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A205971 a(n) = Fibonacci(n)*A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

%I A205971 #14 Dec 25 2017 04:04:53
%S A205971 1,4,4,8,60,120,32,416,1092,136,1320,4272,2880,13048,12064,14640,
%T A205971 114492,114984,10336,334480,811800,350272,850128,2751072,2411136,
%U A205971 9303100,6798008,785672,50849760,61707480,19968960,172322432,531507396,169179744,410607864
%N A205971 a(n) = Fibonacci(n)*A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.
%C A205971 Compare g.f. to the Lambert series of A034896:
%C A205971 1 + 4*Sum_{n>=1} Chi(n,3)*n*x^n/(1 - (-x)^n).
%C A205971 Here Chi(n,3) = principal Dirichlet character of n modulo 3.
%H A205971 G. C. Greubel, <a href="/A205971/b205971.txt">Table of n, a(n) for n = 0..1000</a>
%F A205971 G.f.: 1 + 4*Sum_{n>=1} Fibonacci(n)*Chi(n,3)*n*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
%e A205971 G.f.: A(x) = 1 + 4*x + 4*x^2 + 8*x^3 + 60*x^4 + 120*x^5 + 32*x^6 + ...
%e A205971 where A(x) = 1 + 1*4*x + 1*4*x^2 + 2*4*x^3 + 3*20*x^4 + 5*24*x^5 + 8*4*x^6 + ... + Fibonacci(n)*A034896(n)*x^n + ...
%e A205971 The g.f. is also given by the identity:
%e A205971 A(x) = 1 + 4*( 1*1*x/(1+x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1+11*x^5-x^10) + 13*7*x^7/(1+29*x^7-x^14) + 21*8*x^8/(1-47*x^8+x^16) + ...).
%e A205971 The values of the Dirichlet character Chi(n,3) repeat [1,1,0,...].
%t A205971 A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Join[{1}, Table[Fibonacci[n]*A034896[n], {n, 1, 50}]] (* _G. C. Greubel_, Dec 24 2017 *)
%o A205971 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205971 {a(n)=polcoeff(1 + 4*sum(m=1,n,fibonacci(m)*kronecker(m,3)^2*m*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A205971 for(n=0,61,print1(a(n),", "))
%Y A205971 Cf. A034896, A205882, A205967, A205970, A205972, A203847, A000204 (Lucas).
%Y A205971 Cf. A209451 (Pell variant).
%K A205971 nonn
%O A205971 0,2
%A A205971 _Paul D. Hanna_, Feb 04 2012