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%I #8 Apr 30 2014 01:37:05
%S 1,1,1,2,2,4,5,10,12,23,31,58,79,145,207,374,540,964,1427,2522,3775,
%T 6626,10050,17532,26811,46561,71795,124188,192661,332228,518303,
%U 891340,1396902,2396912,3771822,6459202,10199912,17437727,27622807,47152952
%N G.f. A(x) satisfies both A(-x)*A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x*(1+x)/(1-x).
%C This is a special case of sequences with g.f.s that satisfy the more general functional equation xA(x)^m = B(xA(x^m)) originated by Michael Somos; some other examples are A085748, A091190 and A091200.
%F Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2)) were f(u, v) = u^2 * (1 - v) - v * (1 + v). - _Michael Somos_, Aug 02 2011
%e 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 10*x^7 + 12*x^8 + 23*x^9 + ...
%e q + q^3 + q^5 + 2*q^7 + 2*q^9 + 4*q^11 + 5*q^13 + 10*q^15 + 12*q^17 + ...
%o (PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=2; A = x * subst(A, x, x^2); A = (A *(1 + A) /(1 - A) / x)^(1/2)); polcoeff(A, n))}
%Y Cf. A085748, A091190, A091200, A092869.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Feb 22 2004
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