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A205507 a(n) = Fibonacci(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

%I A205507 #20 Mar 05 2017 17:05:01
%S A205507 1,4,4,0,12,40,0,0,84,136,440,0,0,1864,0,0,3948,12776,10336,0,54120,0,
%T A205507 0,0,0,900300,971144,0,0,4113832,0,0,8713236,0,45623096,0,59721408,
%U A205507 193262536,0,0,818673240,1324641128,0,0,0,9079225360,0,0,0,31114968196
%N A205507 a(n) = Fibonacci(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.
%C A205507 Compare to the g.f. of A004018 given by the Lambert series identity:
%C A205507 1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.
%H A205507 G. C. Greubel, <a href="/A205507/b205507.txt">Table of n, a(n) for n = 0..1000</a>
%F A205507 G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Fibonacci(2*n+1)*x^(2*n+1) / (1 - Lucas(2*n+1)*x^(2*n+1) - x^(4*n+2)), where Lucas(n) = A000204(n).
%e A205507 G.f.: A(x) = 1 + 4*x + 4*x^2 + 12*x^4 + 40*x^5 + 84*x^8 + 136*x^9 + 440*x^10 +...
%e A205507 Compare the g.f to the square of the Jacobi theta_3 series:
%e A205507 theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...
%e A205507 The g.f. equals the sum:
%e A205507 A(x) = 1 + 4*x/(1-x-x^2) - 4*2*x^3/(1-4*x^3-x^6) + 4*5*x^5/(1-11*x^5-x^10) - 4*13*x^7/(1-29*x^7-x^14) + 4*34*x^9/(1-76*x^9-x^18) - 4*89*x^11/(1-199*x^11-x^22) + 4*233*x^13/(1-521*x^13-x^26) - 4*610*x^15/(1-1364*x^15-x^30) +...
%e A205507 which involves odd-indexed Fibonacci and Lucas numbers.
%t A205507 Join[{1}, Table[Fibonacci[n]*SquaresR[2, n], {n,1,50}]] (* _G. C. Greubel_, Mar 05 2017 *)
%o A205507 (PARI) {A004018(n)=polcoeff((1+2*sum(k=1,sqrtint(n+1),x^(k^2),x*O(x^n)))^2,n)}
%o A205507 {a(n)=if(n==0,1,fibonacci(n)*A004018(n))}
%o A205507 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205507 {a(n)=polcoeff((1+4*sum(m=0,n+1,(-1)^m*fibonacci(2*m+1)*x^(2*m+1)/(1-Lucas(2*m+1)*x^(2*m+1)-x^(4*m+2)+x*O(x^n)))),n)}
%Y A205507 Cf. A205508, A204060, A203847, A004018, A000204 (Lucas).
%K A205507 nonn
%O A205507 0,2
%A A205507 _Paul D. Hanna_, Jan 28 2012