd	odd	ADJ
ap-2064	259	8	.	.	PUNCT
ap-2064	260	1	(	(	PUNCT
ap-2064	260	2	3	3	NUM
ap-2064	260	3	.	.	PUNCT
ap-2064	260	4	)	)	PUNCT
ap-2064	261	1	we	we	PRON
ap-2064	261	2	proceed	proceed	VERB
ap-2064	261	3	by	by	ADP
ap-2064	261	4	induction	induction	NOUN
ap-2064	261	5	on	on	ADP
ap-2064	261	6	n.	n.	NOUN
ap-2064	261	7	for	for	ADP
ap-2064	261	8	n	n	NOUN
ap-2064	261	9	=	=	SYM
ap-2064	261	10	3	3	NUM
ap-2064	261	11	,	,	PUNCT
ap-2064	261	12	4	4	NUM
ap-2064	261	13	it	it	PRON
ap-2064	261	14	is	be	AUX
ap-2064	261	15	clear	clear	ADJ
ap-2064	261	16	.	.	PUNCT
ap-2064	262	1	now	now	ADV
ap-2064	262	2	,	,	PUNCT
ap-2064	262	3	assume	assume	VERB
ap-2064	262	4	that	that	SCONJ
ap-2064	262	5	it	it	PRON
ap-2064	262	6	is	be	AUX
ap-2064	262	7	true	true	ADJ
ap-2064	262	8	for	for	ADP
ap-2064	262	9	an	an	DET
ap-2064	262	10	arbitrary	arbitrary	ADJ
ap-2064	262	11	integer	integer	NOUN
ap-2064	262	12	n	n	PRON
ap-2064	262	13	≥	≥	NOUN
ap-2064	262	14	4	4	NUM
ap-2064	262	15	.	.	PUNCT
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ap-2064	263	2	n	n	PRON
ap-2064	263	3	is	be	AUX
ap-2064	263	4	even	even	ADV
ap-2064	263	5	,	,	PUNCT
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ap-2064	263	7	by	by	ADP
ap-2064	263	8	definition	definition	NOUN
ap-2064	263	9	of	of	ADP
ap-2064	263	10	f(a	f(a	PROPN
ap-2064	263	11	,	,	PUNCT
ap-2064	263	12	b	b	NOUN
ap-2064	263	13	,	,	PUNCT
ap-2064	263	14	n	n	CCONJ
ap-2064	263	15	)	)	PUNCT
ap-2064	263	16	,	,	PUNCT
ap-2064	263	17	we	we	PRON
ap-2064	263	18	have	have	VERB
ap-2064	263	19	f(a	f(a	NOUN
ap-2064	263	20	,	,	PUNCT
ap-2064	263	21	b	b	NOUN
ap-2064	263	22	,	,	PUNCT
ap-2064	263	23	n−1)f(a	n−1)f(a	ADJ
ap-2064	263	24	,	,	PUNCT
ap-2064	263	25	b	b	NOUN
ap-2064	263	26	,	,	PUNCT
ap-2064	263	27	n−2	n−2	PROPN
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ap-2064	263	29	=	=	SYM
ap-2064	263	30	f	f	PROPN
ap-2064	263	31	b(a	b(a	PROPN
ap-2064	263	32	,	,	PUNCT
ap-2064	263	33	b	b	NOUN
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ap-2064	263	35	n−2)f(a	n−2)f(a	NOUN
ap-2064	263	36	,	,	PUNCT
ap-2064	263	37	b	b	NOUN
ap-2064	263	38	,	,	PUNCT
ap-2064	263	39	n−3	n−3	PROPN
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ap-2064	263	41	·	·	PUNCT
ap-2064	263	42	fa(a	fa(a	PROPN
ap-2064	263	43	,	,	PUNCT
ap-2064	263	44	b	b	NOUN
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ap-2064	263	46	n−3)f(a	n−3)f(a	PROPN
ap-2064	263	47	,	,	PUNCT
ap-2064	263	48	b	b	PROPN
ap-2064	263	49	,	,	PUNCT
ap-2064	263	50	n−4	n−4	PROPN
ap-2064	263	51	)	)	PUNCT
ap-2064	263	52	=	=	PUNCT
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ap-2064	263	55	,	,	PUNCT
ap-2064	263	56	b	b	NOUN
ap-2064	263	57	,	,	PUNCT
ap-2064	263	58	n−3)f(a	n−3)f(a	PROPN
ap-2064	263	59	,	,	PUNCT
ap-2064	263	60	b	b	PROPN
ap-2064	263	61	,	,	PUNCT
ap-2064	263	62	n−4))b	n−4))b	PROPN
ap-2064	263	63	·	·	PUNCT
ap-2064	263	64	fa(a	fa(a	PROPN
ap-2064	263	65	,	,	PUNCT
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ap-2064	263	67	,	,	PUNCT
ap-2064	263	68	n−3)f(a	n−3)f(a	PROPN
ap-2064	263	69	,	,	PUNCT
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ap-2064	263	71	,	,	PUNCT
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ap-2064	263	73	,	,	PUNCT
ap-2064	263	74	b	b	PROPN
ap-2064	263	75	,	,	PUNCT
ap-2064	263	76	n−4	n−4	PROPN
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ap-2064	263	78	,	,	PUNCT
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ap-2064	263	80	f(a	f(a	PROPN
ap-2064	263	81	,	,	PUNCT
ap-2064	263	82	b	b	NOUN
ap-2064	263	83	,	,	PUNCT
ap-2064	263	84	n−2)f(a	n−2)f(a	NOUN
ap-2064	263	85	,	,	PUNCT
ap-2064	263	86	b	b	NOUN
ap-2064	263	87	,	,	PUNCT
ap-2064	263	88	n−1	n−1	PROPN
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ap-2064	263	90	=	=	PUNCT
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ap-2064	263	92	,	,	PUNCT
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ap-2064	263	98	,	,	PUNCT
ap-2064	263	99	n−4	n−4	PROPN
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ap-2064	264	5	,	,	PUNCT
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ap-2064	264	7	,	,	PUNCT
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ap-2064	264	9	,	,	PUNCT
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ap-2064	264	12	=	=	PUNCT
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ap-2064	264	17	n−3)f(a	n−3)f(a	PROPN
ap-2064	264	18	,	,	PUNCT
ap-2064	264	19	b	b	PROPN
ap-2064	264	20	,	,	PUNCT
ap-2064	264	21	n−4	n−4	PROPN
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ap-2064	264	23	·	·	PUNCT
ap-2064	264	24	(	(	PUNCT
ap-2064	264	25	fa(a	fa(a	PROPN
ap-2064	264	26	,	,	PUNCT
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ap-2064	264	32	,	,	PUNCT
ap-2064	264	33	n−4))b	n−4))b	PROPN
ap-2064	264	34	·	·	SYM
ap-2064	264	35	f(a	f(a	PROPN
ap-2064	264	36	,	,	PUNCT
ap-2064	264	37	b	b	NOUN
ap-2064	264	38	,	,	PUNCT
ap-2064	264	39	n−3	n−3	PROPN
ap-2064	264	40	)	)	PUNCT
ap-2064	264	41	=	=	PUNCT
ap-2064	264	42	(	(	PUNCT
ap-2064	264	43	fa(a	fa(a	PROPN
ap-2064	264	44	,	,	PUNCT
ap-2064	264	45	b	b	NOUN
ap-2064	264	46	,	,	PUNCT
ap-2064	264	47	n−3)f(a	n−3)f(a	PROPN
ap-2064	264	48	,	,	PUNCT
ap-2064	264	49	b	b	PROPN
ap-2064	264	50	,	,	PUNCT
ap-2064	264	51	n−4))b	n−4))b	PROPN
ap-2064	264	52	fa(a	fa(a	PROPN
ap-2064	264	53	,	,	PUNCT
ap-2064	264	54	b	b	NOUN
ap-2064	264	55	,	,	PUNCT
ap-2064	264	56	n−3)f(a	n−3)f(a	PROPN
ap-2064	264	57	,	,	PUNCT
ap-2064	264	58	b	b	PROPN
ap-2064	264	59	,	,	PUNCT
ap-2064	264	60	n−4)f(a	n−4)f(a	PROPN
ap-2064	264	61	,	,	PUNCT
ap-2064	264	62	b	b	NOUN
ap-2064	264	63	,	,	PUNCT
ap-2064	264	64	n−3	n−3	PROPN
ap-2064	264	65	)	)	PUNCT
ap-2064	264	66	.	.	PUNCT
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ap-2064	265	2	the	the	DET
ap-2064	265	3	words	word	NOUN
ap-2064	265	4	have	have	VERB
ap-2064	265	5	a	a	DET
ap-2064	265	6	common	common	ADJ
ap-2064	265	7	prefix	prefix	NOUN
ap-2064	265	8	of	of	ADP
ap-2064	265	9	length	length	NOUN
ap-2064	265	10	b(aqn−3	b(aqn−3	PROPN
ap-2064	265	11	+	+	PUNCT
ap-2064	265	12	qn−4	qn−4	NOUN
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ap-2064	266	1	+	+	CCONJ
ap-2064	267	1	aqn−3	aqn−3	PROPN
ap-2064	267	2	=	=	SYM
ap-2064	267	3	bqn−2	bqn−2	PROPN
ap-2064	267	4	+	+	CCONJ
ap-2064	267	5	aqn−3	aqn−3	PROPN
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ap-2064	267	8	the	the	DET
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ap-2064	267	10	hypothesis	hypothesis	NOUN
ap-2064	267	11	f(a	f(a	PROPN
ap-2064	267	12	,	,	PUNCT
ap-2064	267	13	b	b	NOUN
ap-2064	267	14	,	,	PUNCT
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ap-2064	267	17	b	b	PROPN
ap-2064	267	18	,	,	PUNCT
ap-2064	267	19	n−4	n−4	PROPN
ap-2064	267	20	)	)	PUNCT
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ap-2064	267	22	f(a	f(a	PROPN
ap-2064	267	23	,	,	PUNCT
ap-2064	267	24	b	b	PROPN
ap-2064	267	25	,	,	PUNCT
ap-2064	267	26	n−4)f(a	n−4)f(a	PROPN
ap-2064	267	27	,	,	PUNCT
ap-2064	267	28	b	b	NOUN
ap-2064	267	29	,	,	PUNCT
ap-2064	267	30	n−3	n−3	PROPN
ap-2064	267	31	)	)	PUNCT
ap-2064	267	32	have	have	VERB
ap-2064	267	33	a	a	DET
ap-2064	267	34	common	common	ADJ
ap-2064	267	35	prefix	prefix	NOUN
ap-2064	267	36	of	of	ADP
ap-2064	267	37	length	length	NOUN
ap-2064	267	38	qn−3+qn−4−2	qn−3+qn−4−2	PROPN
ap-2064	267	39	.	.	PUNCT
ap-2064	268	1	therefore	therefore	ADV
ap-2064	268	2	the	the	DET
ap-2064	268	3	words	word	NOUN
ap-2064	268	4	have	have	VERB
ap-2064	268	5	a	a	DET
ap-2064	268	6	common	common	ADJ
ap-2064	268	7	prefix	prefix	NOUN
ap-2064	268	8	of	of	ADP
ap-2064	268	9	length	length	NOUN
ap-2064	268	10	bqn−2	bqn−2	PROPN
ap-2064	268	11	+	+	PROPN
ap-2064	268	12	aqn−3	aqn−3	PROPN
ap-2064	268	13	+	+	CCONJ
ap-2064	268	14	qn−3	qn−3	PROPN
ap-2064	268	15	+	+	CCONJ
ap-2064	268	16	qn−4−2	qn−4−2	PROPN
ap-2064	268	17	=	=	PUNCT
ap-2064	268	18	qn−1	qn−1	PROPN
ap-2064	268	19	+	+	NUM
ap-2064	268	20	qn−2−2	qn−2−2	NOUN
ap-2064	268	21	.	.	PUNCT
ap-2064	269	1	if	if	SCONJ
ap-2064	269	2	n	n	NOUN
ap-2064	269	3	is	be	AUX
ap-2064	269	4	odd	odd	ADJ
ap-2064	269	5	the	the	DET
ap-2064	269	6	proof	proof	NOUN
ap-2064	269	7	is	be	AUX
ap-2064	269	8	analogous	analogous	ADJ
ap-2064	269	9	.	.	PUNCT
ap-2064	270	1	the	the	DET
ap-2064	270	2	above	above	ADJ
ap-2064	270	3	proposition	proposition	NOUN
ap-2064	270	4	is	be	AUX
ap-2064	270	5	a	a	DET
ap-2064	270	6	particular	particular	ADJ
ap-2064	270	7	result	result	NOUN
ap-2064	270	8	related	relate	VERB
ap-2064	270	9	to	to	ADP
ap-2064	270	10	sturmian	sturmian	NOUN
ap-2064	270	11	words	word	NOUN
ap-2064	270	12	,	,	PUNCT
ap-2064	270	13	see	see	VERB
ap-2064	270	14	,	,	PUNCT
ap-2064	270	15	e.g.	e.g.	ADV
ap-2064	270	16	,	,	PUNCT
ap-2064	270	17	[	[	X
ap-2064	270	18	14	14	NUM
ap-2064	270	19	]	]	PUNCT
ap-2064	270	20	.	.	PUNCT
ap-2064	271	1	definition	definition	NOUN
ap-2064	271	2	9	9	NUM
ap-2064	271	3	.	.	PUNCT
ap-2064	272	1	let	let	VERB
ap-2064	272	2	φ	φ	PROPN
ap-2064	272	3	:	:	PUNCT
ap-2064	272	4	{	{	PUNCT
ap-2064	272	5	0	0	NUM
ap-2064	272	6	,	,	PUNCT
ap-2064	272	7	1}∗	1}∗	PROPN
ap-2064	272	8	→	→	SYM
ap-2064	272	9	{	{	PUNCT
ap-2064	272	10	0	0	NUM
ap-2064	272	11	,	,	PUNCT
ap-2064	272	12	1}∗	1}∗	PROPN
ap-2064	272	13	be	be	AUX
ap-2064	272	14	a	a	DET
ap-2064	272	15	map	map	NOUN
ap-2064	272	16	such	such	ADJ
ap-2064	272	17	that	that	SCONJ
ap-2064	272	18	φ	φ	PROPN
ap-2064	272	19	deletes	delete	VERB
ap-2064	272	20	the	the	DET
ap-2064	272	21	last	last	ADJ
ap-2064	272	22	two	two	NUM
ap-2064	272	23	symbols	symbol	NOUN
ap-2064	272	24	,	,	PUNCT
ap-2064	272	25	i.e.	i.e.	X
ap-2064	272	26	,	,	PUNCT
ap-2064	272	27	φ(a1a2	φ(a1a2	INTJ
ap-2064	272	28	·	·	PUNCT
ap-2064	272	29	·	·	PUNCT
ap-2064	272	30	·	·	PUNCT
ap-2064	273	1	an	an	X
ap-2064	273	2	)	)	PUNCT
ap-2064	273	3	=	=	PUNCT
ap-2064	273	4	a1a2	a1a2	PROPN
ap-2064	273	5	·	·	PUNCT
ap-2064	273	6	·	·	PUNCT
ap-2064	273	7	·	·	PUNCT
ap-2064	273	8	an−2	an−2	PROPN
ap-2064	273	9	,	,	PUNCT
ap-2064	273	10	if	if	SCONJ
ap-2064	273	11	n	n	PROPN
ap-2064	273	12	>	>	X
ap-2064	273	13	2	2	NUM
ap-2064	273	14	,	,	PUNCT
ap-2064	273	15	and	and	CCONJ
ap-2064	273	16	φ(a1a2	φ(a1a2	ADJ
ap-2064	273	17	·	·	PUNCT
ap-2064	273	18	·	·	PUNCT
ap-2064	273	19	·	·	PUNCT
ap-2064	273	20	an	an	X
ap-2064	273	21	)	)	PUNCT
ap-2064	273	22	=	=	SYM
ap-2064	273	23	ε	ε	PROPN
ap-2064	273	24	if	if	SCONJ
ap-2064	273	25	n	n	NOUN
ap-2064	273	26	≤	≤	ADV
ap-2064	273	27	2	2	NUM
ap-2064	273	28	.	.	PUNCT
ap-2064	273	29	corollary	corollary	ADJ
ap-2064	273	30	10	10	NUM
ap-2064	273	31	.	.	PUNCT
ap-2064	274	1	the	the	DET
ap-2064	274	2	nth	nth	PROPN
ap-2064	274	3	biperiodic	biperiodic	PROPN
ap-2064	274	4	fibonacci	fibonacci	NOUN
ap-2064	274	5	word	word	NOUN
ap-2064	274	6	satisfies	satisfy	VERB
ap-2064	274	7	for	for	ADP
ap-2064	274	8	all	all	DET
ap-2064	274	9	n	n	PRON
ap-2064	274	10	≥	≥	NOUN
ap-2064	274	11	2	2	NUM
ap-2064	274	12	,	,	PUNCT
ap-2064	274	13	and	and	CCONJ
ap-2064	274	14	x	x	X
ap-2064	274	15	,	,	PUNCT
ap-2064	274	16	y	y	PROPN
ap-2064	274	17	∈	∈	PROPN
ap-2064	274	18	{	{	PUNCT
ap-2064	274	19	0	0	NUM
ap-2064	274	20	,	,	PUNCT
ap-2064	274	21	1	1	NUM
ap-2064	274	22	}	}	PUNCT
ap-2064	274	23	that	that	PRON
ap-2064	274	24	(	(	PUNCT
ap-2064	274	25	1	1	NUM
ap-2064	274	26	.	.	PUNCT
ap-2064	274	27	)	)	PUNCT
ap-2064	275	1	φ(f(a	φ(f(a	PROPN
ap-2064	275	2	,	,	PUNCT
ap-2064	275	3	b	b	NOUN
ap-2064	275	4	,	,	PUNCT
ap-2064	275	5	n−1)f(a	n−1)f(a	ADJ
ap-2064	275	6	,	,	PUNCT
ap-2064	275	7	b	b	NOUN
ap-2064	275	8	,	,	PUNCT
ap-2064	275	9	n−2	n−2	PROPN
ap-2064	275	10	)	)	PUNCT
ap-2064	275	11	)	)	PUNCT
ap-2064	276	1	=	=	SYM
ap-2064	276	2	φ(f(a	φ(f(a	PROPN
ap-2064	276	3	,	,	PUNCT
ap-2064	276	4	b	b	NOUN
ap-2064	276	5	,	,	PUNCT
ap-2064	276	6	n−2)f(a	n−2)f(a	NOUN
ap-2064	276	7	,	,	PUNCT
ap-2064	276	8	b	b	NOUN
ap-2064	276	9	,	,	PUNCT
ap-2064	276	10	n−1	n−1	PROPN
ap-2064	276	11	)	)	PUNCT
ap-2064	276	12	)	)	PUNCT
ap-2064	276	13	.	.	PUNCT
ap-2064	277	1	(	(	PUNCT
ap-2064	277	2	2	2	NUM
ap-2064	277	3	.	.	PUNCT
ap-2064	277	4	)	)	PUNCT
ap-2064	278	1	φ(f(a	φ(f(a	PROPN
ap-2064	278	2	,	,	PUNCT
ap-2064	278	3	b	b	NOUN
ap-2064	278	4	,	,	PUNCT
ap-2064	278	5	n−1)f(a	n−1)f(a	ADJ
ap-2064	278	6	,	,	PUNCT
ap-2064	278	7	b	b	NOUN
ap-2064	278	8	,	,	PUNCT
ap-2064	278	9	n−2	n−2	PROPN
ap-2064	278	10	)	)	PUNCT
ap-2064	278	11	)	)	PUNCT
ap-2064	279	1	=	=	SYM
ap-2064	279	2	f(a	f(a	PROPN
ap-2064	279	3	,	,	PUNCT
ap-2064	279	4	b	b	PROPN
ap-2064	279	5	,	,	PUNCT
ap-2064	279	6	n−2)φ(f(a	n−2)φ(f(a	PROPN
ap-2064	279	7	,	,	PUNCT
ap-2064	279	8	b	b	PROPN
ap-2064	279	9	,	,	PUNCT
ap-2064	279	10	n−1	n−1	PROPN
ap-2064	279	11	)	)	PUNCT
ap-2064	279	12	)	)	PUNCT
ap-2064	280	1	=	=	SYM
ap-2064	280	2	f(a	f(a	PROPN
ap-2064	280	3	,	,	PUNCT
ap-2064	280	4	b	b	NOUN
ap-2064	280	5	,	,	PUNCT
ap-2064	280	6	n−1)φ(f(a	n−1)φ(f(a	PROPN
ap-2064	280	7	,	,	PUNCT
ap-2064	280	8	b	b	NOUN
ap-2064	280	9	,	,	PUNCT
ap-2064	280	10	n−2	n−2	PROPN
ap-2064	280	11	)	)	PUNCT
ap-2064	280	12	)	)	PUNCT
ap-2064	280	13	.	.	PUNCT
ap-2064	281	1	(	(	PUNCT
ap-2064	281	2	3	3	X
ap-2064	281	3	.	.	PUNCT
ap-2064	281	4	)	)	PUNCT
ap-2064	282	1	if	if	SCONJ
