A074937 -id:A074937

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A066932

a(n) is the denominator of b(n) where b(n)=1/b(n-1)+1/b(n-2) with b(1)=1 and b(2)=2.

+10
4

1, 1, 2, 6, 21, 224, 10848, 4843293, 98262557120, 989063619297120960, 197348115975871052843094930213, 380244324677612882673067751880150651746235378560

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Limit_{n->oo} b(n)=sqrt(2) with geometric convergence since abs(b(n)-sqrt(2))<2*2^(-n/2)

LINKS

Table of n, a(n) for n=1..12.

FORMULA

a(n+1) = A057677(n)*A057677(n-1). - Benoit Cloitre, Oct 25 2005

a(n) is the numerator of c(n) where c(n)=1/(c(n-1)+c(n-2)) with c(0)=c(1)=1. - Mark Dols, Jul 17 2009

MATHEMATICA

nxt[{a_, b_}]:={b, 1/a+1/b}; NestList[nxt, {1, 2}, 20][[;; , 1]]//Denominator (* Harvey P. Dale, Apr 02 2024 *)

CROSSREFS

Cf. A057677 (numerator), A074937, A162924, A162926.

KEYWORD

nonn,frac

AUTHOR

Zak Seidov, Oct 24 2002

EXTENSIONS

Edited by Benoit Cloitre, Oct 25 2005

STATUS

approved

A335617

Let c(1) = c(2) = 0, c(3) = 1, and c(n + 3) = (c(n) - 2*c(n + 1) + c(n + 2))/n, then a(n) = ceiling (c(n)).

+10
0

0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Conjectured quasiperiodicity with autocorrelation function R(x) = 1/2 if x = 0, 1/4 if x > 0.

Some other proved or conjectured (or suspected) nonperiodic binary sequences where there are no more than two consecutive 0's or 1's include: A083035, A285305, A190843, A286059, A288213, A288551, A288473, A176405, A188321, A188398, A191162, A272170, A197879, A078588, A272532, A273129, A074937, A188297, A289128. Others?

LINKS

Table of n, a(n) for n=0..127.

MATHEMATICA

c[n_]:=c[n]=(c[n-1]-2c[n-2]+c[n-3])/n;

c[1] = 0; c[2] = 0; c[3] = 1;

Table[Ceiling@c[j], {j, 1, 2^7}]

CROSSREFS

Cf. A083035, A285305, A190843, A286059, A288213, A288551, A288473, A176405, A188321, A188398, A191162, A272170, A197879, A078588, A272532, A273129,A074937, A188297, A289128.

KEYWORD

nonn

AUTHOR

Andres Cicuttin, Oct 11 2020

STATUS

approved

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