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A238438 Expansion of 1/G(0) where G(k) = 1 - q/(1 - q - q^3 / G(k+1) ).

%I A238438 #21 Jun 17 2020 08:55:00
%S A238438 1,1,2,4,9,21,50,121,297,738,1853,4694,11982,30790,79586,206786,
%T A238438 539784,1414905,3722776,9828501,26028969,69129150,184076913,491340306,
%U A238438 1314412198,3523519135,9463563168,25462981484,68626114915,185246103584,500779373140,1355636896041,3674558399538,9972405246294,27095580261125
%N A238438 Expansion of 1/G(0) where G(k) = 1 - q/(1 - q - q^3 / G(k+1) ).
%C A238438 What does this sequence count?
%H A238438 G. C. Greubel, <a href="/A238438/b238438.txt">Table of n, a(n) for n = 0..1000</a>
%F A238438 From _Vaclav Kotesovec_, Mar 01 2014: (Start)
%F A238438 G.f.: 2*(1-x)/(1 - 2*x + x^3 + sqrt(1 - 4*x + 4*x^2 - 2*x^3 + x^6)).
%F A238438 D-finite with Recurrence: (n+3)*a(n) = 2*(2*n+3)*a(n-1) - 4*n*a(n-2) + (2*n-3)*a(n-3) - (n-6)*a(n-6).
%F A238438 a(n) ~ (6*r^2+14*r+17) * sqrt(7*r-2) / (2 * sqrt(Pi) * n^(3/2) * r^(n-1/2)), where r = 1/3*(-2 - 2*(2/(47 + 3*sqrt(249)))^(1/3) + (1/2*(47 + 3*sqrt(249)))^(1/3)) = 0.3532099641993244294831... is the root of the equation r^3 + 2*r^2 + 2*r = 1.
%F A238438 (End)
%F A238438 G.f. A(q) satisfies 0 = -q^3*A(q)^2 + (q^3 - 2*q + 1)*A(q) + (q - 1).
%t A238438 CoefficientList[Series[2*(1-x)/(1 - 2*x + x^3 + Sqrt[1 - 4*x + 4*x^2 - 2*x^3 + x^6]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 01 2014 *)
%o A238438 (PARI) N = 66;  q = 'q + O('q^N);
%o A238438 G(k) = if(k>N, 1,  1 - q/(1 - q - q^3 / G(k+1) ) );
%o A238438 Vec( 1/G(0) )
%Y A238438 Cf. A086581: 1/G(0) where G(k) = 1 - q/(1 - q - q^2 / G(k+1) ).
%Y A238438 Cf. A119370: 1/G(0) where G(k) = 1 - q/(1 - (q + q^2) / G(k+1) ).
%K A238438 nonn
%O A238438 0,3
%A A238438 _Joerg Arndt_, Feb 27 2014