ap-2064	282	2	f(a	f(a	PROPN
ap-2064	282	3	,	,	PUNCT
ap-2064	282	4	b	b	NOUN
ap-2064	282	5	,	,	PUNCT
ap-2064	282	6	n	n	CCONJ
ap-2064	282	7	)	)	PUNCT
ap-2064	282	8	=	=	SYM
ap-2064	282	9	φ(f(a	φ(f(a	PROPN
ap-2064	282	10	,	,	PUNCT
ap-2064	282	11	b	b	PROPN
ap-2064	282	12	,	,	PUNCT
ap-2064	282	13	n))xy	n))xy	PROPN
ap-2064	282	14	,	,	PUNCT
ap-2064	282	15	then	then	ADV
ap-2064	282	16	φ(f(a	φ(f(a	PROPN
ap-2064	282	17	,	,	PUNCT
ap-2064	282	18	b	b	PROPN
ap-2064	282	19	,	,	PUNCT
ap-2064	282	20	n−2))xyφ(f(a	n−2))xyφ(f(a	PROPN
ap-2064	282	21	,	,	PUNCT
ap-2064	282	22	b	b	NOUN
ap-2064	282	23	,	,	PUNCT
ap-2064	282	24	n−1	n−1	PROPN
ap-2064	282	25	)	)	PUNCT
ap-2064	282	26	)	)	PUNCT
ap-2064	282	27	=	=	SYM
ap-2064	282	28	f(a	f(a	PROPN
ap-2064	282	29	,	,	PUNCT
ap-2064	282	30	b	b	NOUN
ap-2064	282	31	,	,	PUNCT
ap-2064	282	32	n−1)φ(f(a	n−1)φ(f(a	PROPN
ap-2064	282	33	,	,	PUNCT
ap-2064	282	34	b	b	NOUN
ap-2064	282	35	,	,	PUNCT
ap-2064	282	36	n−2	n−2	PROPN
ap-2064	282	37	)	)	PUNCT
ap-2064	282	38	)	)	PUNCT
ap-2064	282	39	.	.	PUNCT
ap-2064	283	1	(	(	PUNCT
ap-2064	283	2	4	4	NUM
ap-2064	283	3	.	.	PUNCT
ap-2064	283	4	)	)	PUNCT
ap-2064	284	1	if	if	SCONJ
ap-2064	284	2	f(a	f(a	PROPN
ap-2064	284	3	,	,	PUNCT
ap-2064	284	4	b	b	NOUN
ap-2064	284	5	,	,	PUNCT
ap-2064	284	6	n	n	CCONJ
ap-2064	284	7	)	)	PUNCT
ap-2064	284	8	=	=	SYM
ap-2064	284	9	φ(f(a	φ(f(a	PROPN
ap-2064	284	10	,	,	PUNCT
ap-2064	284	11	b	b	PROPN
ap-2064	284	12	,	,	PUNCT
ap-2064	284	13	n))xy	n))xy	PROPN
ap-2064	284	14	,	,	PUNCT
ap-2064	284	15	then	then	ADV
ap-2064	284	16	φ(f(a	φ(f(a	PROPN
ap-2064	284	17	,	,	PUNCT
ap-2064	284	18	b	b	PROPN
ap-2064	284	19	,	,	PUNCT
ap-2064	284	20	n	n	CCONJ
ap-2064	284	21	)	)	PUNCT
ap-2064	284	22	)	)	PUNCT
ap-2064	284	23	=	=	PUNCT
ap-2064	284	24			PROPN
ap-2064	284	25	φ(f(a	φ(f(a	PROPN
ap-2064	284	26	,	,	PUNCT
ap-2064	284	27	b	b	PROPN
ap-2064	284	28	,	,	PUNCT
ap-2064	284	29	n−2))(10φ(f(a	n−2))(10φ(f(a	PROPN
ap-2064	284	30	,	,	PUNCT
ap-2064	284	31	b	b	PROPN
ap-2064	284	32	,	,	PUNCT
ap-2064	284	33	n−1)))a	n−1)))a	ADJ
ap-2064	284	34	,	,	PUNCT
ap-2064	284	35	if	if	SCONJ
ap-2064	284	36	n	n	PRON
ap-2064	284	37	is	be	AUX
ap-2064	284	38	even	even	ADV
ap-2064	284	39	.	.	PUNCT
ap-2064	285	1	φ(f(a	φ(f(a	PROPN
ap-2064	285	2	,	,	PUNCT
ap-2064	285	3	b	b	PROPN
ap-2064	285	4	,	,	PUNCT
ap-2064	285	5	n−2))(01φ(f(a	n−2))(01φ(f(a	PROPN
ap-2064	285	6	,	,	PUNCT
ap-2064	285	7	b	b	PROPN
ap-2064	285	8	,	,	PUNCT
ap-2064	285	9	n−1)))b	n−1)))b	PROPN
ap-2064	285	10	,	,	PUNCT
ap-2064	285	11	if	if	SCONJ
ap-2064	285	12	n	n	PRON
ap-2064	285	13	is	be	AUX
ap-2064	285	14	odd	odd	ADJ
ap-2064	285	15	.	.	PUNCT
ap-2064	286	1	proof	proof	NOUN
ap-2064	286	2	.	.	PUNCT
ap-2064	287	1	(	(	PUNCT
ap-2064	287	2	1	1	NUM
ap-2064	287	3	.	.	PUNCT
ap-2064	287	4	)	)	PUNCT
ap-2064	288	1	and	and	CCONJ
ap-2064	288	2	(	(	PUNCT
ap-2064	288	3	2	2	NUM
ap-2064	288	4	.	.	PUNCT
ap-2064	288	5	)	)	PUNCT
ap-2064	288	6	follow	follow	VERB
ap-2064	288	7	immediately	immediately	ADV
ap-2064	288	8	from	from	ADP
ap-2064	288	9	proposition	proposition	NOUN
ap-2064	288	10	8.(3	8.(3	NUM
ap-2064	288	11	.	.	PUNCT
ap-2064	288	12	)	)	PUNCT
ap-2064	289	1	and	and	CCONJ
ap-2064	289	2	|f(a	|f(a	VERB
ap-2064	289	3	,	,	PUNCT
ap-2064	289	4	b	b	PROPN
ap-2064	289	5	,	,	PUNCT
ap-2064	289	6	n)|	n)|	PROPN
ap-2064	289	7	≥	≥	NOUN
ap-2064	289	8	2	2	NUM
ap-2064	289	9	for	for	ADP
ap-2064	289	10	all	all	DET
ap-2064	289	11	n	n	PRON
ap-2064	289	12	≥	≥	NOUN
ap-2064	289	13	2	2	NUM
ap-2064	289	14	.	.	PUNCT
ap-2064	290	1	for	for	ADP
ap-2064	290	2	(	(	PUNCT
ap-2064	290	3	3	3	NUM
ap-2064	290	4	.	.	NUM
ap-2064	290	5	)	)	PUNCT
ap-2064	290	6	,	,	PUNCT
ap-2064	290	7	in	in	ADP
ap-2064	290	8	54	54	NUM
ap-2064	290	9	vol	vol	NOUN
ap-2064	290	10	.	.	PUNCT
ap-2064	290	11	55	55	NUM
ap-2064	290	12	no	no	NOUN
ap-2064	290	13	.	.	PUNCT
ap-2064	291	1	1/2015	1/2015	NUM
ap-2064	291	2	biperiodic	biperiodic	PROPN
ap-2064	291	3	fibonacci	fibonacci	NOUN
ap-2064	291	4	word	word	NOUN
ap-2064	291	5	and	and	CCONJ
ap-2064	291	6	its	its	PRON
ap-2064	291	7	fractal	fractal	ADJ
ap-2064	291	8	curve	curve	NOUN
ap-2064	291	9	symbol	symbol	NOUN
ap-2064	291	10	action	action	NOUN
ap-2064	291	11	1	1	NUM
ap-2064	291	12	draw	draw	VERB
ap-2064	291	13	a	a	DET
ap-2064	291	14	line	line	NOUN
ap-2064	291	15	forward	forward	ADV
ap-2064	291	16	.	.	PUNCT
ap-2064	292	1	0	0	PUNCT
ap-2064	292	2	draw	draw	VERB
ap-2064	292	3	a	a	DET
ap-2064	292	4	line	line	NOUN
ap-2064	292	5	forward	forward	ADV
ap-2064	292	6	and	and	CCONJ
ap-2064	292	7	if	if	SCONJ
ap-2064	292	8	the	the	DET
ap-2064	292	9	symbol	symbol	NOUN
ap-2064	292	10	0	0	PUNCT
ap-2064	292	11	is	be	AUX
ap-2064	292	12	in	in	ADP
ap-2064	292	13	a	a	DET
ap-2064	292	14	position	position	NOUN
ap-2064	292	15	even	even	ADV
ap-2064	292	16	,	,	PUNCT
ap-2064	292	17	then	then	ADV
ap-2064	292	18	turn	turn	VERB
ap-2064	292	19	θ	θ	NOUN
ap-2064	292	20	degree	degree	NOUN
ap-2064	292	21	and	and	CCONJ
ap-2064	292	22	if	if	SCONJ
ap-2064	292	23	0	0	NUM
ap-2064	292	24	is	be	AUX
ap-2064	292	25	in	in	ADP
ap-2064	292	26	a	a	DET
ap-2064	292	27	position	position	NOUN
ap-2064	292	28	odd	odd	ADJ
ap-2064	292	29	,	,	PUNCT
ap-2064	292	30	then	then	ADV
ap-2064	292	31	turn	turn	VERB
ap-2064	292	32	−θ	−θ	ADJ
ap-2064	292	33	degrees	degree	NOUN
ap-2064	292	34	.	.	PUNCT
ap-2064	293	1	table	table	NOUN
ap-2064	293	2	4	4	NUM
ap-2064	293	3	.	.	PUNCT
ap-2064	294	1	odd	odd	ADV
ap-2064	294	2	-	-	PUNCT
ap-2064	294	3	even	even	ADV
ap-2064	294	4	drawing	draw	VERB
ap-2064	294	5	rule	rule	NOUN
ap-2064	294	6	with	with	ADP
ap-2064	294	7	turn	turn	NOUN
ap-2064	294	8	angle	angle	PROPN
ap-2064	294	9	θ	θ	PROPN
ap-2064	294	10	.	.	PUNCT
ap-2064	294	11	fact	fact	NOUN
ap-2064	294	12	,	,	PUNCT
ap-2064	294	13	if	if	SCONJ
ap-2064	294	14	f(a	f(a	PROPN
ap-2064	294	15	,	,	PUNCT
ap-2064	294	16	b	b	NOUN
ap-2064	294	17	,	,	PUNCT
ap-2064	294	18	n	n	CCONJ
ap-2064	294	19	)	)	PUNCT
ap-2064	294	20	=	=	SYM
ap-2064	294	21	φ(f(a	φ(f(a	PROPN
ap-2064	294	22	,	,	PUNCT
ap-2064	294	23	b	b	PROPN
ap-2064	294	24	,	,	PUNCT
ap-2064	294	25	n))xy	n))xy	PROPN
ap-2064	294	26	,	,	PUNCT
ap-2064	294	27	then	then	ADV
ap-2064	294	28	from	from	ADP
ap-2064	294	29	proposition	proposition	NOUN
ap-2064	294	30	8.(2	8.(2	NUM
ap-2064	294	31	.	.	PUNCT
ap-2064	294	32	)	)	PUNCT
ap-2064	295	1	we	we	PRON
ap-2064	295	2	have	have	VERB
ap-2064	295	3	f(a	f(a	NOUN
ap-2064	295	4	,	,	PUNCT
ap-2064	295	5	b	b	NOUN
ap-2064	295	6	,	,	PUNCT
ap-2064	295	7	n−2	n−2	PROPN
ap-2064	295	8	)	)	PUNCT
ap-2064	295	9	=	=	SYM
ap-2064	295	10	φ(f(a	φ(f(a	PROPN
ap-2064	295	11	,	,	PUNCT
ap-2064	295	12	b	b	PROPN
ap-2064	295	13	,	,	PUNCT
ap-2064	295	14	n−2))xy	n−2))xy	NOUN
ap-2064	295	15	.	.	PUNCT
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ap-2064	296	2	φ(f(a	φ(f(a	PROPN
ap-2064	296	3	,	,	PUNCT
ap-2064	296	4	b	b	PROPN
ap-2064	296	5	,	,	PUNCT
ap-2064	296	6	n−2))xyφ(f(a	n−2))xyφ(f(a	PROPN
ap-2064	296	7	,	,	PUNCT
ap-2064	296	8	b	b	NOUN
ap-2064	296	9	,	,	PUNCT
ap-2064	296	10	n−1	n−1	PROPN
ap-2064	296	11	)	)	PUNCT
ap-2064	296	12	)	)	PUNCT
ap-2064	297	1	=	=	SYM
ap-2064	297	2	f(a	f(a	PROPN
ap-2064	297	3	,	,	PUNCT
ap-2064	297	4	b	b	PROPN
ap-2064	297	5	,	,	PUNCT
ap-2064	297	6	n−2)φ(f(a	n−2)φ(f(a	PROPN
ap-2064	297	7	,	,	PUNCT
ap-2064	297	8	b	b	PROPN
ap-2064	297	9	,	,	PUNCT
ap-2064	297	10	n−1	n−1	PROPN
ap-2064	297	11	)	)	PUNCT
ap-2064	297	12	)	)	PUNCT
ap-2064	298	1	=	=	SYM
ap-2064	298	2	f(a	f(a	PROPN
ap-2064	298	3	,	,	PUNCT
ap-2064	298	4	b	b	NOUN
ap-2064	298	5	,	,	PUNCT
ap-2064	298	6	n−1)φ(f(a	n−1)φ(f(a	PROPN
ap-2064	298	7	,	,	PUNCT
ap-2064	298	8	b	b	NOUN
ap-2064	298	9	,	,	PUNCT
ap-2064	298	10	n−2	n−2	PROPN
ap-2064	298	11	)	)	PUNCT
ap-2064	298	12	)	)	PUNCT
ap-2064	298	13	.	.	PUNCT
ap-2064	299	1	item	item	NOUN
ap-2064	299	2	(	(	PUNCT
ap-2064	299	3	4	4	NUM
ap-2064	299	4	.	.	PUNCT
ap-2064	299	5	)	)	PUNCT
ap-2064	299	6	is	be	AUX
ap-2064	299	7	clear	clear	ADJ
ap-2064	299	8	from	from	ADP
ap-2064	299	9	(	(	PUNCT
ap-2064	299	10	3	3	NUM
ap-2064	299	11	.	.	PUNCT
ap-2064	299	12	)	)	PUNCT
ap-2064	300	1	and	and	CCONJ
ap-2064	300	2	the	the	DET
ap-2064	300	3	definition	definition	NOUN
ap-2064	300	4	of	of	ADP
ap-2064	300	5	f(a	f(a	PROPN
ap-2064	300	6	,	,	PUNCT
ap-2064	300	7	b	b	NOUN
ap-2064	300	8	,	,	PUNCT
ap-2064	300	9	n	n	CCONJ
ap-2064	300	10	)	)	PUNCT
ap-2064	300	11	.	.	PUNCT
ap-2064	301	1	theorem	theorem	VERB
ap-2064	301	2	11	11	NUM
ap-2064	301	3	.	.	PUNCT
ap-2064	302	1	φ(f(a	φ(f(a	PROPN
ap-2064	302	2	,	,	PUNCT
ap-2064	302	3	b	b	NOUN
ap-2064	302	4	,	,	PUNCT
ap-2064	302	5	n	n	CCONJ
ap-2064	302	6	)	)	PUNCT
ap-2064	302	7	)	)	PUNCT
ap-2064	302	8	is	be	AUX
ap-2064	302	9	a	a	DET
ap-2064	302	10	palindrome	palindrome	NOUN
ap-2064	302	11	for	for	ADP
ap-2064	302	12	all	all	DET
ap-2064	302	13	n	n	CCONJ
ap-2064	302	14	,	,	PUNCT
ap-2064	302	15	a	a	DET
ap-2064	302	16	≥	≥	NOUN
ap-2064	302	17	2	2	NUM
ap-2064	302	18	and	and	CCONJ
ap-2064	302	19	b	b	NOUN
ap-2064	302	20	≥	≥	NUM
ap-2064	302	21	1	1	NUM
ap-2064	302	22	.	.	PUNCT
ap-2064	303	1	proof	proof	NOUN
ap-2064	303	2	.	.	PUNCT
ap-2064	304	1	we	we	PRON
ap-2064	304	2	proceed	proceed	VERB
ap-2064	304	3	by	by	ADP
ap-2064	304	4	induction	induction	NOUN
ap-2064	304	5	on	on	ADP
ap-2064	304	6	n.	n.	NOUN
ap-2064	304	7	if	if	SCONJ
ap-2064	304	8	n	n	NOUN
ap-2064	304	9	=	=	SYM
ap-2064	304	10	2	2	NUM
ap-2064	304	11	,	,	PUNCT
ap-2064	304	12	3	3	NUM
ap-2064	304	13	,	,	PUNCT
ap-2064	304	14	then	then	ADV
ap-2064	304	15	φ(f(a	φ(f(a	PROPN
ap-2064	304	16	,	,	PUNCT
ap-2064	304	17	b,2	b,2	NOUN
ap-2064	304	18	)	)	PUNCT
ap-2064	304	19	)	)	PUNCT
ap-2064	305	1	=	=	SYM
ap-2064	305	2	0a−1	0a−1	PROPN
ap-2064	305	3	and	and	CCONJ
ap-2064	305	4	φ(f(a	φ(f(a	PROPN
ap-2064	305	5	,	,	PUNCT
ap-2064	305	6	b,3	b,3	NUM
ap-2064	305	7	)	)	PUNCT
ap-2064	305	8	)	)	PUNCT
ap-2064	306	1	=	=	PUNCT
ap-2064	306	2	(	(	PUNCT
ap-2064	306	3	0a−11)b0	0a−11)b0	NUM
ap-2064	306	4	are	be	AUX
ap-2064	306	5	palindromes	palindrome	NOUN
ap-2064	306	6	.	.	PUNCT
ap-2064	307	1	now	now	ADV
ap-2064	307	2	,	,	PUNCT
ap-2064	307	3	assume	assume	VERB
ap-2064	307	4	that	that	SCONJ
ap-2064	307	5	it	it	PRON
ap-2064	307	6	is	be	AUX
ap-2064	307	7	true	true	ADJ
ap-2064	307	8	for	for	ADP
ap-2064	307	9	an	an	DET
ap-2064	307	10	arbitrary	arbitrary	ADJ
ap-2064	307	11	integer	integer	NOUN
ap-2064	307	12	n	n	PRON
ap-2064	307	13	≥	≥	NOUN
ap-2064	307	14	3	3	NUM
ap-2064	307	15	.	.	PUNCT
ap-2064	308	1	if	if	SCONJ
ap-2064	308	2	n	n	NOUN
ap-2064	308	3	is	be	AUX
ap-2064	308	4	even	even	ADV
ap-2064	308	5	,	,	PUNCT
ap-2064	308	6	then	then	ADV
ap-2064	308	7	from	from	ADP
ap-2064	308	8	corollary	corollary	ADJ
ap-2064	308	9	10	10	NUM
ap-2064	308	10	(	(	PUNCT
ap-2064	308	11	φ(f(a	φ(f(a	PROPN
ap-2064	308	12	,	,	PUNCT
ap-2064	308	13	b	b	PROPN
ap-2064	308	14	,	,	PUNCT
ap-2064	308	15	n)))r	n)))r	PROPN
ap-2064	308	16	=	=	SYM
ap-2064	308	17	(	(	PUNCT
ap-2064	308	18	φ(fa(a	φ(fa(a	PROPN
ap-2064	308	19	,	,	PUNCT
ap-2064	308	20	b	b	NOUN
ap-2064	308	21	,	,	PUNCT
ap-2064	308	22	n−1)f(a	n−1)f(a	NUM
ap-2064	308	23	,	,	PUNCT
ap-2064	308	24	b	b	NOUN
ap-2064	308	25	,	,	PUNCT
ap-2064	308	26	n−2)))r	n−2)))r	PROPN
ap-2064	308	27	=	=	SYM
ap-2064	308	28	(	(	PUNCT
ap-2064	308	29	fa(a	fa(a	PROPN
ap-2064	308	30	,	,	PUNCT
ap-2064	308	31	b	b	NOUN
ap-2064	308	32	,	,	PUNCT
ap-2064	308	33	n−1)φ(f(a	n−1)φ(f(a	PROPN
ap-2064	308	34	,	,	PUNCT
ap-2064	308	35	b	b	NOUN
ap-2064	308	36	,	,	PUNCT
ap-2064	308	37	n−2)))r	n−2)))r	PROPN
ap-2064	308	38	=	=	SYM
ap-2064	308	39	φ(f(a	φ(f(a	PROPN
ap-2064	308	40	,	,	PUNCT
ap-2064	308	41	b	b	PROPN
ap-2064	308	42	,	,	PUNCT
ap-2064	308	43	n−2))r(fa(a	n−2))r(fa(a	NOUN
ap-2064	308	44	,	,	PUNCT
ap-2064	308	45	b	b	NOUN
ap-2064	308	46	,	,	PUNCT
ap-2064	308	47	n−1))r	n−1))r	PROPN
ap-2064	308	48	=	=	SYM
ap-2064	308	49	φ(f(a	φ(f(a	PROPN
ap-2064	308	50	,	,	PUNCT
ap-2064	308	51	b	b	PROPN
ap-2064	308	52	,	,	PUNCT
ap-2064	308	53	n−2))(fr(a	n−2))(fr(a	ADJ
ap-2064	308	54	,	,	PUNCT
ap-2064	308	55	b	b	NOUN
ap-2064	308	56	,	,	PUNCT
ap-2064	308	57	n−1))a	n−1))a	PROPN
ap-2064	308	58	=	=	SYM
ap-2064	308	59	φ(f(a	φ(f(a	PROPN
ap-2064	308	60	,	,	PUNCT
ap-2064	308	61	b	b	PROPN
ap-2064	308	62	,	,	PUNCT
ap-2064	308	63	n−2))(φ(f(a	n−2))(φ(f(a	PROPN
ap-2064	308	64	,	,	PUNCT
ap-2064	308	65	b	b	NOUN
ap-2064	308	66	,	,	PUNCT
ap-2064	308	67	n−1)10)r)a	n−1)10)r)a	NOUN
ap-2064	308	68	=	=	SYM
ap-2064	308	69	φ(f(a	φ(f(a	PROPN
ap-2064	308	70	,	,	PUNCT
ap-2064	308	71	b	b	PROPN
ap-2064	308	72	,	,	PUNCT
ap-2064	308	73	n−2))(01φ(f(a	n−2))(01φ(f(a	PROPN
ap-2064	308	74	,	,	PUNCT
ap-2064	308	75	b	b	NOUN
ap-2064	308	76	,	,	PUNCT
ap-2064	308	77	n−1)))a	n−1)))a	X
ap-2064	308	78	=	=	SYM
ap-2064	308	79	(	(	PUNCT
ap-2064	308	80	φ(f(a	φ(f(a	PROPN
ap-2064	308	81	,	,	PUNCT
ap-2064	308	82	b	b	PROPN
ap-2064	308	83	,	,	PUNCT
ap-2064	308	84	n	n	CCONJ
ap-2064	308	85	)	)	PUNCT
ap-2064	308	86	)	)	PUNCT
ap-2064	308	87	)	)	PUNCT
ap-2064	308	88	.	.	PUNCT
ap-2064	309	1	if	if	SCONJ
ap-2064	309	2	n	n	NOUN
ap-2064	309	3	is	be	AUX
ap-2064	309	4	odd	odd	ADJ
ap-2064	309	5	,	,	PUNCT
ap-2064	309	6	the	the	DET
