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

%I A226729 #45 Nov 28 2023 11:06:32
%S A226729 1,1,2,4,8,17,36,76,162,345,734,1564,3332,7098,15124,32224,68658,
%T A226729 146291,311704,664152,1415124,3015237,6424636,13689132,29167776,
%U A226729 62148513,132421414,282153672,601192008,1280975135,2729406380,5815615784,12391480916,26402844538,56257214530,119868682488
%N A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ).
%C A226729 What does this sequence count?
%C A226729 Conjectures from _John Tyler Rascoe_, Nov 04 2023: (Start)
%C A226729 a(n) is the number of integer compositions of n into two kinds of odd parts with the following restrictions. Each composition has first part 1a. For all a parts pa_i >= px_{i+1} and for all b parts pb_i >= px_{i+1} or pb_i = (p+2)a_{i+1}.
%C A226729 In general if B(i) = b_1, b_2, ..., b_i is an infinite sequence where b_1 > 0 and b_i <= b_{i+1} for all i, let A(q) = 1/(1-q^b_1/(1-q^b_2/(1-q^b_3/(1-...)))) be a generating function where the exponents of q are the sequence B(i).
%C A226729 Then A(q) counts integer compositions into parts b_i with the following restrictions. Every composition has first part p_1 = b_1 and for every pair of parts (p_j,p_{j+1}), B^-1(p_j) + 1 >= B^-1(p_{j+1}). Where j is the position of the part p_j within the composition itself and B^-1(p_j) is the index of p_j in B(i). (End)
%H A226729 Vaclav Kotesovec, <a href="/A226729/b226729.txt">Table of n, a(n) for n = 0..1000</a>
%F A226729 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 A226729 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 A226729 a(n) ~ c * d^n, where d = 2.13072551790181698200128321720925945740967671226348407873633962907725871... and c = 0.38040216799237980431596440625527448705929594287571043849218282414099437... - _Vaclav Kotesovec_, Sep 05 2017
%F A226729 Conjecture: a(n) = Sum_{i=0..floor((n-sqrt(2*n-1))/2)} A129183(n-(2*i),n-i). - _John Tyler Rascoe_, Nov 04 2023
%t A226729 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 A226729 (PARI) N = 66;  q = 'q + O('q^N);
%o A226729 G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+2) ) );
%o A226729 gf = 1 / G(0)
%o A226729 Vec(gf)
%Y A226729 Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
%Y A226729 Cf. A227309 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
%Y A226729 Cf. A129183.
%K A226729 nonn
%O A226729 0,3
%A A226729 _Joerg Arndt_, Jun 29 2013