A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ).

1, 1, 2, 4, 8, 17, 36, 76, 162, 345, 734, 1564, 3332, 7098, 15124, 32224, 68658, 146291, 311704, 664152, 1415124, 3015237, 6424636, 13689132, 29167776, 62148513, 132421414, 282153672, 601192008, 1280975135, 2729406380, 5815615784, 12391480916, 26402844538, 56257214530, 119868682488

Offset

0,3

Comments

What does this sequence count?

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1/(1-q/(1-q/(1-q^3/(1-q^3/(1-q^5/(1-q^5/(1-q^7/(1-q^7/(1-...))))))))).

G.f.: 1/W(0), where W(k)= 1 - x^(2*k+1)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013

a(n) ~ c * d^n, where d = 2.13072551790181698200128321720925945740967671226348407873633962907725871... and c = 0.38040216799237980431596440625527448705929594287571043849218282414099437... - Vaclav Kotesovec, Sep 05 2017

Mathematica

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(2*Range[nmax + 1] - 2*Floor[Range[nmax + 1]/2] - 1)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

Prog

(PARI) N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+2) ) );
gf = 1 / G(0)
Vec(gf)

Crossrefs

Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).

Cf. A227309 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).

Sequence in context: A008999 A052903 A308745 * A063457 A262735 A190162

Adjacent sequences: A226726 A226727 A226728 * A226730 A226731 A226732

Keyword

nonn

Author

Joerg Arndt, Jun 29 2013

Status

approved