ap-2064	309	7	proof	proof	NOUN
ap-2064	309	8	is	be	AUX
ap-2064	309	9	analogous	analogous	ADJ
ap-2064	309	10	.	.	PUNCT
ap-2064	310	1	corollary	corollary	ADJ
ap-2064	310	2	12	12	NUM
ap-2064	310	3	.	.	PUNCT
ap-2064	311	1	(	(	PUNCT
ap-2064	311	2	1	1	NUM
ap-2064	311	3	.	.	PUNCT
ap-2064	311	4	)	)	PUNCT
ap-2064	312	1	if	if	SCONJ
ap-2064	312	2	f(a	f(a	PROPN
ap-2064	312	3	,	,	PUNCT
ap-2064	312	4	b	b	NOUN
ap-2064	312	5	,	,	PUNCT
ap-2064	312	6	n	n	CCONJ
ap-2064	312	7	)	)	PUNCT
ap-2064	312	8	=	=	SYM
ap-2064	312	9	φ(f(a	φ(f(a	PROPN
ap-2064	312	10	,	,	PUNCT
ap-2064	312	11	b	b	PROPN
ap-2064	312	12	,	,	PUNCT
ap-2064	312	13	n))xy	n))xy	PROPN
ap-2064	312	14	,	,	PUNCT
ap-2064	312	15	then	then	ADV
ap-2064	312	16	yxφ(f(a	yxφ(f(a	PROPN
ap-2064	312	17	,	,	PUNCT
ap-2064	312	18	b	b	PROPN
ap-2064	312	19	,	,	PUNCT
ap-2064	312	20	n))xy	n))xy	NOUN
ap-2064	312	21	is	be	AUX
ap-2064	312	22	a	a	DET
ap-2064	312	23	palindrome	palindrome	NOUN
ap-2064	312	24	.	.	PUNCT
ap-2064	313	1	(	(	PUNCT
ap-2064	313	2	2	2	NUM
ap-2064	313	3	.	.	PUNCT
ap-2064	313	4	)	)	PUNCT
ap-2064	314	1	if	if	SCONJ
ap-2064	314	2	u	u	NOUN
ap-2064	314	3	is	be	AUX
ap-2064	314	4	a	a	DET
ap-2064	314	5	subword	subword	NOUN
ap-2064	314	6	of	of	ADP
ap-2064	314	7	the	the	DET
ap-2064	314	8	biperiodic	biperiodic	ADJ
ap-2064	314	9	fibonacci	fibonacci	PROPN
ap-2064	314	10	word	word	NOUN
ap-2064	314	11	,	,	PUNCT
ap-2064	314	12	then	then	ADV
ap-2064	314	13	so	so	ADV
ap-2064	314	14	is	be	AUX
ap-2064	314	15	its	its	PRON
ap-2064	314	16	reversal	reversal	NOUN
ap-2064	314	17	,	,	PUNCT
ap-2064	314	18	ur	ur	INTJ
ap-2064	314	19	.	.	PUNCT
ap-2064	315	1	the	the	DET
ap-2064	315	2	above	above	ADJ
ap-2064	315	3	propositions	proposition	NOUN
ap-2064	315	4	are	be	AUX
ap-2064	315	5	particular	particular	ADJ
ap-2064	315	6	results	result	NOUN
ap-2064	315	7	related	relate	VERB
ap-2064	315	8	to	to	ADP
ap-2064	315	9	palindromes	palindrome	NOUN
ap-2064	315	10	of	of	ADP
ap-2064	315	11	sturmian	sturmian	NOUN
ap-2064	315	12	words	word	NOUN
ap-2064	315	13	,	,	PUNCT
ap-2064	315	14	see	see	VERB
ap-2064	315	15	,	,	PUNCT
ap-2064	315	16	e.g.	e.g.	ADV
ap-2064	315	17	,	,	PUNCT
ap-2064	315	18	[	[	X
ap-2064	315	19	9	9	NUM
ap-2064	315	20	]	]	PUNCT
ap-2064	315	21	.	.	PUNCT
ap-2064	316	1	theorem	theorem	NOUN
ap-2064	316	2	13	13	NUM
ap-2064	316	3	.	.	PUNCT
ap-2064	317	1	let	let	VERB
ap-2064	317	2	ζ	ζ	NOUN
ap-2064	317	3	=	=	PUNCT
ap-2064	318	1	[	[	X
ap-2064	318	2	0	0	NUM
ap-2064	318	3	,	,	PUNCT
ap-2064	318	4	a	a	DET
ap-2064	318	5	,	,	PUNCT
ap-2064	318	6	b	b	AUX
ap-2064	318	7	]	]	PUNCT
ap-2064	318	8	be	be	AUX
ap-2064	318	9	an	an	DET
ap-2064	318	10	irrational	irrational	ADJ
ap-2064	318	11	number	number	NOUN
ap-2064	318	12	,	,	PUNCT
ap-2064	318	13	with	with	ADP
ap-2064	318	14	a	a	DET
ap-2064	318	15	and	and	CCONJ
ap-2064	318	16	b	b	NOUN
ap-2064	318	17	positive	positive	ADJ
ap-2064	318	18	integers	integer	NOUN
ap-2064	318	19	,	,	PUNCT
ap-2064	318	20	then	then	ADV
ap-2064	318	21	w(ζ	w(ζ	PROPN
ap-2064	318	22	)	)	PUNCT
ap-2064	319	1	=	=	SYM
ap-2064	319	2	f(a	f(a	PROPN
ap-2064	319	3	,	,	PUNCT
ap-2064	319	4	b	b	NOUN
ap-2064	319	5	)	)	PUNCT
ap-2064	319	6	.	.	PUNCT
ap-2064	320	1	proof	proof	NOUN
ap-2064	320	2	.	.	PUNCT
ap-2064	321	1	let	let	VERB
ap-2064	321	2	ζ	ζ	NOUN
ap-2064	321	3	=	=	PUNCT
ap-2064	322	1	[	[	X
ap-2064	322	2	0	0	NUM
ap-2064	322	3	,	,	PUNCT
ap-2064	322	4	a	a	DET
ap-2064	322	5	,	,	PUNCT
ap-2064	322	6	b	b	AUX
ap-2064	322	7	]	]	PUNCT
ap-2064	322	8	be	be	AUX
ap-2064	322	9	an	an	DET
ap-2064	322	10	irrational	irrational	ADJ
ap-2064	322	11	number	number	NOUN
ap-2064	322	12	,	,	PUNCT
ap-2064	322	13	then	then	ADV
ap-2064	322	14	its	its	PRON
ap-2064	322	15	associated	associated	ADJ
ap-2064	322	16	standard	standard	ADJ
ap-2064	322	17	sequence	sequence	NOUN
ap-2064	322	18	is	be	AUX
ap-2064	322	19	s−1	s−1	PROPN
ap-2064	322	20	=	=	SYM
ap-2064	322	21	1	1	NUM
ap-2064	322	22	,	,	PUNCT
ap-2064	322	23	s0	s0	PROPN
ap-2064	322	24	=	=	SYM
ap-2064	322	25	0	0	NUM
ap-2064	322	26	,	,	PUNCT
ap-2064	322	27	s1	s1	NOUN
ap-2064	322	28	=	=	SYM
ap-2064	322	29	sa−1	sa−1	NOUN
ap-2064	322	30	0	0	PUNCT
ap-2064	323	1	s−1	s−1	PROPN
ap-2064	323	2	=	=	SYM
ap-2064	323	3	0a−11	0a−11	NOUN
ap-2064	323	4	,	,	PUNCT
ap-2064	323	5	and	and	CCONJ
ap-2064	323	6	sn	sn	PROPN
ap-2064	323	7	=	=	SYM
ap-2064	323	8	{	{	PUNCT
ap-2064	323	9	sbn−1sn−2	sbn−1sn−2	ADJ
ap-2064	323	10	,	,	PUNCT
ap-2064	323	11	if	if	SCONJ
ap-2064	323	12	n	n	PRON
ap-2064	323	13	≥	≥	NOUN
ap-2064	323	14	2	2	NUM
ap-2064	323	15	is	be	AUX
ap-2064	323	16	even	even	ADV
ap-2064	323	17	,	,	PUNCT
ap-2064	323	18	san−1sn−2	san−1sn−2	ADV
ap-2064	323	19	,	,	PUNCT
ap-2064	323	20	if	if	SCONJ
ap-2064	323	21	n	n	DET
ap-2064	323	22	≥	≥	NOUN
ap-2064	323	23	2	2	NUM
ap-2064	323	24	is	be	AUX
ap-2064	323	25	odd	odd	ADJ
ap-2064	323	26	.	.	PUNCT
ap-2064	324	1	hence	hence	ADV
ap-2064	324	2	{	{	PUNCT
ap-2064	324	3	sn}n≥0	sn}n≥0	X
ap-2064	324	4	=	=	SYM
ap-2064	324	5	{	{	PUNCT
ap-2064	324	6	f(a	f(a	NOUN
ap-2064	324	7	,	,	PUNCT
ap-2064	324	8	b	b	NOUN
ap-2064	324	9	,	,	PUNCT
ap-2064	324	10	n+1)}n≥0	n+1)}n≥0	PROPN
ap-2064	324	11	and	and	CCONJ
ap-2064	324	12	from	from	ADP
ap-2064	324	13	equation	equation	NOUN
ap-2064	324	14	(	(	PUNCT
ap-2064	324	15	2	2	NUM
ap-2064	324	16	)	)	PUNCT
ap-2064	324	17	,	,	PUNCT
ap-2064	324	18	we	we	PRON
ap-2064	324	19	have	have	VERB
ap-2064	324	20	w(ζ	w(ζ	PROPN
ap-2064	324	21	)	)	PUNCT
ap-2064	325	1	=	=	PROPN
ap-2064	325	2	lim	lim	PROPN
ap-2064	325	3	n→∞	n→∞	X
ap-2064	325	4	sn	sn	PROPN
ap-2064	325	5	=	=	SYM
ap-2064	325	6	f(a	f(a	PROPN
ap-2064	325	7	,	,	PUNCT
ap-2064	325	8	b	b	NOUN
ap-2064	325	9	)	)	PUNCT
ap-2064	325	10	.	.	PUNCT
ap-2064	326	1	remark	remark	PROPN
ap-2064	326	2	.	.	PUNCT
ap-2064	327	1	note	note	VERB
ap-2064	327	2	that	that	SCONJ
ap-2064	327	3	ζ	ζ	NOUN
ap-2064	327	4	=	=	PUNCT
ap-2064	328	1	[	[	X
ap-2064	328	2	0	0	NUM
ap-2064	328	3	,	,	PUNCT
ap-2064	328	4	a	a	DET
ap-2064	328	5	,	,	PUNCT
ap-2064	328	6	b	b	NOUN
ap-2064	328	7	]	]	X
ap-2064	328	8	=	=	SYM
ap-2064	328	9	1	1	NUM
ap-2064	328	10	a+	a+	SYM
ap-2064	328	11	1	1	NUM
ap-2064	328	12	b+	b+	ADP
ap-2064	328	13	1	1	NUM
ap-2064	328	14	a+	a+	SYM
ap-2064	328	15	1	1	NUM
ap-2064	328	16	·	·	PUNCT
ap-2064	328	17	·	·	PUNCT
ap-2064	328	18	·	·	PUNCT
ap-2064	328	19	=	=	PUNCT
ap-2064	328	20	−ab+	−ab+	NOUN
ap-2064	328	21	√	√	PROPN
ap-2064	328	22	(	(	PUNCT
ap-2064	328	23	ab)2	ab)2	PROPN
ap-2064	328	24	+	+	NOUN
ap-2064	328	25	4ab	4ab	ADJ
ap-2064	328	26	2a	2a	NUM
ap-2064	328	27	=	=	SYM
ap-2064	328	28	−α2	−α2	PROPN
ap-2064	328	29	.	.	PUNCT
ap-2064	329	1	from	from	ADP
ap-2064	329	2	the	the	DET
ap-2064	329	3	above	above	ADJ
ap-2064	329	4	theorem	theorem	NOUN
ap-2064	329	5	,	,	PUNCT
ap-2064	329	6	we	we	PRON
ap-2064	329	7	conclude	conclude	VERB
ap-2064	329	8	that	that	SCONJ
ap-2064	329	9	biperiodic	biperiodic	ADJ
ap-2064	329	10	fibonacci	fibonacci	NOUN
ap-2064	329	11	words	word	NOUN
ap-2064	329	12	are	be	AUX
ap-2064	329	13	sturmian	sturmian	NOUN
ap-2064	329	14	words	word	NOUN
ap-2064	329	15	.	.	PUNCT
ap-2064	330	1	a	a	DET
ap-2064	330	2	fractional	fractional	ADJ
ap-2064	330	3	power	power	NOUN
ap-2064	330	4	is	be	AUX
ap-2064	330	5	a	a	DET
ap-2064	330	6	word	word	NOUN
ap-2064	330	7	of	of	ADP
ap-2064	330	8	the	the	DET
ap-2064	330	9	form	form	NOUN
ap-2064	330	10	z	z	NOUN
ap-2064	330	11	=	=	SYM
ap-2064	330	12	xny	xny	PROPN
ap-2064	330	13	,	,	PUNCT
ap-2064	330	14	where	where	SCONJ
ap-2064	330	15	n	n	X
ap-2064	330	16	∈	∈	PROPN
ap-2064	330	17	z+	z+	NUM
ap-2064	330	18	,	,	PUNCT
ap-2064	330	19	x	x	SYM
ap-2064	330	20	∈	∈	NOUN
ap-2064	330	21	σ+	σ+	NOUN
ap-2064	330	22	and	and	CCONJ
ap-2064	330	23	y	y	PROPN
ap-2064	330	24	is	be	AUX
ap-2064	330	25	a	a	DET
ap-2064	330	26	prefix	prefix	NOUN
ap-2064	330	27	of	of	ADP
ap-2064	330	28	x.	x.	NOUN
ap-2064	330	29	if	if	SCONJ
ap-2064	330	30	|z|	|z|	VERB
ap-2064	330	31	=	=	SYM
ap-2064	330	32	p	p	NOUN
ap-2064	330	33	and	and	CCONJ
ap-2064	330	34	|x|	|x|	PROPN
ap-2064	330	35	=	=	SYM
ap-2064	331	1	q	q	NOUN
ap-2064	331	2	,	,	PUNCT
ap-2064	331	3	we	we	PRON
ap-2064	331	4	say	say	VERB
ap-2064	331	5	that	that	SCONJ
ap-2064	331	6	z	z	PROPN
ap-2064	331	7	is	be	AUX
ap-2064	331	8	a	a	DET
ap-2064	331	9	p	p	X
ap-2064	331	10	/	/	SYM
ap-2064	331	11	q	q	NOUN
ap-2064	331	12	-	-	PUNCT
ap-2064	331	13	power	power	NOUN
ap-2064	331	14	,	,	PUNCT
ap-2064	331	15	or	or	CCONJ
ap-2064	331	16	z	z	NOUN
ap-2064	331	17	=	=	SYM
ap-2064	331	18	xp	xp	PROPN
ap-2064	331	19	/	/	SYM
ap-2064	331	20	q.	q.	PROPN
ap-2064	331	21	in	in	ADP
ap-2064	331	22	the	the	DET
ap-2064	331	23	expression	expression	NOUN
ap-2064	331	24	xp	xp	PROPN
ap-2064	331	25	/	/	SYM
ap-2064	331	26	q	q	NOUN
ap-2064	331	27	,	,	PUNCT
ap-2064	331	28	the	the	DET
ap-2064	331	29	number	number	NOUN
ap-2064	331	30	p	p	NOUN
ap-2064	331	31	/	/	X
ap-2064	331	32	q	q	NOUN
ap-2064	331	33	is	be	AUX
ap-2064	331	34	the	the	DET
ap-2064	331	35	power	power	NOUN
ap-2064	331	36	’s	’s	PART
ap-2064	331	37	exponent	exponent	NOUN
ap-2064	331	38	.	.	PUNCT
ap-2064	332	1	for	for	ADP
ap-2064	332	2	example	example	NOUN
ap-2064	332	3	,	,	PUNCT
ap-2064	332	4	01201201	01201201	NUM
ap-2064	332	5	is	be	AUX
ap-2064	332	6	an	an	DET
ap-2064	332	7	8/3	8/3	NUM
ap-2064	332	8	-	-	PUNCT
ap-2064	332	9	power	power	NOUN
ap-2064	332	10	,	,	PUNCT
ap-2064	332	11	01201201	01201201	NUM
ap-2064	332	12	=	=	SYM
ap-2064	332	13	(	(	PUNCT
ap-2064	332	14	012)8/3	012)8/3	PROPN
ap-2064	332	15	.	.	PUNCT
ap-2064	333	1	the	the	DET
ap-2064	333	2	index	index	NOUN
ap-2064	333	3	of	of	ADP
ap-2064	333	4	an	an	DET
ap-2064	333	5	infinite	infinite	ADJ
ap-2064	333	6	word	word	NOUN
ap-2064	333	7	w	w	PROPN
ap-2064	333	8	∈	∈	PROPN
ap-2064	333	9	σω	σω	NOUN
ap-2064	333	10	is	be	AUX
ap-2064	333	11	defined	define	VERB
ap-2064	333	12	by	by	ADP
ap-2064	333	13	ind(w	ind(w	NOUN
ap-2064	333	14	)	)	PUNCT
ap-2064	333	15	:	:	PUNCT
ap-2064	334	1	=	=	SYM
ap-2064	334	2	sup{r	sup{r	NOUN
ap-2064	334	3	∈	∈	PROPN
ap-2064	334	4	q≥1	q≥1	NOUN
ap-2064	334	5	:	:	PUNCT
ap-2064	334	6	w	w	NOUN
ap-2064	334	7	contains	contain	VERB
ap-2064	334	8	an	an	DET
ap-2064	334	9	r	r	NOUN
ap-2064	334	10	-	-	PUNCT
ap-2064	334	11	power	power	NOUN
ap-2064	334	12	.	.	PUNCT
ap-2064	334	13	}	}	PUNCT
ap-2064	335	1	for	for	ADP
ap-2064	335	2	example	example	NOUN
ap-2064	335	3	,	,	PUNCT
ap-2064	335	4	ind(f	ind(f	PROPN
ap-2064	335	5	)	)	PUNCT
ap-2064	335	6	>	>	X
ap-2064	335	7	3	3	NUM
ap-2064	335	8	because	because	SCONJ
ap-2064	335	9	the	the	DET
ap-2064	335	10	cube	cube	NOUN
ap-2064	335	11	(	(	PUNCT
ap-2064	335	12	010)3	010)3	NUM
ap-2064	335	13	occurs	occur	VERB
ap-2064	335	14	in	in	ADP
ap-2064	335	15	f	f	PROPN
ap-2064	335	16	at	at	ADP
ap-2064	335	17	position	position	NOUN
ap-2064	335	18	6	6	NUM
ap-2064	335	19	.	.	PUNCT
ap-2064	335	20	mignosi	mignosi	NOUN
ap-2064	335	21	and	and	CCONJ
ap-2064	335	22	pirillo	pirillo	VERB
ap-2064	336	1	[	[	X
ap-2064	336	2	15	15	NUM
ap-2064	336	3	]	]	PUNCT
ap-2064	336	4	proved	prove	VERB
ap-2064	336	5	that	that	SCONJ
ap-2064	336	6	ind(f	ind(f	PROPN
ap-2064	336	7	)	)	PUNCT
ap-2064	336	8	=	=	SYM
ap-2064	336	9	2	2	NUM
ap-2064	336	10	+	+	NUM
ap-2064	336	11	φ	φ	PROPN
ap-2064	336	12	≈	≈	PROPN
ap-2064	336	13	3.618	3.618	NUM
ap-2064	336	14	,	,	PUNCT
ap-2064	336	15	where	where	SCONJ
ap-2064	336	16	φ	φ	PROPN
ap-2064	336	17	is	be	AUX
ap-2064	336	18	the	the	DET
ap-2064	336	19	golden	golden	ADJ
ap-2064	336	20	ratio	ratio	NOUN
ap-2064	336	21	.	.	PUNCT
ap-2064	337	1	ramírez	ramírez	PROPN
ap-2064	337	2	and	and	CCONJ
ap-2064	337	3	rubiano	rubiano	VERB
ap-2064	338	1	[	[	X
ap-2064	338	2	20	20	NUM
ap-2064	338	3	]	]	PUNCT
ap-2064	338	4	proved	prove	VERB
ap-2064	338	5	that	that	SCONJ
ap-2064	338	6	the	the	DET
ap-2064	338	7	index	index	NOUN
ap-2064	338	8	of	of	ADP
ap-2064	338	9	the	the	DET
ap-2064	338	10	k	k	PROPN
ap-2064	338	11	-	-	PUNCT
ap-2064	338	12	fibonacci	fibonacci	NOUN
ap-2064	338	13	word	word	NOUN
ap-2064	338	14	is	be	AUX
ap-2064	338	15	given	give	VERB
ap-2064	338	16	by	by	ADP
ap-2064	338	17	ind(fk	ind(fk	NOUN
ap-2064	338	18	)	)	PUNCT
ap-2064	338	19	=	=	SYM
ap-2064	338	20	2+k+1	2+k+1	NUM
ap-2064	338	21	/	/	SYM
ap-2064	338	22	rk,1	rk,1	NOUN
ap-2064	338	23	,	,	PUNCT
ap-2064	338	24	where	where	SCONJ
ap-2064	338	25	rk,1	rk,1	NOUN
ap-2064	338	26	=	=	SYM
ap-2064	338	27	(	(	PUNCT
ap-2064	338	28	k+	k+	NOUN
ap-2064	338	29	√	√	PROPN
ap-2064	338	30	k2	k2	NOUN
ap-2064	338	31	+	+	CCONJ
ap-2064	338	32	4)/2	4)/2	PROPN
ap-2064	338	33	.	.	PUNCT
ap-2064	339	1	a	a	DET
ap-2064	339	2	general	general	ADJ
ap-2064	339	3	formula	formula	NOUN
ap-2064	339	4	for	for	ADP
ap-2064	339	5	the	the	DET
ap-2064	339	6	index	index	NOUN
ap-2064	339	7	of	of	ADP
ap-2064	339	8	a	a	DET
ap-2064	339	9	sturmian	sturmian	NOUN
ap-2064	339	10	word	word	NOUN
ap-2064	339	11	was	be	AUX
ap-2064	339	12	given	give	VERB
ap-2064	339	13	by	by	ADP
ap-2064	339	14	damanik	damanik	NOUN
ap-2064	339	15	and	and	CCONJ
ap-2064	339	16	lenz	lenz	PROPN
ap-2064	340	1	[	[	X
ap-2064	340	2	11	11	NUM
ap-2064	340	3	]	]	PUNCT
ap-2064	340	4	.	.	PUNCT
ap-2064	341	1	theorem	theorem	VERB
