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A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ). 5

%I

%S 1,1,2,4,8,17,36,76,162,345,734,1564,3332,7098,15124,32224,68658,

%T 146291,311704,664152,1415124,3015237,6424636,13689132,29167776,

%U 62148513,132421414,282153672,601192008,1280975135,2729406380,5815615784,12391480916,26402844538,56257214530,119868682488

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

%C What does this sequence count?

%H Vaclav Kotesovec, <a href="/A226729/b226729.txt">Table of n, a(n) for n = 0..1000</a>

%F 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-...))))))))).

%F 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

%F a(n) ~ c * d^n, where d = 2.13072551790181698200128321720925945740967671226348407873633962907725871... and c = 0.38040216799237980431596440625527448705929594287571043849218282414099437... - _Vaclav Kotesovec_, Sep 05 2017

%t 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 *)

%o (PARI) N = 66; q = 'q + O('q^N);

%o G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+2) ) );

%o gf = 1 / G(0)

%o Vec(gf)

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

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

%K nonn

%O 0,3

%A _Joerg Arndt_, Jun 29 2013

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Last modified October 14 04:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)