cp's OEIS Frontend

A205963 a(n) = Fibonacci(n)*A000118(n) for n>=1 with a(0)=1, where A000118(n) is the number of ways of writing n as a sum of 4 squares.

%I A205963 #20 Mar 09 2017 05:25:08
%S A205963 1,8,24,64,72,240,768,832,504,3536,7920,8544,13824,26096,72384,117120,
%T A205963 23688,229968,806208,668960,974160,2802176,5100768,5502144,4451328,
%U A205963 18606200,40788048,62853760,61019712,123414960,479255040,344644864,52279416,1353437952,2463647184
%N A205963 a(n) = Fibonacci(n)*A000118(n) for n>=1 with a(0)=1, where A000118(n) is the number of ways of writing n as a sum of 4 squares.
%C A205963 Compare g.f. to the Lambert series of A000118: 1 + 8*Sum_{n>=1} n*x^n/(1 + (-x)^n).
%H A205963 G. C. Greubel, <a href="/A205963/b205963.txt">Table of n, a(n) for n = 0..1000</a>
%F A205963 G.f.: 1 + 8*Sum_{n>=1} Fibonacci(n)*n*x^n/(1 + Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
%e A205963 G.f.: A(x) = 1 + 8*x + 24*x^2 + 64*x^3 + 72*x^4 + 240*x^5 + 768*x^6 +...
%e A205963 where A(x) = 1 + 1*8*x + 1*24*x^2 + 2*32*x^3 + 3*24*x^4 + 5*48*x^5 + 8*96*x^6 + 13*64*x^7 + 21*24*x^8 +...+ Fibonacci(n)*A000118(n)*x^n +...
%e A205963 The g.f. is also given by the identity:
%e A205963 A(x) = 1 + 8*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1+3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1+7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1+18*x^6+x^12) + 13*7*x^7/(1-29*x^7-x^14) +...).
%t A205963 Join[{1}, Table[Fibonacci[n]*SquaresR[4, n], {n,1,50}]] (* _G. C. Greubel_, Mar 09 2017 *)
%o A205963 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205963 {a(n)=polcoeff(1+8*sum(m=1,n,fibonacci(m)*m*x^m/(1+Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%o A205963 for(n=0,31,print1(a(n),", "))
%Y A205963 Cf. A000118, A205507, A205964, A203847, A000204 (Lucas).
%Y A205963 Cf. A209443 (Pell variant).
%K A205963 nonn
%O A205963 0,2
%A A205963 _Paul D. Hanna_, Feb 03 2012