ap-2064	341	2	14	14	NUM
ap-2064	342	1	[	[	X
ap-2064	342	2	11	11	NUM
ap-2064	342	3	]	]	PUNCT
ap-2064	342	4	.	.	PUNCT
ap-2064	343	1	if	if	SCONJ
ap-2064	343	2	w	w	NOUN
ap-2064	343	3	is	be	AUX
ap-2064	343	4	a	a	DET
ap-2064	343	5	sturmian	sturmian	ADJ
ap-2064	343	6	word	word	NOUN
ap-2064	343	7	of	of	ADP
ap-2064	343	8	slope	slope	NOUN
ap-2064	343	9	θ	θ	PROPN
ap-2064	343	10	=	=	PUNCT
ap-2064	344	1	[	[	X
ap-2064	344	2	0	0	NUM
ap-2064	344	3	,	,	PUNCT
ap-2064	344	4	a1	a1	NOUN
ap-2064	344	5	,	,	PUNCT
ap-2064	344	6	a2	a2	PROPN
ap-2064	344	7	,	,	PUNCT
ap-2064	344	8	a3	a3	NOUN
ap-2064	344	9	,	,	PUNCT
ap-2064	344	10	.	.	PUNCT
ap-2064	344	11	.	.	PUNCT
ap-2064	344	12	.	.	PUNCT
ap-2064	345	1	]	]	X
ap-2064	345	2	,	,	PUNCT
ap-2064	345	3	then	then	ADV
ap-2064	345	4	ind(w	ind(w	NUM
ap-2064	345	5	)	)	PUNCT
ap-2064	345	6	=	=	PUNCT
ap-2064	345	7	sup	sup	NUM
ap-2064	345	8	n≥0	n≥0	PROPN
ap-2064	345	9	{	{	PUNCT
ap-2064	345	10	2	2	NUM
ap-2064	345	11	+	+	NUM
ap-2064	345	12	an+1	an+1	NOUN
ap-2064	345	13	+	+	CCONJ
ap-2064	345	14	rn−1	rn−1	PROPN
ap-2064	345	15	−	−	PROPN
ap-2064	345	16	2	2	NUM
ap-2064	345	17	rn	rn	PROPN
ap-2064	345	18	}	}	PUNCT
ap-2064	345	19	,	,	PUNCT
ap-2064	345	20	where	where	SCONJ
ap-2064	345	21	rn	rn	PROPN
ap-2064	345	22	is	be	AUX
ap-2064	345	23	the	the	DET
ap-2064	345	24	denominator	denominator	NOUN
ap-2064	345	25	of	of	ADP
ap-2064	345	26	θ	θ	PROPN
ap-2064	345	27	=	=	PUNCT
ap-2064	346	1	[	[	X
ap-2064	346	2	0	0	NUM
ap-2064	346	3	,	,	PUNCT
ap-2064	346	4	a1	a1	NOUN
ap-2064	346	5	,	,	PUNCT
ap-2064	346	6	a2	a2	PROPN
ap-2064	346	7	,	,	PUNCT
ap-2064	346	8	a3	a3	NOUN
ap-2064	346	9	,	,	PUNCT
ap-2064	346	10	.	.	PUNCT
ap-2064	346	11	.	.	PUNCT
ap-2064	347	1	.	.	PUNCT
ap-2064	348	1	,	,	PUNCT
ap-2064	348	2	an	an	DET
ap-2064	348	3	]	]	X
ap-2064	348	4	and	and	CCONJ
ap-2064	348	5	satisfies	satisfy	VERB
ap-2064	348	6	r−1	r−1	PROPN
ap-2064	348	7	=	=	SYM
ap-2064	348	8	0	0	NUM
ap-2064	348	9	,	,	PUNCT
ap-2064	348	10	r0	r0	NOUN
ap-2064	348	11	=	=	SYM
ap-2064	348	12	1	1	NUM
ap-2064	348	13	,	,	PUNCT
ap-2064	348	14	rn+1	rn+1	X
ap-2064	348	15	=	=	SYM
ap-2064	348	16	an+1rn	an+1rn	PROPN
ap-2064	348	17	+	+	X
ap-2064	348	18	rn−1	rn−1	NOUN
ap-2064	348	19	.	.	PUNCT
ap-2064	349	1	corollary	corollary	ADJ
ap-2064	349	2	15	15	NUM
ap-2064	349	3	.	.	PUNCT
ap-2064	350	1	the	the	DET
ap-2064	350	2	index	index	NOUN
ap-2064	350	3	of	of	ADP
ap-2064	350	4	the	the	DET
ap-2064	350	5	biperiodic	biperiodic	ADJ
ap-2064	350	6	fibonacci	fibonacci	NOUN
ap-2064	350	7	word	word	NOUN
ap-2064	350	8	is	be	AUX
ap-2064	350	9	ind(f(a	ind(f(a	ADJ
ap-2064	350	10	,	,	PUNCT
ap-2064	350	11	b	b	NOUN
ap-2064	350	12	)	)	PUNCT
ap-2064	350	13	)	)	PUNCT
ap-2064	351	1	=	=	SYM
ap-2064	351	2	max	max	PROPN
ap-2064	351	3	{	{	PUNCT
ap-2064	351	4	2	2	NUM
ap-2064	351	5	+	+	CCONJ
ap-2064	351	6	a+	a+	PUNCT
ap-2064	351	7	a	a	DET
ap-2064	351	8	α	α	NOUN
ap-2064	351	9	,	,	PUNCT
ap-2064	351	10	2	2	NUM
ap-2064	351	11	+	+	CCONJ
ap-2064	351	12	b+	b+	X
ap-2064	351	13	b	b	PROPN
ap-2064	351	14	α	α	PROPN
ap-2064	351	15	}	}	PUNCT
ap-2064	351	16	,	,	PUNCT
ap-2064	351	17	(	(	PUNCT
ap-2064	351	18	7	7	X
ap-2064	351	19	)	)	PUNCT
ap-2064	351	20	where	where	SCONJ
ap-2064	351	21	α	α	NOUN
ap-2064	351	22	=	=	X
ap-2064	351	23	(	(	PUNCT
ap-2064	351	24	ab+	ab+	PROPN
ap-2064	351	25	√	√	PROPN
ap-2064	351	26	(	(	PUNCT
ap-2064	351	27	ab)2	ab)2	PROPN
ap-2064	351	28	+	+	PROPN
ap-2064	351	29	4ab)/2	4ab)/2	NOUN
ap-2064	351	30	.	.	PUNCT
ap-2064	352	1	proof	proof	NOUN
ap-2064	352	2	.	.	PUNCT
ap-2064	353	1	the	the	DET
ap-2064	353	2	word	word	NOUN
ap-2064	353	3	f(a	f(a	PROPN
ap-2064	353	4	,	,	PUNCT
ap-2064	353	5	b	b	NOUN
ap-2064	353	6	)	)	PUNCT
ap-2064	353	7	is	be	AUX
ap-2064	353	8	a	a	DET
ap-2064	353	9	sturmian	sturmian	ADJ
ap-2064	353	10	word	word	NOUN
ap-2064	353	11	of	of	ADP
ap-2064	353	12	slope	slope	NOUN
ap-2064	353	13	ζ	ζ	NOUN
ap-2064	353	14	=	=	SYM
ap-2064	354	1	[	[	X
ap-2064	354	2	0	0	NUM
ap-2064	354	3	,	,	PUNCT
ap-2064	354	4	a	a	DET
ap-2064	354	5	,	,	PUNCT
ap-2064	354	6	b	b	NOUN
ap-2064	354	7	]	]	X
ap-2064	354	8	,	,	PUNCT
ap-2064	354	9	then	then	ADV
ap-2064	354	10	from	from	ADP
ap-2064	354	11	the	the	DET
ap-2064	354	12	above	above	ADJ
ap-2064	354	13	theorem	theorem	PROPN
ap-2064	354	14	rn	rn	PROPN
ap-2064	354	15	=	=	PROPN
ap-2064	354	16	qn+1	qn+1	PROPN
ap-2064	354	17	,	,	PUNCT
ap-2064	354	18	and	and	CCONJ
ap-2064	354	19	ind(f(a	ind(f(a	NOUN
ap-2064	354	20	,	,	PUNCT
ap-2064	354	21	b	b	NOUN
ap-2064	354	22	)	)	PUNCT
ap-2064	354	23	)	)	PUNCT
ap-2064	355	1	=	=	SYM
ap-2064	355	2	supn≥0{hn	supn≥0{hn	NOUN
ap-2064	355	3	}	}	PUNCT
ap-2064	355	4	,	,	PUNCT
ap-2064	355	5	where	where	SCONJ
ap-2064	355	6	hn	hn	PROPN
ap-2064	355	7	=	=	X
ap-2064	355	8	{	{	PUNCT
ap-2064	355	9	2	2	NUM
ap-2064	355	10	+	+	CCONJ
ap-2064	355	11	b+	b+	ADP
ap-2064	355	12	qn−1−2	qn−1−2	PROPN
ap-2064	355	13	qn	qn	INTJ
ap-2064	355	14	,	,	PUNCT
ap-2064	355	15	if	if	SCONJ
ap-2064	355	16	n	n	PRON
ap-2064	355	17	is	be	AUX
ap-2064	355	18	even	even	ADV
ap-2064	355	19	,	,	PUNCT
ap-2064	355	20	2	2	NUM
ap-2064	355	21	+	+	CCONJ
ap-2064	355	22	a+	a+	PUNCT
ap-2064	355	23	qn−1−2	qn−1−2	NOUN
ap-2064	355	24	qn	qn	INTJ
ap-2064	355	25	,	,	PUNCT
ap-2064	355	26	if	if	SCONJ
ap-2064	355	27	n	n	PRON
ap-2064	355	28	is	be	AUX
ap-2064	355	29	odd	odd	ADJ
ap-2064	355	30	.	.	PUNCT
ap-2064	356	1	then	then	ADV
ap-2064	356	2	ind(f(a	ind(f(a	NOUN
ap-2064	356	3	,	,	PUNCT
ap-2064	356	4	b	b	NOUN
ap-2064	356	5	)	)	PUNCT
ap-2064	356	6	)	)	PUNCT
ap-2064	357	1	=	=	SYM
ap-2064	357	2	max	max	PROPN
ap-2064	357	3	{	{	PUNCT
ap-2064	357	4	sup	sup	PROPN
ap-2064	357	5	n≥0	n≥0	PROPN
ap-2064	357	6	{	{	PUNCT
ap-2064	357	7	2	2	NUM
ap-2064	357	8	+	+	CCONJ
ap-2064	357	9	a+	a+	PUNCT
ap-2064	357	10	q2n−2	q2n−2	PROPN
ap-2064	357	11	q2n+1	q2n+1	PROPN
ap-2064	357	12	}	}	PUNCT
ap-2064	357	13	,	,	PUNCT
ap-2064	357	14	sup	sup	NOUN
ap-2064	357	15	n≥0	n≥0	PROPN
ap-2064	357	16	{	{	PUNCT
ap-2064	357	17	2	2	NUM
ap-2064	357	18	+	+	NUM
ap-2064	357	19	b+	b+	X
ap-2064	357	20	q2n−1−2	q2n−1−2	PROPN
ap-2064	357	21	q2n	q2n	CCONJ
ap-2064	357	22	}	}	PUNCT
ap-2064	357	23	}	}	PUNCT
ap-2064	357	24	.	.	PUNCT
ap-2064	358	1	since	since	SCONJ
ap-2064	358	2	q2n+1	q2n+1	PROPN
ap-2064	358	3	/	/	SYM
ap-2064	358	4	q2n	q2n	PROPN
ap-2064	358	5	→	→	SYM
ap-2064	358	6	α	α	X
ap-2064	358	7	/	/	SYM
ap-2064	358	8	a	a	PRON
ap-2064	358	9	and	and	CCONJ
ap-2064	358	10	q2n	q2n	CCONJ
ap-2064	358	11	/	/	SYM
ap-2064	358	12	q2n−1	q2n−1	PROPN
ap-2064	358	13	→	→	SYM
ap-2064	358	14	α	α	X
ap-2064	358	15	/	/	SYM
ap-2064	358	16	b	b	PROPN
ap-2064	358	17	as	as	ADP
ap-2064	358	18	n→	n→	PROPN
ap-2064	358	19	∞	∞	PROPN
ap-2064	358	20	,	,	PUNCT
ap-2064	358	21	then	then	ADV
ap-2064	358	22	equation	equation	NOUN
ap-2064	358	23	(	(	PUNCT
ap-2064	358	24	7	7	X
ap-2064	358	25	)	)	PUNCT
ap-2064	358	26	follows	follow	VERB
ap-2064	358	27	.	.	PUNCT
ap-2064	359	1	55	55	NUM
ap-2064	359	2	josé	josé	PROPN
ap-2064	359	3	l.	l.	PROPN
ap-2064	359	4	ramírez	ramírez	PROPN
ap-2064	359	5	,	,	PUNCT
ap-2064	359	6	gustavo	gustavo	PROPN
ap-2064	359	7	n.	n.	PROPN
ap-2064	359	8	rubiano	rubiano	PROPN
ap-2064	359	9	acta	acta	PROPN
ap-2064	359	10	polytechnica	polytechnica	PROPN
ap-2064	359	11	f	f	PROPN
ap-2064	359	12	(	(	PUNCT
ap-2064	359	13	2,5	2,5	NUM
ap-2064	359	14	)	)	PUNCT
ap-2064	359	15	9	9	NUM
ap-2064	359	16	,	,	PUNCT
ap-2064	359	17	θ	θ	PROPN
ap-2064	359	18	=	=	SYM
ap-2064	359	19	90	90	NUM
ap-2064	359	20	◦	◦	NOUN
ap-2064	359	21	f	f	X
ap-2064	359	22	(	(	PUNCT
ap-2064	359	23	5,5	5,5	NUM
ap-2064	359	24	)	)	PUNCT
ap-2064	359	25	7	7	NUM
ap-2064	359	26	,	,	PUNCT
ap-2064	359	27	θ	θ	NOUN
ap-2064	359	28	=	=	SYM
ap-2064	359	29	90	90	NUM
ap-2064	359	30	◦	◦	NOUN
ap-2064	359	31	f	f	X
ap-2064	359	32	(	(	PUNCT
ap-2064	359	33	6,6	6,6	NUM
ap-2064	359	34	)	)	PUNCT
ap-2064	359	35	6	6	NUM
ap-2064	359	36	,	,	PUNCT
ap-2064	359	37	θ	θ	PROPN
ap-2064	359	38	=	=	SYM
ap-2064	359	39	90	90	NUM
ap-2064	359	40	◦	◦	NOUN
ap-2064	359	41	f	f	X
ap-2064	359	42	(	(	PUNCT
ap-2064	359	43	2,5	2,5	NUM
ap-2064	359	44	)	)	PUNCT
ap-2064	359	45	10	10	NUM
ap-2064	359	46	,	,	PUNCT
ap-2064	359	47	θ	θ	PROPN
ap-2064	359	48	=	=	SYM
ap-2064	359	49	90	90	NUM
ap-2064	359	50	◦	◦	NOUN
ap-2064	359	51	f	f	X
ap-2064	359	52	(	(	PUNCT
ap-2064	359	53	2,3	2,3	NUM
ap-2064	359	54	)	)	PUNCT
ap-2064	359	55	10	10	NUM
ap-2064	359	56	,	,	PUNCT
ap-2064	359	57	θ	θ	X
ap-2064	359	58	=	=	SYM
ap-2064	359	59	60	60	NUM
ap-2064	359	60	◦	◦	NOUN
ap-2064	359	61	f	f	X
ap-2064	359	62	(	(	PUNCT
ap-2064	359	63	6,6	6,6	NUM
ap-2064	359	64	)	)	PUNCT
ap-2064	359	65	5	5	NUM
ap-2064	359	66	,	,	PUNCT
ap-2064	359	67	θ	θ	PROPN
ap-2064	359	68	=	=	SYM
ap-2064	359	69	60	60	NUM
ap-2064	359	70	◦	◦	NOUN
ap-2064	359	71	table	table	NOUN
ap-2064	359	72	5	5	NUM
ap-2064	359	73	.	.	PUNCT
ap-2064	360	1	some	some	DET
ap-2064	360	2	curves	curve	NOUN
ap-2064	360	3	f	f	X
ap-2064	360	4	(	(	PUNCT
ap-2064	360	5	a	a	DET
ap-2064	360	6	,	,	PUNCT
ap-2064	360	7	b	b	NOUN
ap-2064	360	8	)	)	PUNCT
ap-2064	360	9	n	n	NOUN
ap-2064	360	10	with	with	ADP
ap-2064	360	11	an	an	DET
ap-2064	360	12	angle	angle	NOUN
ap-2064	360	13	θ	θ	PROPN
ap-2064	360	14	.	.	PROPN
ap-2064	360	15	4	4	NUM
ap-2064	360	16	.	.	PUNCT
ap-2064	361	1	the	the	DET
ap-2064	361	2	biperiodic	biperiodic	ADJ
ap-2064	361	3	fibonacci	fibonacci	PROPN
ap-2064	361	4	word	word	NOUN
ap-2064	361	5	curve	curve	VERB
ap-2064	361	6	the	the	DET
ap-2064	361	7	odd	odd	ADV
ap-2064	361	8	-	-	PUNCT
ap-2064	361	9	even	even	ADV
ap-2064	361	10	drawing	draw	VERB
ap-2064	361	11	rule	rule	NOUN
ap-2064	361	12	can	can	AUX
ap-2064	361	13	be	be	AUX
ap-2064	361	14	extended	extend	VERB
ap-2064	361	15	from	from	ADP
ap-2064	361	16	a	a	DET
ap-2064	361	17	parameter	parameter	NOUN
ap-2064	361	18	θ	θ	NOUN
ap-2064	361	19	,	,	PUNCT
ap-2064	361	20	where	where	SCONJ
ap-2064	361	21	θ	θ	PROPN
ap-2064	361	22	is	be	AUX
ap-2064	361	23	the	the	DET
ap-2064	361	24	turn	turn	NOUN
ap-2064	361	25	angle	angle	NOUN
ap-2064	361	26	,	,	PUNCT
ap-2064	361	27	see	see	VERB
ap-2064	361	28	table	table	NOUN
ap-2064	361	29	4	4	NUM
ap-2064	361	30	.	.	PUNCT
ap-2064	362	1	if	if	SCONJ
ap-2064	362	2	θ	θ	PROPN
ap-2064	362	3	=	=	SYM
ap-2064	362	4	90	90	NUM
ap-2064	362	5	◦	◦	NOUN
ap-2064	362	6	,	,	PUNCT
ap-2064	362	7	then	then	ADV
ap-2064	362	8	we	we	PRON
ap-2064	362	9	obtain	obtain	VERB
ap-2064	362	10	the	the	DET
ap-2064	362	11	drawing	drawing	NOUN
ap-2064	362	12	rule	rule	NOUN
ap-2064	362	13	in	in	ADP
ap-2064	362	14	table	table	NOUN
ap-2064	362	15	1	1	NUM
ap-2064	362	16	.	.	PUNCT
ap-2064	363	1	definition	definition	NOUN
ap-2064	363	2	16	16	NUM
ap-2064	363	3	.	.	PUNCT
ap-2064	364	1	the	the	DET
ap-2064	364	2	nth	nth	NOUN
ap-2064	364	3	-	-	PUNCT
ap-2064	364	4	biperiodic	biperiodic	ADJ
ap-2064	364	5	curve	curve	NOUN
ap-2064	364	6	of	of	ADP
ap-2064	364	7	fibonacci	fibonacci	NOUN
ap-2064	364	8	,	,	PUNCT
ap-2064	364	9	denoted	denote	VERB
ap-2064	364	10	by	by	ADP
ap-2064	364	11	f	f	PROPN
ap-2064	364	12	(	(	PUNCT
ap-2064	364	13	a	a	DET
ap-2064	364	14	,	,	PUNCT
ap-2064	364	15	b	b	NOUN
ap-2064	364	16	)	)	PUNCT
ap-2064	364	17	n	n	CCONJ
ap-2064	364	18	,	,	PUNCT
ap-2064	364	19	is	be	AUX
ap-2064	364	20	obtained	obtain	VERB
ap-2064	364	21	by	by	ADP
ap-2064	364	22	applying	apply	VERB
ap-2064	364	23	the	the	DET
ap-2064	364	24	odd	odd	ADV
ap-2064	364	25	-	-	PUNCT
ap-2064	364	26	even	even	ADV
ap-2064	364	27	drawing	draw	VERB
ap-2064	364	28	rule	rule	NOUN
ap-2064	364	29	to	to	ADP
ap-2064	364	30	the	the	DET
ap-2064	364	31	word	word	NOUN
ap-2064	364	32	f(a	f(a	PROPN
ap-2064	364	33	,	,	PUNCT
ap-2064	364	34	b	b	NOUN
ap-2064	364	35	,	,	PUNCT
ap-2064	364	36	n	n	CCONJ
ap-2064	364	37	)	)	PUNCT
ap-2064	364	38	.	.	PUNCT
ap-2064	365	1	the	the	DET
ap-2064	365	2	biperiodic	biperiodic	ADJ
ap-2064	365	3	fibonacci	fibonacci	PROPN
ap-2064	365	4	word	word	PROPN
ap-2064	365	5	fractal	fractal	PROPN
ap-2064	365	6	f	f	PROPN
ap-2064	365	7	(	(	PUNCT
ap-2064	365	8	a	a	DET
ap-2064	365	9	,	,	PUNCT
ap-2064	365	10	b	b	NOUN
ap-2064	365	11	)	)	PUNCT
ap-2064	365	12	,	,	PUNCT
ap-2064	365	13	is	be	AUX
ap-2064	365	14	defined	define	VERB
ap-2064	365	15	as	as	ADP
ap-2064	365	16	f	f	PROPN
ap-2064	365	17	(	(	PUNCT
ap-2064	365	18	a	a	DET
ap-2064	365	19	,	,	PUNCT
ap-2064	365	20	b	b	NOUN
ap-2064	365	21	)	)	PUNCT
ap-2064	365	22	:	:	PUNCT
ap-2064	366	1	=	=	SYM
ap-2064	366	2	limn→∞	limn→∞	SYM
ap-2064	366	3	f	f	X
ap-2064	366	4	(	(	PUNCT
ap-2064	366	5	a	a	DET
ap-2064	366	6	,	,	PUNCT
ap-2064	366	7	b	b	NOUN
ap-2064	366	8	)	)	PUNCT
ap-2064	366	9	n	n	NOUN
ap-2064	366	10	.	.	PUNCT
ap-2064	367	1	in	in	ADP
ap-2064	367	2	table	table	NOUN
ap-2064	367	3	5	5	NUM
ap-2064	367	4	,	,	PUNCT
ap-2064	367	5	we	we	PRON
ap-2064	367	6	show	show	VERB
ap-2064	367	7	some	some	DET
ap-2064	367	8	curves	curve	NOUN
ap-2064	367	9	f	f	NOUN
ap-2064	367	10	(	(	PUNCT
ap-2064	367	11	a	a	DET
ap-2064	367	12	,	,	PUNCT
ap-2064	367	13	b	b	NOUN
ap-2064	367	14	)	)	PUNCT
ap-2064	367	15	n	n	NOUN
ap-2064	367	16	with	with	ADP
ap-2064	367	17	a	a	DET
ap-2064	367	18	given	give	VERB
ap-2064	367	19	angle	angle	NOUN
