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A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

%I A227045 #30 Sep 05 2017 04:19:37
%S A227045 1,1,2,5,13,35,95,260,713,1959,5386,14815,40759,112151,308609,849240,
%T A227045 2337009,6431246,17698332,48704714,134032593,368850417,1015056867,
%U A227045 2793383746,7687248186,21154913043,58217239536,160210872557,440892153268,1213312738702,3338974845151,9188688696438
%N A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).
%H A227045 Vaclav Kotesovec, <a href="/A227045/b227045.txt">Table of n, a(n) for n = 0..1000</a>
%F A227045 G.f.: 1/(1-q/ (1-q/(1-q/ (1-q^2/(1-q^2/ (1-q^3/(1-q^3/ (1-q^4/(1-q^4/ (1-q^5/(1-q^5/ (1-...))))))))))) ).
%F A227045 G.f. A(x) = 1/(1 - B(x)) where B(x) is the g.f. of A006958.
%F A227045 a(n) ~ c * d^n, where d = 2.751949072495748078279227332764623096815571855905843246297955690122791154... and c = 0.215973947378529032758849789768859077066690378163074586384819930605436492... - _Vaclav Kotesovec_, Sep 05 2017
%t A227045 nmax = 40; CoefficientList[Series[1/(1 - x/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 2] - Floor[Range[nmax + 2]/2])]]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 05 2017 *)
%o A227045 (PARI) N = 66;  q = 'q + O('q^N);
%o A227045 G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ) );
%o A227045 gf = 1 /(1- q/G(0));
%o A227045 Vec(gf)
%Y A227045 Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
%Y A227045 Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
%Y A227045 Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
%Y A227045 Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
%Y A227045 Cf. A227310 (g.f.: 1/G(0), where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).
%K A227045 nonn
%O A227045 0,3
%A A227045 _Joerg Arndt_, Jul 06 2013