ap-2064	367	20	θ	θ	PROPN
ap-2064	367	21	.	.	PUNCT
ap-2064	367	22	proposition	proposition	NOUN
ap-2064	367	23	17	17	NUM
ap-2064	367	24	.	.	PUNCT
ap-2064	368	1	the	the	DET
ap-2064	368	2	biperiodic	biperiodic	ADJ
ap-2064	368	3	fibonacci	fibonacci	PROPN
ap-2064	368	4	word	word	NOUN
ap-2064	368	5	curve	curve	NOUN
ap-2064	368	6	and	and	CCONJ
ap-2064	368	7	the	the	DET
ap-2064	368	8	curve	curve	NOUN
ap-2064	368	9	f	f	PROPN
ap-2064	368	10	(	(	PUNCT
ap-2064	368	11	a	a	DET
ap-2064	368	12	,	,	PUNCT
ap-2064	368	13	b	b	NOUN
ap-2064	368	14	)	)	PUNCT
ap-2064	368	15	n	n	VERB
ap-2064	368	16	have	have	VERB
ap-2064	368	17	the	the	DET
ap-2064	368	18	following	follow	VERB
ap-2064	368	19	properties	property	NOUN
ap-2064	368	20	:	:	PUNCT
ap-2064	368	21	(	(	PUNCT
ap-2064	368	22	1	1	NUM
ap-2064	368	23	.	.	PUNCT
ap-2064	368	24	)	)	PUNCT
ap-2064	369	1	the	the	DET
ap-2064	369	2	biperiodic	biperiodic	PROPN
ap-2064	369	3	fibonacci	fibonacci	PROPN
ap-2064	369	4	curve	curve	PROPN
ap-2064	369	5	f	f	PROPN
ap-2064	369	6	(	(	PUNCT
ap-2064	369	7	a	a	DET
ap-2064	369	8	,	,	PUNCT
ap-2064	369	9	b	b	NOUN
ap-2064	369	10	)	)	PUNCT
ap-2064	369	11	is	be	AUX
ap-2064	369	12	composed	compose	VERB
ap-2064	369	13	only	only	ADV
ap-2064	369	14	of	of	ADP
ap-2064	369	15	segments	segment	NOUN
ap-2064	369	16	of	of	ADP
ap-2064	369	17	length	length	NOUN
ap-2064	369	18	1	1	NUM
ap-2064	369	19	or	or	CCONJ
ap-2064	369	20	2	2	NUM
ap-2064	369	21	.	.	PUNCT
ap-2064	370	1	(	(	PUNCT
ap-2064	370	2	2	2	NUM
ap-2064	370	3	.	.	PUNCT
ap-2064	370	4	)	)	PUNCT
ap-2064	371	1	the	the	DET
ap-2064	371	2	curve	curve	NOUN
ap-2064	371	3	f	f	PROPN
ap-2064	371	4	(	(	PUNCT
ap-2064	371	5	a	a	DET
ap-2064	371	6	,	,	PUNCT
ap-2064	371	7	b	b	NOUN
ap-2064	371	8	)	)	PUNCT
ap-2064	371	9	n	n	PRON
ap-2064	371	10	is	be	AUX
ap-2064	371	11	symmetric	symmetric	ADJ
ap-2064	371	12	.	.	PUNCT
ap-2064	372	1	(	(	PUNCT
ap-2064	372	2	3	3	NUM
ap-2064	372	3	.	.	PUNCT
ap-2064	372	4	)	)	PUNCT
ap-2064	373	1	the	the	DET
ap-2064	373	2	number	number	NOUN
ap-2064	373	3	of	of	ADP
ap-2064	373	4	turns	turn	NOUN
ap-2064	373	5	in	in	ADP
ap-2064	373	6	the	the	DET
ap-2064	373	7	curve	curve	NOUN
ap-2064	373	8	f	f	PROPN
ap-2064	373	9	(	(	PUNCT
ap-2064	373	10	a	a	DET
ap-2064	373	11	,	,	PUNCT
ap-2064	373	12	b	b	NOUN
ap-2064	373	13	)	)	PUNCT
ap-2064	373	14	n	n	PRON
ap-2064	373	15	is	be	AUX
ap-2064	373	16	hn	hn	PROPN
ap-2064	373	17	.	.	PUNCT
ap-2064	373	18	proof	proof	NOUN
ap-2064	373	19	.	.	PUNCT
ap-2064	374	1	(	(	PUNCT
ap-2064	374	2	1	1	NUM
ap-2064	374	3	.	.	PUNCT
ap-2064	374	4	)	)	PUNCT
ap-2064	375	1	it	it	PRON
ap-2064	375	2	is	be	AUX
ap-2064	375	3	clear	clear	ADJ
ap-2064	375	4	from	from	ADP
ap-2064	375	5	proposition	proposition	NOUN
ap-2064	375	6	8.(1	8.(1	NUM
ap-2064	375	7	.	.	PUNCT
ap-2064	375	8	)	)	PUNCT
ap-2064	375	9	,	,	PUNCT
ap-2064	375	10	because	because	SCONJ
ap-2064	375	11	110	110	NUM
ap-2064	375	12	and	and	CCONJ
ap-2064	375	13	111	111	NUM
ap-2064	375	14	are	be	AUX
ap-2064	375	15	not	not	PART
ap-2064	375	16	subwords	subword	NOUN
ap-2064	375	17	of	of	ADP
ap-2064	375	18	f	f	PROPN
ap-2064	375	19	(	(	PUNCT
ap-2064	375	20	a	a	DET
ap-2064	375	21	,	,	PUNCT
ap-2064	375	22	b	b	NOUN
ap-2064	375	23	)	)	PUNCT
ap-2064	375	24	n	n	NOUN
ap-2064	375	25	.	.	PUNCT
ap-2064	376	1	(	(	PUNCT
ap-2064	376	2	2	2	NUM
ap-2064	376	3	.	.	PUNCT
ap-2064	376	4	)	)	PUNCT
ap-2064	377	1	it	it	PRON
ap-2064	377	2	is	be	AUX
ap-2064	377	3	clear	clear	ADJ
ap-2064	377	4	from	from	ADP
ap-2064	377	5	theorem	theorem	ADJ
ap-2064	377	6	11	11	NUM
ap-2064	377	7	,	,	PUNCT
ap-2064	377	8	because	because	SCONJ
ap-2064	377	9	f(a	f(a	PROPN
ap-2064	377	10	,	,	PUNCT
ap-2064	377	11	b	b	NOUN
ap-2064	377	12	,	,	PUNCT
ap-2064	377	13	n	n	CCONJ
ap-2064	377	14	)	)	PUNCT
ap-2064	377	15	=	=	SYM
ap-2064	377	16	φ(f(a	φ(f(a	PROPN
ap-2064	377	17	,	,	PUNCT
ap-2064	377	18	b	b	PROPN
ap-2064	377	19	,	,	PUNCT
ap-2064	377	20	n))xy	n))xy	PROPN
ap-2064	377	21	,	,	PUNCT
ap-2064	377	22	where	where	SCONJ
ap-2064	377	23	φ(f(a	φ(f(a	PROPN
ap-2064	377	24	,	,	PUNCT
ap-2064	377	25	b	b	PROPN
ap-2064	377	26	,	,	PUNCT
ap-2064	377	27	n	n	CCONJ
ap-2064	377	28	)	)	PUNCT
ap-2064	377	29	)	)	PUNCT
ap-2064	377	30	is	be	AUX
ap-2064	377	31	a	a	DET
ap-2064	377	32	palindrome	palindrome	NOUN
ap-2064	377	33	.	.	PUNCT
ap-2064	378	1	(	(	PUNCT
ap-2064	378	2	3	3	NUM
ap-2064	378	3	.	.	PUNCT
ap-2064	378	4	)	)	PUNCT
ap-2064	379	1	it	it	PRON
ap-2064	379	2	is	be	AUX
ap-2064	379	3	clear	clear	ADJ
ap-2064	379	4	from	from	ADP
ap-2064	379	5	the	the	DET
ap-2064	379	6	definition	definition	NOUN
ap-2064	379	7	of	of	ADP
ap-2064	379	8	the	the	DET
ap-2064	379	9	odd	odd	ADV
ap-2064	379	10	-	-	PUNCT
ap-2064	379	11	even	even	ADV
ap-2064	379	12	drawn	draw	VERB
ap-2064	379	13	rule	rule	NOUN
ap-2064	379	14	and	and	CCONJ
ap-2064	379	15	because	because	SCONJ
ap-2064	379	16	|f(a	|f(a	PROPN
ap-2064	379	17	,	,	PUNCT
ap-2064	379	18	b	b	NOUN
ap-2064	379	19	,	,	PUNCT
ap-2064	379	20	n)|0	n)|0	NOUN
ap-2064	379	21	=	=	SYM
ap-2064	380	1	hn	hn	NOUN
ap-2064	380	2	;	;	PUNCT
ap-2064	380	3	see	see	VERB
ap-2064	380	4	proposition	proposition	NOUN
ap-2064	380	5	6	6	NUM
ap-2064	380	6	.	.	PUNCT
ap-2064	380	7	proposition	proposition	NOUN
ap-2064	380	8	18	18	NUM
ap-2064	380	9	(	(	PUNCT
ap-2064	380	10	monnerot	monnerot	NOUN
ap-2064	380	11	-	-	PUNCT
ap-2064	380	12	dumaine	dumaine	NOUN
ap-2064	380	13	[	[	X
ap-2064	380	14	16	16	NUM
ap-2064	380	15	]	]	PUNCT
ap-2064	380	16	)	)	PUNCT
ap-2064	380	17	.	.	PUNCT
ap-2064	381	1	the	the	DET
ap-2064	381	2	curve	curve	NOUN
ap-2064	381	3	f	f	PROPN
ap-2064	381	4	(	(	PUNCT
ap-2064	381	5	1,1	1,1	NUM
ap-2064	381	6	)	)	PUNCT
ap-2064	381	7	n	n	NOUN
ap-2064	381	8	=	=	PRON
ap-2064	381	9	fn	fn	NOUN
ap-2064	381	10	is	be	AUX
ap-2064	381	11	similar	similar	ADJ
ap-2064	381	12	to	to	ADP
ap-2064	381	13	the	the	DET
ap-2064	381	14	curve	curve	NOUN
ap-2064	381	15	f	f	PROPN
ap-2064	381	16	(	(	PUNCT
ap-2064	381	17	1,1	1,1	NUM
ap-2064	381	18	)	)	PUNCT
ap-2064	382	1	n+3	n+3	PROPN
ap-2064	383	1	=	=	SYM
ap-2064	384	1	fn+3	fn+3	PROPN
ap-2064	384	2	.	.	PUNCT
ap-2064	385	1	proposition	proposition	NOUN
ap-2064	385	2	19	19	NUM
ap-2064	385	3	(	(	PUNCT
ap-2064	385	4	ramírez	ramírez	NOUN
ap-2064	385	5	and	and	CCONJ
ap-2064	385	6	rubiano	rubiano	VERB
ap-2064	385	7	[	[	X
ap-2064	385	8	20	20	NUM
ap-2064	385	9	]	]	PUNCT
ap-2064	385	10	)	)	PUNCT
ap-2064	385	11	.	.	PUNCT
ap-2064	386	1	if	if	SCONJ
ap-2064	386	2	a	a	PRON
ap-2064	386	3	is	be	AUX
ap-2064	386	4	even	even	ADV
ap-2064	386	5	,	,	PUNCT
ap-2064	386	6	then	then	ADV
ap-2064	386	7	the	the	DET
ap-2064	386	8	curve	curve	NOUN
ap-2064	386	9	f	f	PROPN
ap-2064	386	10	(	(	PUNCT
ap-2064	386	11	a	a	PRON
ap-2064	386	12	,	,	PUNCT
ap-2064	386	13	a	a	NOUN
ap-2064	386	14	)	)	PUNCT
ap-2064	386	15	n	n	NOUN
ap-2064	386	16	=	=	SYM
ap-2064	386	17	fa	fa	PROPN
ap-2064	386	18	,	,	PUNCT
ap-2064	386	19	n	n	PROPN
ap-2064	386	20	is	be	AUX
ap-2064	386	21	similar	similar	ADJ
ap-2064	386	22	to	to	ADP
ap-2064	386	23	the	the	DET
ap-2064	386	24	curve	curve	NOUN
ap-2064	386	25	f	f	PROPN
ap-2064	386	26	(	(	PUNCT
ap-2064	386	27	a	a	DET
ap-2064	386	28	,	,	PUNCT
ap-2064	386	29	a	a	NOUN
ap-2064	386	30	)	)	PUNCT
ap-2064	386	31	n+2	n+2	NUM
ap-2064	387	1	=	=	SYM
ap-2064	387	2	fa	fa	PROPN
ap-2064	387	3	,	,	PUNCT
ap-2064	387	4	n+2	n+2	PRON
ap-2064	387	5	.	.	PUNCT
ap-2064	388	1	if	if	SCONJ
ap-2064	388	2	n	n	NOUN
ap-2064	388	3	is	be	AUX
ap-2064	388	4	odd	odd	ADJ
ap-2064	388	5	,	,	PUNCT
ap-2064	388	6	then	then	ADV
ap-2064	388	7	the	the	DET
ap-2064	388	8	curve	curve	NOUN
ap-2064	388	9	f	f	PROPN
ap-2064	388	10	(	(	PUNCT
ap-2064	388	11	a	a	PRON
ap-2064	388	12	,	,	PUNCT
ap-2064	388	13	a	a	NOUN
ap-2064	388	14	)	)	PUNCT
ap-2064	388	15	n	n	NOUN
ap-2064	388	16	=	=	SYM
ap-2064	388	17	fa	fa	PROPN
ap-2064	388	18	,	,	PUNCT
ap-2064	388	19	n	n	PROPN
ap-2064	388	20	is	be	AUX
ap-2064	388	21	similar	similar	ADJ
ap-2064	388	22	to	to	ADP
ap-2064	388	23	the	the	DET
ap-2064	388	24	curve	curve	NOUN
ap-2064	388	25	f	f	PROPN
ap-2064	388	26	(	(	PUNCT
ap-2064	388	27	a	a	PRON
ap-2064	388	28	,	,	PUNCT
ap-2064	388	29	a	a	NOUN
ap-2064	388	30	)	)	PUNCT
ap-2064	388	31	n	n	NOUN
ap-2064	388	32	=	=	SYM
ap-2064	388	33	fa	fa	PROPN
ap-2064	388	34	,	,	PUNCT
ap-2064	388	35	n+3	n+3	PROPN
ap-2064	388	36	.	.	PUNCT
ap-2064	388	37	theorem	theorem	VERB
ap-2064	388	38	20	20	NUM
ap-2064	388	39	.	.	PUNCT
ap-2064	389	1	(	(	PUNCT
ap-2064	389	2	1	1	NUM
ap-2064	389	3	.	.	PUNCT
ap-2064	389	4	)	)	PUNCT
ap-2064	390	1	if	if	SCONJ
ap-2064	390	2	a	a	PRON
ap-2064	390	3	is	be	AUX
ap-2064	390	4	even	even	ADV
ap-2064	390	5	,	,	PUNCT
ap-2064	390	6	then	then	ADV
ap-2064	390	7	the	the	DET
ap-2064	390	8	curve	curve	NOUN
ap-2064	390	9	f	f	PROPN
ap-2064	390	10	(	(	PUNCT
ap-2064	390	11	a	a	DET
ap-2064	390	12	,	,	PUNCT
ap-2064	390	13	b	b	NOUN
ap-2064	390	14	)	)	PUNCT
ap-2064	390	15	n	n	PRON
ap-2064	390	16	is	be	AUX
ap-2064	390	17	similar	similar	ADJ
ap-2064	390	18	to	to	ADP
ap-2064	390	19	the	the	DET
ap-2064	390	20	curve	curve	NOUN
ap-2064	390	21	f	f	PROPN
ap-2064	390	22	(	(	PUNCT
ap-2064	390	23	a	a	DET
ap-2064	390	24	,	,	PUNCT
ap-2064	390	25	b	b	NOUN
ap-2064	390	26	)	)	PUNCT
ap-2064	390	27	n+2	n+2	PRON
ap-2064	390	28	,	,	PUNCT
ap-2064	390	29	i.e.	i.e.	X
ap-2064	390	30	,	,	PUNCT
ap-2064	390	31	they	they	PRON
ap-2064	390	32	have	have	VERB
ap-2064	390	33	the	the	DET
ap-2064	390	34	same	same	ADJ
ap-2064	390	35	shape	shape	NOUN
ap-2064	390	36	except	except	SCONJ
ap-2064	390	37	for	for	ADP
ap-2064	390	38	the	the	DET
ap-2064	390	39	number	number	NOUN
ap-2064	390	40	of	of	ADP
ap-2064	390	41	segments	segment	NOUN
ap-2064	390	42	.	.	PUNCT
ap-2064	391	1	(	(	PUNCT
ap-2064	391	2	2	2	NUM
ap-2064	391	3	.	.	PUNCT
ap-2064	391	4	)	)	PUNCT
ap-2064	392	1	if	if	SCONJ
ap-2064	392	2	a	a	DET
ap-2064	392	3	6=	6=	ADP
ap-2064	392	4	b	b	NOUN
ap-2064	392	5	,	,	PUNCT
ap-2064	392	6	and	and	CCONJ
ap-2064	392	7	a	a	DET
ap-2064	392	8	,	,	PUNCT
ap-2064	392	9	b	b	NOUN
ap-2064	392	10	are	be	AUX
ap-2064	392	11	odd	odd	ADJ
ap-2064	392	12	,	,	PUNCT
ap-2064	392	13	then	then	ADV
ap-2064	392	14	the	the	DET
ap-2064	392	15	curve	curve	NOUN
ap-2064	392	16	f	f	PROPN
ap-2064	392	17	(	(	PUNCT
ap-2064	392	18	a	a	DET
ap-2064	392	19	,	,	PUNCT
ap-2064	392	20	b	b	NOUN
ap-2064	392	21	)	)	PUNCT
ap-2064	392	22	n	n	PRON
ap-2064	392	23	is	be	AUX
ap-2064	392	24	similar	similar	ADJ
ap-2064	392	25	to	to	ADP
ap-2064	392	26	the	the	DET
ap-2064	392	27	curve	curve	NOUN
ap-2064	392	28	f	f	PROPN
ap-2064	392	29	(	(	PUNCT
ap-2064	392	30	a	a	DET
ap-2064	392	31	,	,	PUNCT
ap-2064	392	32	b	b	NOUN
ap-2064	392	33	)	)	PUNCT
ap-2064	392	34	n+6	n+6	NUM
ap-2064	392	35	.	.	PUNCT
ap-2064	393	1	(	(	PUNCT
ap-2064	393	2	3	3	NUM
ap-2064	393	3	.	.	PUNCT
ap-2064	393	4	)	)	PUNCT
ap-2064	394	1	if	if	SCONJ
ap-2064	394	2	a	a	PRON
ap-2064	394	3	is	be	AUX
ap-2064	394	4	odd	odd	ADJ
ap-2064	394	5	,	,	PUNCT
ap-2064	394	6	and	and	CCONJ
ap-2064	394	7	b	b	X
ap-2064	394	8	is	be	AUX
ap-2064	394	9	even	even	ADV
ap-2064	394	10	,	,	PUNCT
ap-2064	394	11	then	then	ADV
ap-2064	394	12	the	the	DET
ap-2064	394	13	curve	curve	NOUN
ap-2064	394	14	f	f	PROPN
ap-2064	394	15	(	(	PUNCT
ap-2064	394	16	a	a	DET
ap-2064	394	17	,	,	PUNCT
ap-2064	394	18	b	b	NOUN
ap-2064	394	19	)	)	PUNCT
ap-2064	394	20	n	n	PRON
ap-2064	394	21	is	be	AUX
ap-2064	394	22	similar	similar	ADJ
ap-2064	394	23	to	to	ADP
ap-2064	394	24	the	the	DET
ap-2064	394	25	curve	curve	NOUN
ap-2064	394	26	f	f	PROPN
ap-2064	394	27	(	(	PUNCT
ap-2064	394	28	a	a	DET
ap-2064	394	29	,	,	PUNCT
ap-2064	394	30	b	b	NOUN
ap-2064	394	31	)	)	PUNCT
ap-2064	394	32	n+4	n+4	NUM
ap-2064	394	33	.	.	PUNCT
ap-2064	395	1	proof	proof	NOUN
ap-2064	395	2	.	.	PUNCT
ap-2064	396	1	suppose	suppose	VERB
ap-2064	396	2	a	a	PRON
ap-2064	396	3	is	be	AUX
ap-2064	396	4	even	even	ADV
ap-2064	396	5	.	.	PUNCT
ap-2064	397	1	then	then	ADV
ap-2064	397	2	it	it	PRON
ap-2064	397	3	is	be	AUX
ap-2064	397	4	clear	clear	ADJ
ap-2064	397	5	that	that	SCONJ
ap-2064	397	6	σ(a	σ(a	PROPN
ap-2064	397	7	,	,	PUNCT
ap-2064	397	8	b)(f(a	b)(f(a	PROPN
ap-2064	397	9	,	,	PUNCT
ap-2064	397	10	b	b	NOUN
ap-2064	397	11	,	,	PUNCT
ap-2064	397	12	n	n	CCONJ
ap-2064	397	13	)	)	PUNCT
ap-2064	397	14	)	)	PUNCT
ap-2064	398	1	=	=	SYM
ap-2064	398	2	f(a	f(a	PROPN
ap-2064	398	3	,	,	PUNCT
ap-2064	398	4	b	b	NOUN
ap-2064	398	5	,	,	PUNCT
ap-2064	398	6	n+2	n+2	NUM
ap-2064	398	7	)	)	PUNCT
ap-2064	398	8	;	;	PUNCT
ap-2064	398	9	see	see	VERB
ap-2064	398	10	theorem	theorem	NOUN
ap-2064	398	11	4	4	X
ap-2064	398	12	.	.	PUNCT
ap-2064	399	1	we	we	PRON
ap-2064	399	2	are	be	AUX
ap-2064	399	3	going	go	VERB
ap-2064	399	4	to	to	PART
ap-2064	399	5	prove	prove	VERB
ap-2064	399	6	that	that	SCONJ
ap-2064	399	7	σ(a	σ(a	PROPN
ap-2064	399	8	,	,	PUNCT
ap-2064	399	9	b	b	NOUN
ap-2064	399	10	)	)	PUNCT
ap-2064	399	11	preserves	preserve	VERB
ap-2064	399	12	the	the	DET
ap-2064	399	13	odd	odd	ADV
ap-2064	399	14	-	-	PUNCT
ap-2064	399	15	even	even	ADV
ap-2064	399	16	alternation	alternation	NOUN
ap-2064	399	17	required	require	VERB
ap-2064	399	18	by	by	ADP
ap-2064	399	19	the	the	DET
ap-2064	399	20	odd	odd	ADV
ap-2064	399	21	-	-	PUNCT
ap-2064	399	22	even	even	ADV
ap-2064	399	23	drawing	draw	VERB
ap-2064	399	24	rule	rule	NOUN
ap-2064	399	25	.	.	PUNCT
ap-2064	400	1	in	in	ADP
ap-2064	400	2	fact	fact	NOUN
ap-2064	400	3	,	,	PUNCT
ap-2064	400	4	σ(a	σ(a	PROPN
ap-2064	400	5	,	,	PUNCT
ap-2064	400	6	b)(0	b)(0	NUM
ap-2064	400	7	)	)	PUNCT
ap-2064	400	8	=	=	SYM
ap-2064	400	9	(	(	PUNCT
ap-2064	400	10	0a−11)b0	0a−11)b0	NOUN
ap-2064	400	11	,	,	PUNCT
ap-2064	400	12	and	and	CCONJ
ap-2064	400	13	σ(a	σ(a	PROPN
ap-2064	400	14	,	,	PUNCT
ap-2064	400	15	b)(1	b)(1	NOUN
ap-2064	400	16	)	)	PUNCT
ap-2064	400	17	=	=	PUNCT
ap-2064	400	18	(	(	PUNCT
ap-2064	400	19	0a−11)b0a1	0a−11)b0a1	NOUN
ap-2064	400	20	.	.	PUNCT
ap-2064	401	1	as	as	SCONJ
ap-2064	401	2	a	a	PRON
ap-2064	401	3	is	be	AUX
ap-2064	401	4	even	even	ADV
ap-2064	401	5	,	,	PUNCT
ap-2064	401	6	then	then	ADV
ap-2064	401	7	|σ(a	|σ(a	PROPN
ap-2064	401	8	,	,	PUNCT
ap-2064	401	9	b)(0)|	b)(0)|	NOUN
ap-2064	401	10	and	and	CCONJ
ap-2064	401	11	|σ(a	|σ(a	PROPN
ap-2064	401	12	,	,	PUNCT
ap-2064	401	13	b)(1)|	b)(1)|	PROPN
ap-2064	401	14	are	be	AUX
ap-2064	401	15	odd	odd	ADJ
ap-2064	401	16	.	.	PUNCT
ap-2064	402	1	56	56	NUM
ap-2064	402	2	vol	vol	NOUN
ap-2064	402	3	.	.	PUNCT
ap-2064	403	1	55	55	NUM
ap-2064	403	2	no	no	NOUN
ap-2064	403	3	.	.	PUNCT
ap-2064	404	1	1/2015	1/2015	NUM
ap-2064	404	2	biperiodic	biperiodic	PROPN
ap-2064	404	3	fibonacci	fibonacci	NOUN
ap-2064	404	4	word	word	NOUN
ap-2064	404	5	and	and	CCONJ
ap-2064	404	6	its	its	PRON
ap-2064	404	7	fractal	fractal	ADJ
ap-2064	404	8	curve	curve	NOUN
ap-2064	404	9	figure	figure	NOUN
ap-2064	404	10	2	2	NUM
ap-2064	404	11	.	.	PUNCT
ap-2064	405	1	curves	curve	NOUN
ap-2064	405	2	f	f	PROPN
ap-2064	405	3	(	(	PUNCT
ap-2064	405	4	2,6	2,6	NUM
ap-2064	405	5	)	)	PUNCT
ap-2064	405	6	7	7	NUM
ap-2064	405	7	,	,	PUNCT
ap-2064	405	8	f	f	X
ap-2064	405	9	(	(	PUNCT
ap-2064	405	10	2,6	2,6	NUM
ap-2064	405	11	)	)	PUNCT
ap-2064	405	12	9	9	NUM
ap-2064	405	13	,	,	PUNCT
ap-2064	405	14	f	f	X
ap-2064	405	15	(	(	PUNCT
ap-2064	405	16	2,6	2,6	NUM
ap-2064	405	17	)	)	PUNCT
ap-2064	405	18	11	11	NUM
ap-2064	405	19	with	with	ADP
ap-2064	405	20	θ	θ	PROPN
ap-2064	405	21	=	=	SYM
ap-2064	405	22	72	72	NUM
ap-2064	405	23	◦	◦	NOUN
ap-2064	405	24	.	.	PUNCT
ap-2064	406	1	hence	hence	ADV
ap-2064	406	2	if	if	SCONJ
ap-2064	406	3	|w|	|w|	ADJ
ap-2064	406	4	is	be	AUX
ap-2064	406	5	even	even	ADV
ap-2064	406	6	(	(	PUNCT
ap-2064	406	7	odd	odd	ADJ
ap-2064	406	8	)	)	PUNCT
ap-2064	406	9	,	,	PUNCT
ap-2064	406	10	then	then	ADV
ap-2064	406	11	|σ(a	|σ(a	PROPN
ap-2064	406	12	,	,	PUNCT
ap-2064	406	13	b)(w)|	b)(w)|	PROPN
ap-2064	406	14	is	be	AUX
ap-2064	406	15	even	even	ADV
ap-2064	406	16	(	(	PUNCT
ap-2064	406	17	odd	odd	ADJ
ap-2064	406	18	)	)	PUNCT
ap-2064	406	19	.	.	PUNCT
ap-2064	407	1	since	since	SCONJ
ap-2064	407	2	σ(a	σ(a	PROPN
ap-2064	407	3	,	,	PUNCT
ap-2064	407	4	b	b	NOUN
ap-2064	407	5	)	)	PUNCT
ap-2064	407	6	preserves	preserve	VERB
ap-2064	407	7	parity	parity	NOUN
ap-2064	407	8	of	of	ADP
ap-2064	407	9	length	length	NOUN
ap-2064	407	10	then	then	ADV
ap-2064	407	11	any	any	DET
ap-2064	407	12	subword	subword	NOUN
ap-2064	407	13	in	in	ADP
ap-2064	407	14	a	a	DET
ap-2064	407	15	biperiodic	biperiodic	ADJ
ap-2064	407	16	fibonacci	fibonacci	NOUN
ap-2064	407	17	word	word	NOUN
ap-2064	407	18	preserves	preserve	VERB
ap-2064	407	19	parity	parity	NOUN
ap-2064	407	20	of	of	ADP
ap-2064	407	21	position	position	NOUN
ap-2064	407	22	.	.	PUNCT
ap-2064	408	1	finally	finally	ADV
ap-2064	408	2	,	,	PUNCT
ap-2064	408	3	let	let	VERB
ap-2064	408	4	a(w	a(w	PROPN
ap-2064	408	5	)	)	PUNCT
ap-2064	408	6	be	be	VERB
ap-2064	408	7	the	the	DET
ap-2064	408	8	function	function	NOUN
ap-2064	408	9	that	that	PRON
ap-2064	408	10	gives	give	VERB
ap-2064	408	11	the	the	DET
ap-2064	408	12	ending	ending	NOUN
ap-2064	408	13	or	or	CCONJ
ap-2064	408	14	resulting	result	VERB
ap-2064	408	15	angle	angle	NOUN
ap-2064	408	16	of	of	ADP
ap-2064	408	17	an	an	DET
ap-2064	408	18	associate	associate	ADJ
ap-2064	408	19	curve	curve	NOUN
ap-2064	408	20	to	to	ADP
ap-2064	408	21	the	the	DET
ap-2064	408	22	word	word	NOUN
ap-2064	408	23	w	w	NOUN
ap-2064	408	24	through	through	ADP
ap-2064	408	25	the	the	DET
ap-2064	408	26	odd	odd	ADV
ap-2064	408	27	-	-	PUNCT
ap-2064	408	28	even	even	ADV
ap-2064	408	29	drawing	draw	VERB
ap-2064	408	30	rule	rule	NOUN
ap-2064	408	31	of	of	ADP
ap-2064	408	32	angle	angle	NOUN
ap-2064	408	33	θ	θ	PROPN
ap-2064	408	34	.	.	PUNCT
ap-2064	409	1	we	we	PRON
ap-2064	409	2	have	have	VERB
ap-2064	409	3	to	to	PART
ap-2064	409	4	prove	prove	VERB
ap-2064	409	5	that	that	SCONJ
ap-2064	409	6	the	the	DET
ap-2064	409	7	resulting	result	VERB
ap-2064	409	8	angle	angle	NOUN
ap-2064	409	9	of	of	ADP
ap-2064	409	10	a	a	DET
ap-2064	409	11	curve	curve	NOUN
ap-2064	409	12	must	must	AUX
ap-2064	409	13	be	be	AUX
ap-2064	409	14	preserved	preserve	VERB
ap-2064	409	15	or	or	CCONJ
ap-2064	409	16	inverted	invert	VERB
ap-2064	409	17	by	by	ADP
ap-2064	409	18	σ(a	σ(a	PROPN
ap-2064	409	19	,	,	PUNCT
ap-2064	409	20	b	b	NOUN
ap-2064	409	21	)	)	PUNCT
ap-2064	409	22	.	.	PUNCT
ap-2064	410	1	note	note	VERB
ap-2064	410	2	that	that	SCONJ
ap-2064	410	3	a(00	a(00	NOUN
ap-2064	410	4	)	)	PUNCT
ap-2064	410	5	=	=	SYM
ap-2064	410	6	0	0	NUM
ap-2064	410	7	,	,	PUNCT
ap-2064	410	8	a(01	a(01	NUM
ap-2064	410	9	)	)	PUNCT
ap-2064	410	10	=	=	SYM
ap-2064	410	11	−θ	−θ	ADJ
ap-2064	410	12	and	and	CCONJ
ap-2064	410	13	a(10	a(10	ADJ
ap-2064	410	14	)	)	PUNCT
ap-2064	410	15	=	=	PUNCT
ap-2064	411	1	+	+	NUM
ap-2064	411	2	θ	θ	PROPN
ap-2064	411	3	.	.	PUNCT
ap-2064	411	4	therefore	therefore	ADV
ap-2064	411	5	•	•	NUM
ap-2064	411	6	a(σ(a	a(σ(a	PROPN
ap-2064	411	7	,	,	PUNCT
ap-2064	411	8	b)(00	b)(00	ADJ
ap-2064	411	9	)	)	PUNCT
ap-2064	411	10	)	)	PUNCT
ap-2064	411	11	=	=	SYM
ap-2064	411	12	a((0a−11)b0(0a−11)b0	a((0a−11)b0(0a−11)b0	PROPN
ap-2064	411	13	)	)	PUNCT
ap-2064	411	14	=	=	SYM
ap-2064	411	15	a((0a−11)b	a((0a−11)b	PROPN
ap-2064	411	16	)	)	PUNCT
ap-2064	411	17	+	+	NUM
ap-2064	411	18	a(0a	a(0a	PRON
ap-2064	411	19	)	)	PUNCT
ap-2064	411	20	+	+	X
ap-2064	411	21	a((10a−1)b−1	a((10a−1)b−1	NOUN
ap-2064	411	22	)	)	PUNCT
ap-2064	412	1	+	+	CCONJ
ap-2064	412	2	a(10	a(10	NOUN
ap-2064	412	3	)	)	PUNCT
ap-2064	412	4	=	=	SYM
ap-2064	412	5	a((0a−11)b	a((0a−11)b	PROPN
ap-2064	412	6	)	)	PUNCT
ap-2064	412	7	+	+	NOUN
ap-2064	412	8	a(0a	a(0a	X
ap-2064	412	9	)	)	PUNCT
ap-2064	413	1	+	+	ADJ
ap-2064	413	2	a(10	a(10	NOUN
ap-2064	413	3	)	)	PUNCT
ap-2064	413	4	+	+	NOUN
ap-2064	413	5	a(0a−2	a(0a−2	NOUN
ap-2064	413	6	)	)	PUNCT
ap-2064	413	7	+	+	NOUN
ap-2064	413	8	a(10	a(10	NOUN
ap-2064	413	9	)	)	PUNCT
ap-2064	413	10	+	+	NUM
ap-2064	413	11	a(0a−2	a(0a−2	NOUN
ap-2064	413	12	)	)	PUNCT
ap-2064	414	1	+	+	CCONJ
ap-2064	414	2	·	·	PUNCT
ap-2064	414	3	·	·	PUNCT
ap-2064	414	4	·	·	PUNCT
ap-2064	414	5	+	+	NOUN
ap-2064	414	6	a(10	a(10	NOUN
ap-2064	414	7	)	)	PUNCT
ap-2064	414	8	=	=	SYM
ap-2064	414	9	b(−θ	b(−θ	X
ap-2064	414	10	)	)	PUNCT
ap-2064	415	1	+	+	CCONJ
ap-2064	415	2	0	0	NUM
ap-2064	416	1	+	+	CCONJ
ap-2064	416	2	θ+	θ+	NUM
ap-2064	416	3	0	0	NUM
ap-2064	416	4	+	+	CCONJ
ap-2064	416	5	θ+	θ+	X
ap-2064	416	6	·	·	PUNCT
ap-2064	416	7	·	·	PUNCT
ap-2064	416	8	·	·	PUNCT
ap-2064	417	1	+	+	NUM
ap-2064	417	2	θ	θ	X
ap-2064	417	3	=	=	SYM
ap-2064	417	4	0	0	NUM
ap-2064	417	5	.	.	NOUN
ap-2064	417	6	•	•	NUM
ap-2064	417	7	a(σ(a	a(σ(a	PROPN
ap-2064	417	8	,	,	PUNCT
ap-2064	417	9	b)(01	b)(01	NOUN
ap-2064	417	10	)	)	PUNCT
ap-2064	417	11	)	)	PUNCT
ap-2064	417	12	=	=	PUNCT
ap-2064	417	13	a((0a−11)b0(0a−11)b0a1	a((0a−11)b0(0a−11)b0a1	PRON
ap-2064	417	14	)	)	PUNCT
ap-2064	417	15	=	=	SYM
ap-2064	417	16	a((0a−11)b	a((0a−11)b	PROPN
ap-2064	417	17	)	)	PUNCT
ap-2064	417	18	+	+	NUM
ap-2064	417	19	a(0a	a(0a	VERB
ap-2064	417	20	)	)	PUNCT
ap-2064	417	21	+	+	CCONJ
ap-2064	417	22	a(10	a(10	ADJ
ap-2064	417	23	)	)	PUNCT
ap-2064	417	24	+	+	NUM
ap-2064	417	25	a(0a−2	a(0a−2	NOUN
ap-2064	417	26	)	)	PUNCT
ap-2064	417	27	+	+	CCONJ
ap-2064	417	28	·	·	PUNCT
ap-2064	417	29	·	·	PUNCT
ap-2064	417	30	·	·	PUNCT
ap-2064	417	31	+	+	PUNCT
ap-2064	417	32	a(10)a(0a−11	a(10)a(0a−11	NOUN
ap-2064	417	33	)	)	PUNCT
ap-2064	417	34	=	=	PUNCT
ap-2064	417	35	0−	0−	NUM
ap-2064	417	36	θ	θ	X
ap-2064	417	37	=	=	SYM
ap-2064	417	38	−θ	−θ	NOUN
ap-2064	417	39	.	.	PUNCT
ap-2064	418	1	•	•	NUM
ap-2064	418	2	a(σ(a	a(σ(a	PROPN
ap-2064	418	3	,	,	PUNCT
ap-2064	418	4	b)(10	b)(10	NOUN
ap-2064	418	5	)	)	PUNCT
ap-2064	418	6	)	)	PUNCT
ap-2064	419	1	=	=	SYM
ap-2064	419	2	a((0a−11)b0a1(0a−11)b0	a((0a−11)b0a1(0a−11)b0	NOUN
ap-2064	419	3	)	)	PUNCT
ap-2064	419	4	=	=	SYM
ap-2064	419	5	a((0a−11)b	a((0a−11)b	PROPN
ap-2064	419	6	)	)	PUNCT
ap-2064	419	7	+	+	NOUN
ap-2064	419	8	a(0a	a(0a	X
ap-2064	419	9	)	)	PUNCT
ap-2064	419	10	+	+	ADJ
ap-2064	419	11	a(10	a(10	NOUN
ap-2064	419	12	)	)	PUNCT
ap-2064	419	13	+	+	NOUN
ap-2064	419	14	a(0a−2	a(0a−2	NOUN
ap-2064	419	15	)	)	PUNCT
ap-2064	419	16	+	+	NOUN
ap-2064	419	17	a(10	a(10	NOUN
ap-2064	419	18	)	)	PUNCT
ap-2064	419	19	+	+	CCONJ
ap-2064	419	20	a(0a−2)+	a(0a−2)+	ADV
ap-2064	419	21	·	·	PUNCT
ap-2064	419	22	·	·	PUNCT
ap-2064	419	23	·	·	PUNCT
ap-2064	420	1	+	+	NOUN
ap-2064	420	2	a(10	a(10	NOUN
ap-2064	420	3	)	)	PUNCT
ap-2064	420	4	=	=	NOUN
ap-2064	420	5	bθ+0+θ+0+θ+	bθ+0+θ+0+θ+	NOUN
ap-2064	420	6	·	·	PUNCT
ap-2064	420	7	·	·	PUNCT
ap-2064	420	8	·	·	PUNCT
ap-2064	420	9	+	+	NUM
ap-2064	420	10	θ	θ	X
ap-2064	420	11	=	=	SYM
ap-2064	420	12	bθ	bθ	PROPN
ap-2064	420	13	+	+	X
ap-2064	420	14	(	(	PUNCT
ap-2064	420	15	b+	b+	X
ap-2064	420	16	1)θ	1)θ	X
ap-2064	420	17	=	=	SYM
ap-2064	420	18	+	+	NUM
ap-2064	420	19	θ	θ	PROPN
ap-2064	420	20	.	.	PUNCT
ap-2064	420	21	then	then	ADV
ap-2064	420	22	σ(a	σ(a	PROPN
ap-2064	420	23	,	,	PUNCT
ap-2064	420	24	b	b	NOUN
ap-2064	420	25	)	)	PUNCT
ap-2064	420	26	preserves	preserve	VERB
ap-2064	420	27	the	the	DET
ap-2064	420	28	resulting	result	VERB
ap-2064	420	29	angle	angle	NOUN
ap-2064	420	30	,	,	PUNCT
ap-2064	420	31	i.e.	i.e.	X
ap-2064	420	32	,	,	PUNCT
ap-2064	420	33	a(w	a(w	PROPN
ap-2064	420	34	)	)	PUNCT
ap-2064	420	35	=	=	SYM
ap-2064	420	36	a(σ(a	a(σ(a	PROPN
ap-2064	420	37	,	,	PUNCT
ap-2064	420	38	b)(w	b)(w	NOUN
ap-2064	420	39	)	)	PUNCT
ap-2064	420	40	)	)	PUNCT
ap-2064	420	41	for	for	ADP
ap-2064	420	42	any	any	DET
ap-2064	420	43	word	word	NOUN
ap-2064	420	44	w	w	ADP
ap-2064	420	45	of	of	ADP
ap-2064	420	46	even	even	ADV
ap-2064	420	47	length	length	NOUN
ap-2064	420	48	.	.	PUNCT
ap-2064	421	1	therefore	therefore	ADV
ap-2064	421	2	the	the	DET
ap-2064	421	3	image	image	NOUN
ap-2064	421	4	of	of	ADP
ap-2064	421	5	a	a	DET
ap-2064	421	6	pattern	pattern	NOUN
ap-2064	421	7	by	by	ADP
ap-2064	421	8	σ(a	σ(a	PROPN
ap-2064	421	9	,	,	PUNCT
ap-2064	421	10	b	b	NOUN
ap-2064	421	11	)	)	PUNCT
ap-2064	421	12	is	be	AUX
ap-2064	421	13	the	the	DET
ap-2064	421	14	rotation	rotation	NOUN
ap-2064	421	15	of	of	ADP
ap-2064	421	16	this	this	DET
ap-2064	421	17	pattern	pattern	NOUN
ap-2064	421	18	by	by	ADP
ap-2064	421	19	an	an	DET
ap-2064	421	20	angle	angle	NOUN
ap-2064	421	21	of	of	ADP
ap-2064	421	22	+	+	NOUN
ap-2064	421	23	θ	θ	NOUN
ap-2064	421	24	.	.	PUNCT
ap-2064	421	25	since	since	SCONJ
ap-2064	421	26	σ(a	σ(a	PROPN
ap-2064	421	27	,	,	PUNCT
ap-2064	421	28	b)(f(a	b)(f(a	PROPN
ap-2064	421	29	,	,	PUNCT
ap-2064	421	30	b	b	NOUN
ap-2064	421	31	,	,	PUNCT
ap-2064	421	32	n	n	CCONJ
ap-2064	421	33	)	)	PUNCT
ap-2064	421	34	)	)	PUNCT
ap-2064	422	1	=	=	SYM
ap-2064	422	2	f(a	f(a	PROPN
ap-2064	422	3	,	,	PUNCT
ap-2064	422	4	b	b	NOUN
ap-2064	422	5	,	,	PUNCT
ap-2064	422	6	n+2	n+2	NUM
ap-2064	422	7	)	)	PUNCT
ap-2064	422	8	,	,	PUNCT
ap-2064	422	9	then	then	ADV
ap-2064	422	10	the	the	DET
ap-2064	422	11	curve	curve	NOUN
ap-2064	422	12	f(a	f(a	PROPN
ap-2064	422	13	,	,	PUNCT
ap-2064	422	14	b	b	NOUN
ap-2064	422	15	,	,	PUNCT
ap-2064	422	16	n	n	CCONJ
ap-2064	422	17	)	)	PUNCT
ap-2064	422	18	is	be	AUX
ap-2064	422	19	similar	similar	ADJ
ap-2064	422	20	to	to	ADP
ap-2064	422	21	the	the	DET
ap-2064	422	22	curve	curve	NOUN
ap-2064	422	23	f(a	f(a	PROPN
ap-2064	422	24	,	,	PUNCT
ap-2064	422	25	b	b	NOUN
ap-2064	422	26	,	,	PUNCT
ap-2064	422	27	n+2	n+2	NUM
ap-2064	422	28	)	)	PUNCT
ap-2064	422	29	.	.	PUNCT
ap-2064	423	1	if	if	SCONJ
ap-2064	423	2	a	a	DET
ap-2064	423	3	6=	6=	ADP
ap-2064	423	4	b	b	NOUN
ap-2064	423	5	,	,	PUNCT
ap-2064	423	6	and	and	CCONJ
ap-2064	423	7	a	a	DET
ap-2064	423	8	,	,	PUNCT
ap-2064	423	9	b	b	NOUN
ap-2064	423	10	are	be	AUX
ap-2064	423	11	odd	odd	ADJ
ap-2064	423	12	the	the	DET
ap-2064	423	13	proof	proof	NOUN
ap-2064	423	14	is	be	AUX
ap-2064	423	15	similar	similar	ADJ
ap-2064	423	16	,	,	PUNCT
ap-2064	423	17	but	but	CCONJ
ap-2064	423	18	using	use	VERB
ap-2064	423	19	σ3	σ3	PROPN
ap-2064	423	20	(	(	PUNCT
ap-2064	423	21	a	a	DET
ap-2064	423	22	,	,	PUNCT
ap-2064	423	23	b	b	NOUN
ap-2064	423	24	)	)	PUNCT
ap-2064	423	25	.	.	PUNCT
ap-2064	424	1	if	if	SCONJ
ap-2064	424	2	a	a	PRON
ap-2064	424	3	is	be	AUX
ap-2064	424	4	odd	odd	ADJ
ap-2064	424	5	,	,	PUNCT
ap-2064	424	6	and	and	CCONJ
ap-2064	424	7	b	b	X
ap-2064	424	8	is	be	AUX
ap-2064	424	9	even	even	ADV
ap-2064	424	10	the	the	DET
ap-2064	424	11	proof	proof	NOUN
ap-2064	424	12	is	be	AUX
ap-2064	424	13	similar	similar	ADJ
ap-2064	424	14	,	,	PUNCT
ap-2064	424	15	but	but	CCONJ
ap-2064	424	16	using	use	VERB
ap-2064	424	17	σ2	σ2	PROPN
ap-2064	424	18	(	(	PUNCT
ap-2064	424	19	a	a	DET
ap-2064	424	20	,	,	PUNCT
ap-2064	424	21	b	b	NOUN
ap-2064	424	22	)	)	PUNCT
ap-2064	424	23	.	.	PUNCT
ap-2064	425	1	an	an	DET
ap-2064	425	2	open	open	ADJ
ap-2064	425	3	problem	problem	NOUN
ap-2064	425	4	is	be	AUX
ap-2064	425	5	try	try	VERB
ap-2064	425	6	to	to	PART
ap-2064	425	7	find	find	VERB
ap-2064	425	8	a	a	DET
ap-2064	425	9	characterization	characterization	NOUN
ap-2064	425	10	of	of	ADP
ap-2064	425	11	similar	similar	ADJ
ap-2064	425	12	word	word	NOUN
ap-2064	425	13	curves	curve	NOUN
ap-2064	425	14	in	in	ADP
ap-2064	425	15	terms	term	NOUN
ap-2064	425	16	of	of	ADP
ap-2064	425	17	words	word	NOUN
ap-2064	425	18	.	.	PUNCT
ap-2064	426	1	example	example	NOUN
ap-2064	426	2	21	21	NUM
ap-2064	426	3	.	.	PUNCT
ap-2064	427	1	in	in	ADP
ap-2064	427	2	figure	figure	NOUN
ap-2064	427	3	2	2	NUM
ap-2064	427	4	,	,	PUNCT
ap-2064	427	5	f	f	X
ap-2064	427	6	(	(	PUNCT
ap-2064	427	7	2,6	2,6	NUM
ap-2064	427	8	)	)	PUNCT
ap-2064	427	9	7	7	NUM
ap-2064	427	10	is	be	AUX
ap-2064	427	11	similar	similar	ADJ
ap-2064	427	12	to	to	ADP
ap-2064	427	13	f	f	PROPN
ap-2064	427	14	(	(	PUNCT
ap-2064	427	15	2,6	2,6	NUM
ap-2064	427	16	)	)	PUNCT
ap-2064	427	17	9	9	NUM
ap-2064	427	18	,	,	PUNCT
ap-2064	427	19	f	f	X
ap-2064	427	20	(	(	PUNCT
ap-2064	427	21	2,6	2,6	NUM
ap-2064	427	22	)	)	PUNCT
ap-2064	427	23	11	11	NUM
ap-2064	427	24	and	and	CCONJ
ap-2064	427	25	so	so	ADV
ap-2064	427	26	on	on	ADV
ap-2064	427	27	.	.	PUNCT
ap-2064	428	1	in	in	ADP
ap-2064	428	2	figure	figure	NOUN
ap-2064	428	3	3	3	NUM
ap-2064	428	4	,	,	PUNCT
ap-2064	428	5	f	f	X
ap-2064	428	6	(	(	PUNCT
ap-2064	428	7	3,4	3,4	NUM
ap-2064	428	8	)	)	PUNCT
ap-2064	428	9	5	5	NUM
ap-2064	428	10	is	be	AUX
ap-2064	428	11	similar	similar	ADJ
ap-2064	428	12	to	to	ADP
ap-2064	428	13	f	f	PROPN
ap-2064	428	14	(	(	PUNCT
ap-2064	428	15	3,4	3,4	NUM
ap-2064	428	16	)	)	PUNCT
ap-2064	428	17	9	9	NUM
ap-2064	428	18	and	and	CCONJ
ap-2064	428	19	so	so	ADV
ap-2064	428	20	on	on	ADV
ap-2064	428	21	.	.	PUNCT
ap-2064	429	1	acknowledgements	acknowledgement	NOUN
ap-2064	429	2	the	the	DET
ap-2064	429	3	authors	author	NOUN
ap-2064	429	4	thank	thank	VERB
ap-2064	429	5	the	the	DET
ap-2064	429	6	anonymous	anonymous	ADJ
ap-2064	429	7	referees	referee	NOUN
ap-2064	429	8	for	for	ADP
ap-2064	429	9	their	their	PRON
ap-2064	429	10	careful	careful	ADJ
ap-2064	429	11	reading	reading	NOUN
ap-2064	429	12	of	of	ADP
ap-2064	429	13	the	the	DET
ap-2064	429	14	manuscript	manuscript	NOUN
ap-2064	429	15	and	and	CCONJ
ap-2064	429	16	their	their	PRON
ap-2064	429	17	fruitful	fruitful	ADJ
ap-2064	429	18	comments	comment	NOUN
ap-2064	429	19	and	and	CCONJ
ap-2064	429	20	suggestions	suggestion	NOUN
ap-2064	429	21	.	.	PUNCT
ap-2064	430	1	this	this	DET
ap-2064	430	2	research	research	NOUN
ap-2064	430	3	is	be	AUX
ap-2064	430	4	for	for	ADP
ap-2064	430	5	the	the	DET
ap-2064	430	6	observatorio	observatorio	PROPN
ap-2064	430	7	de	de	PROPN
ap-2064	430	8	restituciãşn	restituciãşn	PROPN
ap-2064	430	9	y	y	PROPN
ap-2064	430	10	regulaciãşn	regulaciãşn	PROPN
ap-2064	430	11	de	de	PROPN
ap-2064	430	12	derechos	derechos	PROPN
ap-2064	430	13	de	de	PROPN
ap-2064	430	14	propiedad	propiedad	PROPN
ap-2064	430	15	agraria	agraria	PROPN
ap-2064	430	16	.	.	PUNCT
ap-2064	431	1	moreover	moreover	ADV
ap-2064	431	2	,	,	PUNCT
ap-2064	431	3	it	it	PRON
ap-2064	431	4	was	be	AUX
ap-2064	431	5	supported	support	VERB
ap-2064	431	6	in	in	ADP
ap-2064	431	7	part	part	NOUN
ap-2064	431	8	by	by	ADP
ap-2064	431	9	the	the	DET
ap-2064	431	10	equipment	equipment	NOUN
ap-2064	431	11	donation	donation	NOUN
ap-2064	431	12	from	from	ADP
ap-2064	431	13	the	the	DET
ap-2064	431	14	german	german	ADJ
ap-2064	431	15	academic	academic	ADJ
ap-2064	431	16	exchange	exchange	NOUN
ap-2064	431	17	service	service	NOUN
ap-2064	431	18	-	-	PUNCT
ap-2064	431	19	daad	daad	NOUN
ap-2064	431	20	to	to	ADP
ap-2064	431	21	the	the	DET
ap-2064	431	22	faculty	faculty	NOUN
ap-2064	431	23	of	of	ADP
ap-2064	431	24	science	science	NOUN
ap-2064	431	25	at	at	ADP
ap-2064	431	26	the	the	DET
ap-2064	431	27	universidad	universidad	PROPN
ap-2064	431	28	nacional	nacional	PROPN
ap-2064	431	29	de	de	X
ap-2064	431	30	colombia	colombia	PROPN
ap-2064	431	31	.	.	PUNCT
ap-2064	432	1	57	57	NUM
ap-2064	432	2	josé	josé	PROPN
ap-2064	432	3	l.	l.	PROPN
ap-2064	432	4	ramírez	ramírez	PROPN
ap-2064	432	5	,	,	PUNCT
ap-2064	432	6	gustavo	gustavo	PROPN
ap-2064	432	7	n.	n.	PROPN
ap-2064	432	8	rubiano	rubiano	PROPN
ap-2064	432	9	acta	acta	PROPN
ap-2064	432	10	polytechnica	polytechnica	PROPN
ap-2064	432	11	figure	figure	NOUN
ap-2064	432	12	3	3	NUM
ap-2064	432	13	.	.	PUNCT
ap-2064	432	14	curves	curve	NOUN
ap-2064	432	15	f	f	PROPN
ap-2064	432	16	(	(	PUNCT
ap-2064	432	17	3,4	3,4	NUM
ap-2064	432	18	)	)	PUNCT
ap-2064	432	19	5	5	NUM
ap-2064	432	20	,	,	PUNCT
ap-2064	432	21	f	f	X
ap-2064	432	22	(	(	PUNCT
ap-2064	432	23	3,4	3,4	NUM
ap-2064	432	24	)	)	PUNCT
ap-2064	432	25	9	9	NUM
ap-2064	432	26	with	with	ADP
ap-2064	432	27	θ	θ	PROPN
ap-2064	432	28	=	=	SYM
ap-2064	432	29	120	120	NUM
ap-2064	432	30	◦	◦	NOUN
ap-2064	432	31	.	.	PUNCT
ap-2064	433	1	references	reference	NOUN
ap-2064	433	2	[	[	X
ap-2064	433	3	1	1	NUM
ap-2064	433	4	]	]	PUNCT
ap-2064	433	5	allouche	allouche	PROPN
ap-2064	433	6	,	,	PUNCT
ap-2064	433	7	j.	j.	PROPN
ap-2064	433	8	,	,	PUNCT
ap-2064	433	9	shallit	shallit	ADJ
ap-2064	433	10	,	,	PUNCT
ap-2064	433	11	j.	j.	PROPN
ap-2064	433	12	,	,	PUNCT
ap-2064	433	13	automatic	automatic	ADJ
ap-2064	433	14	sequences	sequence	NOUN
ap-2064	433	15	,	,	PUNCT
ap-2064	433	16	cambridge	cambridge	PROPN
ap-2064	433	17	university	university	PROPN
ap-2064	433	18	press	press	PROPN
ap-2064	433	19	,	,	PUNCT
ap-2064	433	20	cambridge	cambridge	PROPN
ap-2064	433	21	,	,	PUNCT
ap-2064	433	22	2003	2003	NUM
ap-2064	433	23	.	.	PUNCT
ap-2064	434	1	[	[	X
ap-2064	434	2	2	2	NUM
ap-2064	434	3	]	]	X
ap-2064	434	4	baláži	baláži	NOUN
ap-2064	434	5	,	,	PUNCT
ap-2064	434	6	p.	p.	NOUN
ap-2064	434	7	,	,	PUNCT
ap-2064	434	8	various	various	ADJ
ap-2064	434	9	properties	property	NOUN
ap-2064	434	10	of	of	ADP
ap-2064	434	11	sturmian	sturmian	NOUN
ap-2064	434	12	words	word	NOUN
ap-2064	434	13	,	,	PUNCT
ap-2064	434	14	acta	acta	PROPN
ap-2064	434	15	polytech	polytech	PROPN
ap-2064	434	16	.	.	PUNCT
ap-2064	435	1	45(5	45(5	NUM
ap-2064	435	2	)	)	PUNCT
ap-2064	435	3	,	,	PUNCT
ap-2064	435	4	2002	2002	NUM
ap-2064	435	5	,	,	PUNCT
ap-2064	435	6	19–23	19–23	NUM
ap-2064	435	7	.	.	PUNCT
ap-2064	436	1	[	[	X
ap-2064	436	2	3	3	NUM
ap-2064	436	3	]	]	X
ap-2064	436	4	berstel	berstel	NOUN
ap-2064	436	5	,	,	PUNCT
ap-2064	436	6	j.	j.	PROPN
ap-2064	436	7	,	,	PUNCT
ap-2064	436	8	fibonacci	fibonacci	PROPN
ap-2064	436	9	words	word	NOUN
ap-2064	436	10	-	-	PUNCT
ap-2064	436	11	a	a	DET
ap-2064	436	12	survey	survey	NOUN
ap-2064	436	13	,	,	PUNCT
ap-2064	436	14	in	in	ADP
ap-2064	436	15	:	:	PUNCT
ap-2064	436	16	g.	g.	PROPN
ap-2064	436	17	rosenberg	rosenberg	PROPN
ap-2064	436	18	,	,	PUNCT
ap-2064	436	19	a.	a.	NOUN
ap-2064	436	20	salomaa	salomaa	PROPN
ap-2064	436	21	(	(	PUNCT
ap-2064	436	22	eds	ed	NOUN
ap-2064	436	23	.	.	PUNCT
ap-2064	436	24	)	)	PUNCT
ap-2064	436	25	,	,	PUNCT
ap-2064	436	26	the	the	DET
ap-2064	436	27	book	book	NOUN
ap-2064	436	28	of	of	ADP
ap-2064	436	29	l	l	PROPN
ap-2064	436	30	,	,	PUNCT
ap-2064	436	31	springer	springer	NOUN
ap-2064	436	32	,	,	PUNCT
ap-2064	436	33	berlin	berlin	PROPN
ap-2064	436	34	,	,	PUNCT
ap-2064	436	35	1986	1986	NUM
ap-2064	436	36	,	,	PUNCT
ap-2064	436	37	11–26	11–26	NUM
ap-2064	436	38	.	.	PUNCT
ap-2064	437	1	[	[	X
ap-2064	437	2	4	4	X
ap-2064	437	3	]	]	X
ap-2064	437	4	blondin	blondin	PROPN
ap-2064	437	5	-	-	PUNCT
ap-2064	437	6	massé	massé	PROPN
ap-2064	437	7	,	,	PUNCT
ap-2064	437	8	a.	a.	NOUN
ap-2064	437	9	,	,	PUNCT
ap-2064	437	10	brlek	brlek	PROPN
ap-2064	437	11	,	,	PUNCT
ap-2064	437	12	s.	s.	PROPN
ap-2064	437	13	,	,	PUNCT
ap-2064	437	14	garon	garon	PROPN
ap-2064	437	15	,	,	PUNCT
ap-2064	437	16	a.	a.	PROPN
ap-2064	437	17	,	,	PUNCT
ap-2064	437	18	labbé	labbé	PROPN
ap-2064	437	19	,	,	PUNCT
ap-2064	437	20	s.	s.	PROPN
ap-2064	437	21	,	,	PUNCT
ap-2064	437	22	two	two	NUM
ap-2064	437	23	infinite	infinite	ADJ
ap-2064	437	24	families	family	NOUN
ap-2064	437	25	of	of	ADP
ap-2064	437	26	polyominoes	polyominoe	NOUN
ap-2064	437	27	that	that	PRON
ap-2064	437	28	tile	tile	VERB
ap-2064	437	29	the	the	DET
ap-2064	437	30	plane	plane	NOUN
ap-2064	437	31	by	by	ADP
ap-2064	437	32	translation	translation	NOUN
ap-2064	437	33	in	in	ADP
ap-2064	437	34	two	two	NUM
ap-2064	437	35	distinct	distinct	ADJ
ap-2064	437	36	ways	way	NOUN
ap-2064	437	37	,	,	PUNCT
ap-2064	437	38	theoret	theoret	ADJ
ap-2064	437	39	.	.	PUNCT
ap-2064	438	1	comput	comput	NOUN
ap-2064	438	2	.	.	PUNCT
ap-2064	439	1	sci	sci	PROPN
ap-2064	439	2	.	.	PROPN
ap-2064	439	3	412	412	NUM
ap-2064	439	4	,	,	PUNCT
ap-2064	439	5	2011	2011	NUM
ap-2064	439	6	,	,	PUNCT
ap-2064	439	7	4778–4786	4778–4786	NUM
ap-2064	439	8	.	.	PUNCT
ap-2064	440	1	[	[	X
ap-2064	440	2	5	5	NUM
ap-2064	440	3	]	]	X
ap-2064	440	4	cassaigne	cassaigne	PROPN
ap-2064	440	5	,	,	PUNCT
ap-2064	440	6	j.	j.	PROPN
ap-2064	440	7	,	,	PUNCT
ap-2064	440	8	on	on	ADP
ap-2064	440	9	extremal	extremal	ADJ
ap-2064	440	10	properties	property	NOUN
ap-2064	440	11	of	of	ADP
ap-2064	440	12	the	the	DET
ap-2064	440	13	fibonacci	fibonacci	NOUN
ap-2064	440	14	word	word	NOUN
ap-2064	440	15	,	,	PUNCT
ap-2064	440	16	rairo	rairo	PROPN
ap-2064	440	17	theor	theor	PROPN
ap-2064	440	18	.	.	PUNCT
ap-2064	441	1	inf	inf	PROPN
ap-2064	441	2	.	.	PUNCT
ap-2064	441	3	appl	appl	PROPN
ap-2064	441	4	.	.	PROPN
ap-2064	441	5	,	,	PUNCT
ap-2064	441	6	42(4	42(4	PROPN
ap-2064	441	7	)	)	PUNCT
ap-2064	441	8	,	,	PUNCT
ap-2064	441	9	2008	2008	NUM
ap-2064	441	10	,	,	PUNCT
ap-2064	441	11	701–715	701–715	NUM
ap-2064	441	12	.	.	PUNCT
ap-2064	442	1	[	[	X
ap-2064	442	2	6	6	NUM
ap-2064	442	3	]	]	X
ap-2064	442	4	chuan	chuan	PROPN
ap-2064	442	5	,	,	PUNCT
ap-2064	442	6	w.	w.	PROPN
ap-2064	442	7	,	,	PUNCT
ap-2064	442	8	fibonacci	fibonacci	NOUN
ap-2064	442	9	words	word	NOUN
ap-2064	442	10	,	,	PUNCT
ap-2064	442	11	fibonacci	fibonacci	NOUN
ap-2064	442	12	quart	quart	NOUN
ap-2064	442	13	.	.	PUNCT
ap-2064	442	14	,	,	PUNCT
ap-2064	442	15	30(1	30(1	NUM
ap-2064	442	16	)	)	PUNCT
ap-2064	442	17	,	,	PUNCT
ap-2064	442	18	1992	1992	NUM
ap-2064	442	19	,	,	PUNCT
ap-2064	442	20	68–76	68–76	NUM
ap-2064	442	21	.	.	PUNCT
ap-2064	443	1	[	[	X
ap-2064	443	2	7	7	NUM
ap-2064	443	3	]	]	X
ap-2064	443	4	fuchs	fuch	NOUN
ap-2064	443	5	,	,	PUNCT
ap-2064	443	6	c.	c.	NOUN
ap-2064	443	7	,	,	PUNCT
ap-2064	443	8	tijdeman	tijdeman	NOUN
ap-2064	443	9	,	,	PUNCT
ap-2064	443	10	r.	r.	PROPN
ap-2064	443	11	,	,	PUNCT
ap-2064	443	12	substitutions	substitution	NOUN
ap-2064	443	13	,	,	PUNCT
ap-2064	443	14	abstract	abstract	ADJ
ap-2064	443	15	number	number	NOUN
ap-2064	443	16	systems	system	NOUN
ap-2064	443	17	and	and	CCONJ
ap-2064	443	18	the	the	DET
ap-2064	443	19	space	space	NOUN
ap-2064	443	20	filling	fill	VERB
ap-2064	443	21	property	property	NOUN
ap-2064	443	22	,	,	PUNCT
ap-2064	443	23	ann	ann	PROPN
ap-2064	443	24	.	.	PROPN
ap-2064	443	25	inst	inst	PROPN
ap-2064	443	26	.	.	PUNCT
ap-2064	444	1	fourier	fourier	PROPN
ap-2064	444	2	,	,	PUNCT
ap-2064	444	3	56(7	56(7	NUM
ap-2064	444	4	)	)	PUNCT
ap-2064	444	5	,	,	PUNCT
ap-2064	444	6	2006	2006	NUM
ap-2064	444	7	,	,	PUNCT
ap-2064	444	8	2345–2389	2345–2389	NUM
ap-2064	444	9	.	.	PUNCT
ap-2064	445	1	[	[	X
ap-2064	445	2	8	8	NUM
ap-2064	445	3	]	]	X
ap-2064	445	4	de	de	X
ap-2064	445	5	luca	luca	PROPN
ap-2064	445	6	,	,	PUNCT
ap-2064	445	7	a.	a.	PROPN
ap-2064	445	8	,	,	PUNCT
ap-2064	445	9	a	a	DET
ap-2064	445	10	division	division	NOUN
ap-2064	445	11	property	property	NOUN
ap-2064	445	12	of	of	ADP
ap-2064	445	13	the	the	DET
ap-2064	445	14	fibonacci	fibonacci	NOUN
ap-2064	445	15	word	word	NOUN
ap-2064	445	16	,	,	PUNCT
ap-2064	445	17	inform	inform	NOUN
ap-2064	445	18	.	.	PUNCT
ap-2064	445	19	process	process	NOUN
ap-2064	445	20	.	.	PUNCT
ap-2064	446	1	lett	lett	PROPN
ap-2064	446	2	.	.	PROPN
ap-2064	446	3	,	,	PUNCT
ap-2064	446	4	54	54	NUM
ap-2064	446	5	,	,	PUNCT
ap-2064	446	6	1995	1995	NUM
ap-2064	446	7	,	,	PUNCT
ap-2064	446	8	307–312	307–312	NUM
ap-2064	446	9	.	.	PUNCT
ap-2064	447	1	[	[	X
ap-2064	447	2	9	9	NUM
ap-2064	447	3	]	]	X
ap-2064	447	4	de	de	X
ap-2064	447	5	luca	luca	PROPN
ap-2064	447	6	,	,	PUNCT
ap-2064	447	7	a.	a.	PROPN
ap-2064	447	8	,	,	PUNCT
ap-2064	447	9	mignosi	mignosi	PROPN
ap-2064	447	10	,	,	PUNCT
ap-2064	447	11	f.	f.	PROPN
ap-2064	447	12	,	,	PUNCT
ap-2064	447	13	some	some	DET
ap-2064	447	14	combinatorial	combinatorial	ADJ
ap-2064	447	15	properties	property	NOUN
ap-2064	447	16	of	of	ADP
ap-2064	447	17	sturmian	sturmian	NOUN
ap-2064	447	18	words	word	NOUN
ap-2064	447	19	,	,	PUNCT
ap-2064	447	20	theoret	theoret	ADJ
ap-2064	447	21	.	.	PUNCT
ap-2064	448	1	comput	comput	NOUN
ap-2064	448	2	.	.	PUNCT
ap-2064	449	1	sci	sci	PROPN
ap-2064	449	2	.	.	PROPN
ap-2064	449	3	136	136	NUM
ap-2064	449	4	,	,	PUNCT
ap-2064	449	5	1994	1994	NUM
ap-2064	449	6	,	,	PUNCT
ap-2064	449	7	361–385	361–385	NUM
ap-2064	449	8	.	.	PUNCT
ap-2064	450	1	[	[	X
ap-2064	450	2	10	10	NUM
ap-2064	450	3	]	]	PUNCT
ap-2064	450	4	droubay	droubay	ADJ
ap-2064	450	5	x.	x.	PROPN
ap-2064	450	6	,	,	PUNCT
ap-2064	450	7	palindromes	palindrome	NOUN
ap-2064	450	8	in	in	ADP
ap-2064	450	9	the	the	DET
ap-2064	450	10	fibonacci	fibonacci	NOUN
ap-2064	450	11	word	word	NOUN
ap-2064	450	12	,	,	PUNCT
ap-2064	450	13	inform	inform	NOUN
ap-2064	450	14	.	.	PUNCT
ap-2064	450	15	process	process	NOUN
ap-2064	450	16	.	.	PUNCT
ap-2064	451	1	lett	lett	PROPN
ap-2064	451	2	.	.	PROPN
ap-2064	451	3	,	,	PUNCT
ap-2064	451	4	55	55	NUM
ap-2064	451	5	,	,	PUNCT
ap-2064	451	6	1995	1995	NUM
ap-2064	451	7	,	,	PUNCT
ap-2064	451	8	217–221	217–221	NUM
ap-2064	451	9	.	.	PUNCT
ap-2064	452	1	[	[	X
ap-2064	452	2	11	11	NUM
ap-2064	452	3	]	]	SYM
ap-2064	452	4	damanik	damanik	X
ap-2064	452	5	,	,	PUNCT
ap-2064	452	6	d.	d.	PROPN
ap-2064	452	7	,	,	PUNCT
ap-2064	452	8	lenz	lenz	PROPN
ap-2064	452	9	,	,	PUNCT
ap-2064	452	10	d.	d.	PROPN
ap-2064	452	11	,	,	PUNCT
ap-2064	452	12	the	the	DET
ap-2064	452	13	index	index	NOUN
ap-2064	452	14	of	of	ADP
ap-2064	452	15	sturmian	sturmian	ADJ
ap-2064	452	16	sequences	sequence	NOUN
ap-2064	452	17	,	,	PUNCT
ap-2064	452	18	european	european	PROPN
ap-2064	452	19	j.	j.	PROPN
ap-2064	452	20	combin	combin	PROPN
ap-2064	452	21	.	.	PROPN
ap-2064	452	22	,	,	PUNCT
ap-2064	452	23	23(1	23(1	X
ap-2064	452	24	)	)	PUNCT
ap-2064	452	25	,	,	PUNCT
ap-2064	452	26	2002	2002	NUM
ap-2064	452	27	,	,	PUNCT
ap-2064	452	28	23–29	23–29	NUM
ap-2064	452	29	.	.	PUNCT
ap-2064	453	1	[	[	X
ap-2064	453	2	12	12	NUM
ap-2064	453	3	]	]	X
ap-2064	453	4	edson	edson	PROPN
ap-2064	453	5	,	,	PUNCT
ap-2064	453	6	m.	m.	NOUN
ap-2064	453	7	,	,	PUNCT
ap-2064	453	8	yayenie	yayenie	PROPN
ap-2064	453	9	,	,	PUNCT
ap-2064	453	10	o.	o.	PROPN
ap-2064	453	11	,	,	PUNCT
ap-2064	453	12	a	a	DET
ap-2064	453	13	new	new	ADJ
ap-2064	453	14	generalization	generalization	NOUN
ap-2064	453	15	of	of	ADP
ap-2064	453	16	fibonacci	fibonacci	NOUN
ap-2064	453	17	sequence	sequence	NOUN
ap-2064	453	18	and	and	CCONJ
ap-2064	453	19	extended	extended	ADJ
ap-2064	453	20	binet	binet	NOUN
ap-2064	453	21	’s	’s	PART
ap-2064	453	22	formula	formula	NOUN
ap-2064	453	23	,	,	PUNCT
ap-2064	453	24	integers	integer	NOUN
ap-2064	453	25	,	,	PUNCT
ap-2064	453	26	9(6	9(6	NUM
ap-2064	453	27	)	)	PUNCT
ap-2064	453	28	,	,	PUNCT
ap-2064	453	29	2009	2009	NUM
ap-2064	453	30	,	,	PUNCT
ap-2064	453	31	639–654	639–654	NUM
ap-2064	453	32	.	.	PUNCT
ap-2064	454	1	[	[	X
ap-2064	454	2	13	13	NUM
ap-2064	454	3	]	]	X
ap-2064	454	4	koshy	koshy	ADJ
ap-2064	454	5	,	,	PUNCT
ap-2064	454	6	t.	t.	PROPN
ap-2064	454	7	,	,	PUNCT
ap-2064	454	8	fibonacci	fibonacci	PROPN
ap-2064	454	9	and	and	CCONJ
ap-2064	454	10	lucas	lucas	PROPN
ap-2064	454	11	numbers	number	NOUN
ap-2064	454	12	with	with	ADP
ap-2064	454	13	applications	application	NOUN
ap-2064	454	14	,	,	PUNCT
ap-2064	454	15	a	a	DET
ap-2064	454	16	wiley	wiley	NOUN
ap-2064	454	17	-	-	PUNCT
ap-2064	454	18	interscience	interscience	NOUN
ap-2064	454	19	publication	publication	NOUN
ap-2064	454	20	,	,	PUNCT
ap-2064	454	21	2001	2001	NUM
ap-2064	454	22	.	.	PUNCT
ap-2064	455	1	[	[	X
ap-2064	455	2	14	14	NUM
ap-2064	455	3	]	]	SYM
ap-2064	455	4	lothaire	lothaire	NOUN
ap-2064	455	5	,	,	PUNCT
ap-2064	455	6	m.	m.	NOUN
ap-2064	455	7	,	,	PUNCT
ap-2064	455	8	algebraic	algebraic	ADJ
ap-2064	455	9	combinatorics	combinatoric	NOUN
ap-2064	455	10	on	on	ADP
ap-2064	455	11	words	word	NOUN
ap-2064	455	12	,	,	PUNCT
ap-2064	455	13	encyclopedia	encyclopedia	NOUN
ap-2064	455	14	of	of	ADP
ap-2064	455	15	mathematics	mathematic	NOUN
ap-2064	455	16	and	and	CCONJ
ap-2064	455	17	its	its	PRON
ap-2064	455	18	applications	application	NOUN
ap-2064	455	19	,	,	PUNCT
ap-2064	455	20	cambridge	cambridge	PROPN
ap-2064	455	21	university	university	PROPN
ap-2064	455	22	press	press	PROPN
ap-2064	455	23	,	,	PUNCT
ap-2064	455	24	cambridge	cambridge	PROPN
ap-2064	455	25	,	,	PUNCT
ap-2064	455	26	2002	2002	NUM
ap-2064	455	27	.	.	PUNCT
ap-2064	456	1	[	[	X
ap-2064	456	2	15	15	NUM
ap-2064	456	3	]	]	X
ap-2064	456	4	mignosi	mignosi	NOUN
ap-2064	456	5	,	,	PUNCT
ap-2064	456	6	f.	f.	PROPN
ap-2064	456	7	,	,	PUNCT
ap-2064	456	8	pirillo	pirillo	PROPN
ap-2064	456	9	,	,	PUNCT
ap-2064	456	10	g.	g.	PROPN
ap-2064	456	11	,	,	PUNCT
ap-2064	456	12	repetitions	repetition	NOUN
ap-2064	456	13	in	in	ADP
ap-2064	456	14	the	the	DET
ap-2064	456	15	fibonacci	fibonacci	NOUN
ap-2064	456	16	infinite	infinite	PROPN
ap-2064	456	17	word	word	NOUN
ap-2064	456	18	,	,	PUNCT
ap-2064	456	19	rairo	rairo	NOUN
ap-2064	456	20	inform	inform	NOUN
ap-2064	456	21	.	.	PUNCT
ap-2064	457	1	theor	theor	PROPN
ap-2064	457	2	.	.	PUNCT
ap-2064	457	3	appl	appl	PROPN
ap-2064	457	4	.	.	PROPN
ap-2064	457	5	,	,	PUNCT
ap-2064	457	6	26	26	NUM
ap-2064	457	7	,	,	PUNCT
ap-2064	457	8	1992	1992	NUM
ap-2064	457	9	,	,	PUNCT
ap-2064	457	10	199–204	199–204	NUM
ap-2064	457	11	.	.	PUNCT
ap-2064	458	1	[	[	X
ap-2064	458	2	16	16	NUM
ap-2064	458	3	]	]	PUNCT
ap-2064	458	4	monnerot	monnerot	NOUN
ap-2064	458	5	-	-	PUNCT
ap-2064	458	6	dumaine	dumaine	NOUN
ap-2064	458	7	,	,	PUNCT
ap-2064	458	8	a.	a.	NOUN
ap-2064	458	9	,	,	PUNCT
ap-2064	458	10	the	the	DET
ap-2064	458	11	fibonacci	fibonacci	NOUN
ap-2064	458	12	word	word	NOUN
ap-2064	458	13	fractal	fractal	NOUN
ap-2064	458	14	,	,	PUNCT
ap-2064	458	15	preprint	preprint	NOUN
ap-2064	458	16	,	,	PUNCT
ap-2064	458	17	http	http	PROPN
ap-2064	458	18	:	:	PUNCT
ap-2064	458	19	//hal.archives	//hal.archive	NOUN
ap-2064	458	20	-	-	PUNCT
ap-2064	458	21	ouvertes.fr	ouvertes.fr	PROPN
ap-2064	458	22	/	/	SYM
ap-2064	458	23	hal-00367972	hal-00367972	NOUN
ap-2064	458	24	/	/	SYM
ap-2064	458	25	fr/	fr/	NOUN
ap-2064	458	26	,	,	PUNCT
ap-2064	458	27	2009	2009	NUM
ap-2064	458	28	[	[	X
ap-2064	458	29	2014	2014	NUM
ap-2064	458	30	-	-	SYM
ap-2064	458	31	12	12	NUM
ap-2064	458	32	-	-	SYM
ap-2064	458	33	01	01	NUM
ap-2064	458	34	]	]	PUNCT
ap-2064	458	35	.	.	PUNCT
ap-2064	459	1	[	[	X
ap-2064	459	2	17	17	NUM
ap-2064	459	3	]	]	X
ap-2064	459	4	pirillo	pirillo	PROPN
ap-2064	459	5	,	,	PUNCT
ap-2064	459	6	g.	g.	PROPN
ap-2064	459	7	,	,	PUNCT
ap-2064	459	8	fibonacci	fibonacci	NOUN
ap-2064	459	9	numbers	number	NOUN
ap-2064	459	10	and	and	CCONJ
ap-2064	459	11	words	word	NOUN
ap-2064	459	12	,	,	PUNCT
ap-2064	459	13	discrete	discrete	ADJ
ap-2064	459	14	math	math	NOUN
ap-2064	459	15	.	.	PUNCT
ap-2064	459	16	,	,	PUNCT
ap-2064	459	17	173	173	NUM
ap-2064	459	18	,	,	PUNCT
ap-2064	459	19	1997	1997	NUM
ap-2064	459	20	,	,	PUNCT
ap-2064	459	21	197–207	197–207	NUM
ap-2064	459	22	.	.	PUNCT
ap-2064	460	1	[	[	X
ap-2064	460	2	18	18	NUM
ap-2064	460	3	]	]	X
ap-2064	460	4	prusinkiewicz	prusinkiewicz	NOUN
ap-2064	460	5	,	,	PUNCT
ap-2064	460	6	p.	p.	NOUN
ap-2064	460	7	,	,	PUNCT
ap-2064	460	8	lindenmayer	lindenmayer	PROPN
ap-2064	460	9	,	,	PUNCT
ap-2064	460	10	a.	a.	NOUN
ap-2064	460	11	:	:	PUNCT
ap-2064	460	12	the	the	DET
ap-2064	460	13	algorithmic	algorithmic	ADJ
ap-2064	460	14	beauty	beauty	NOUN
ap-2064	460	15	of	of	ADP
ap-2064	460	16	plants	plant	NOUN
ap-2064	460	17	,	,	PUNCT
ap-2064	460	18	springer	springer	NOUN
ap-2064	460	19	-	-	PUNCT
ap-2064	460	20	verlag	verlag	PROPN
ap-2064	460	21	.	.	PUNCT
ap-2064	460	22	nueva	nueva	PROPN
ap-2064	460	23	york	york	PROPN
ap-2064	460	24	,	,	PUNCT
ap-2064	460	25	2004	2004	NUM
ap-2064	460	26	.	.	PUNCT
ap-2064	461	1	[	[	X
ap-2064	461	2	19	19	NUM
ap-2064	461	3	]	]	X
ap-2064	461	4	ramírez	ramírez	NOUN
ap-2064	461	5	,	,	PUNCT
ap-2064	461	6	j.	j.	PROPN
ap-2064	461	7	,	,	PUNCT
ap-2064	461	8	rubiano	rubiano	PROPN
ap-2064	461	9	,	,	PUNCT
ap-2064	461	10	g.	g.	PROPN
ap-2064	461	11	,	,	PUNCT
ap-2064	461	12	generating	generate	VERB
ap-2064	461	13	fractals	fractal	NOUN
ap-2064	461	14	curves	curve	NOUN
ap-2064	461	15	from	from	ADP
ap-2064	461	16	homomorphisms	homomorphism	NOUN
ap-2064	461	17	between	between	ADP
ap-2064	461	18	languages	language	NOUN
ap-2064	461	19	[	[	X
ap-2064	461	20	with	with	ADP
ap-2064	461	21	mathematicar	mathematicar	NOUN
ap-2064	461	22	]	]	PUNCT
ap-2064	461	23	(	(	PUNCT
ap-2064	461	24	in	in	ADP
ap-2064	461	25	spanish	spanish	ADJ
ap-2064	461	26	)	)	PUNCT
ap-2064	461	27	,	,	PUNCT
ap-2064	461	28	revista	revista	PROPN
ap-2064	461	29	integración	integración	PROPN
ap-2064	461	30	30(2	30(2	NUM
ap-2064	461	31	)	)	PUNCT
ap-2064	461	32	,	,	PUNCT
ap-2064	461	33	2012	2012	NUM
ap-2064	461	34	,	,	PUNCT
ap-2064	461	35	129–150	129–150	NUM
ap-2064	461	36	.	.	PUNCT
ap-2064	462	1	[	[	X
ap-2064	462	2	20	20	NUM
ap-2064	462	3	]	]	X
ap-2064	462	4	ramírez	ramírez	NOUN
ap-2064	462	5	,	,	PUNCT
ap-2064	462	6	j.	j.	PROPN
ap-2064	462	7	,	,	PUNCT
ap-2064	462	8	rubiano	rubiano	PROPN
ap-2064	462	9	,	,	PUNCT
ap-2064	462	10	g.	g.	PROPN
ap-2064	462	11	,	,	PUNCT
ap-2064	462	12	on	on	ADP
ap-2064	462	13	the	the	DET
ap-2064	462	14	k	k	PROPN
ap-2064	462	15	-	-	PUNCT
ap-2064	462	16	fibonacci	fibonacci	NOUN
ap-2064	462	17	words	word	NOUN
ap-2064	462	18	,	,	PUNCT
ap-2064	462	19	acta	acta	PROPN
ap-2064	462	20	univ	univ	PROPN
ap-2064	462	21	.	.	PUNCT
ap-2064	462	22	sapientiae	sapientiae	PROPN
ap-2064	462	23	infor	infor	PROPN
ap-2064	462	24	.	.	PROPN
ap-2064	462	25	,	,	PUNCT
ap-2064	462	26	5(2	5(2	NUM
ap-2064	462	27	)	)	PUNCT
ap-2064	462	28	,	,	PUNCT
ap-2064	462	29	2013	2013	NUM
ap-2064	462	30	,	,	PUNCT
ap-2064	462	31	212–226	212–226	NUM
ap-2064	462	32	.	.	PUNCT
ap-2064	463	1	[	[	X
ap-2064	463	2	21	21	NUM
ap-2064	463	3	]	]	X
ap-2064	463	4	ramírez	ramírez	NOUN
ap-2064	463	5	,	,	PUNCT
ap-2064	463	6	j.	j.	PROPN
ap-2064	463	7	,	,	PUNCT
ap-2064	463	8	rubiano	rubiano	PROPN
ap-2064	463	9	,	,	PUNCT
ap-2064	463	10	g.	g.	PROPN
ap-2064	463	11	,	,	PUNCT
ap-2064	463	12	properties	property	NOUN
ap-2064	463	13	and	and	CCONJ
ap-2064	463	14	generalizations	generalization	NOUN
ap-2064	463	15	of	of	ADP
ap-2064	463	16	the	the	DET
ap-2064	463	17	fibonacci	fibonacci	NOUN
ap-2064	463	18	word	word	NOUN
ap-2064	463	19	fractal	fractal	PROPN
ap-2064	463	20	.	.	PUNCT
ap-2064	464	1	exploring	explore	VERB
ap-2064	464	2	fractal	fractal	ADJ
ap-2064	464	3	curves	curve	NOUN
ap-2064	464	4	,	,	PUNCT
ap-2064	464	5	the	the	DET
ap-2064	464	6	mathematica	mathematica	PROPN
ap-2064	464	7	journal	journal	PROPN
ap-2064	464	8	,	,	PUNCT
ap-2064	464	9	16	16	NUM
ap-2064	464	10	,	,	PUNCT
ap-2064	464	11	2014	2014	NUM
ap-2064	464	12	.	.	PUNCT
ap-2064	465	1	[	[	X
ap-2064	465	2	22	22	NUM
ap-2064	465	3	]	]	X
ap-2064	465	4	ramírez	ramírez	NOUN
ap-2064	465	5	,	,	PUNCT
ap-2064	465	6	j.	j.	PROPN
ap-2064	465	7	,	,	PUNCT
ap-2064	465	8	rubiano	rubiano	PROPN
ap-2064	465	9	,	,	PUNCT
ap-2064	465	10	g.	g.	PROPN
ap-2064	465	11	,	,	PUNCT
ap-2064	465	12	de	de	X
ap-2064	465	13	castro	castro	PROPN
ap-2064	465	14	,	,	PUNCT
ap-2064	465	15	r.	r.	PROPN
ap-2064	465	16	,	,	PUNCT
ap-2064	465	17	a	a	DET
ap-2064	465	18	generalization	generalization	NOUN
ap-2064	465	19	of	of	ADP
ap-2064	465	20	the	the	DET
ap-2064	465	21	fibonacci	fibonacci	NOUN
ap-2064	465	22	word	word	NOUN
ap-2064	465	23	fractal	fractal	ADJ
ap-2064	465	24	and	and	CCONJ
ap-2064	465	25	the	the	DET
ap-2064	465	26	fibonacci	fibonacci	NOUN
ap-2064	465	27	snowflake	snowflake	NOUN
ap-2064	465	28	,	,	PUNCT
ap-2064	465	29	theoret	theoret	ADJ
ap-2064	465	30	.	.	PUNCT
ap-2064	466	1	comput	comput	NOUN
ap-2064	466	2	.	.	PUNCT
ap-2064	467	1	sci	sci	PROPN
ap-2064	467	2	.	.	PROPN
ap-2064	467	3	,	,	PUNCT
ap-2064	467	4	528	528	NUM
ap-2064	467	5	,	,	PUNCT
ap-2064	467	6	2014	2014	NUM
ap-2064	467	7	,	,	PUNCT
ap-2064	467	8	40–56	40–56	NUM
ap-2064	467	9	.	.	PUNCT
ap-2064	468	1	58	58	NUM
ap-2064	468	2	http://hal.archives-ouvertes.fr/hal-00367972/fr/	http://hal.archives-ouvertes.fr/hal-00367972/fr/	NOUN
ap-2064	468	3	http://hal.archives-ouvertes.fr/hal-00367972/fr/	http://hal.archives-ouvertes.fr/hal-00367972/fr/	PROPN
ap-2064	468	4	acta	acta	PROPN
ap-2064	468	5	polytechnica	polytechnica	PROPN
ap-2064	468	6	55(1):50–58	55(1):50–58	PROPN
ap-2064	468	7	,	,	PUNCT
ap-2064	468	8	2015	2015	NUM
ap-2064	468	9	1	1	NUM
ap-2064	468	10	introduction	introduction	NOUN
ap-2064	468	11	2	2	NUM
ap-2064	468	12	definitions	definition	NOUN
ap-2064	468	13	and	and	CCONJ
ap-2064	468	14	notation	notation	NOUN
ap-2064	468	15	3	3	NUM
ap-2064	468	16	biperiodic	biperiodic	ADJ
ap-2064	468	17	fibonacci	fibonacci	NOUN
ap-2064	468	18	words	word	VERB
ap-2064	468	19	4	4	NUM
ap-2064	468	20	the	the	DET
ap-2064	468	21	biperiodic	biperiodic	ADJ
ap-2064	468	22	fibonacci	fibonacci	PROPN
ap-2064	468	23	word	word	NOUN
ap-2064	468	24	curve	curve	NOUN
ap-2064	468	25	acknowledgements	acknowledgement	NOUN
ap-2064	468	26	references	reference	NOUN