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A205964 a(n) = Fibonacci(n)*A000143(n) for n>=1 with a(0)=1, where A000143(n) is the number of ways of writing n as a sum of 8 squares.

%I A205964 #22 Aug 03 2025 03:21:26
%S A205964 1,16,112,896,3408,10080,25088,71552,195888,411808,776160,1896768,
%T A205964 4580352,8194144,14525056,34433280,73890768,125562528,219081856,
%U A205964 458906560,968315040,1686909952,2642197824,5579174016,12110579712,18907500400,29884043168,64236542720
%N A205964 a(n) = Fibonacci(n)*A000143(n) for n>=1 with a(0)=1, where A000143(n) is the number of ways of writing n as a sum of 8 squares.
%C A205964 Compare g.f. to the Lambert series of A000143: 1 + 16*Sum_{n>=1} n^3*x^n/(1 - (-x)^n).
%H A205964 G. C. Greubel, <a href="/A205964/b205964.txt">Table of n, a(n) for n = 0..1000</a>
%F A205964 G.f.: 1 + 16*Sum_{n>=1} Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
%e A205964 G.f.: A(x) = 1 + 16*x + 112*x^2 + 896*x^3 + 3408*x^4 + 10080*x^5 +...
%e A205964 where A(x) = 1 + 1*16*x + 1*112*x^2 + 2*448*x^3 + 3*1136*x^4 + 5*2016*x^5 + 8*3136*x^6 + 13*5504*x^7 + 21*9328*x^8 +...+ Fibonacci(n)*A000143(n)*x^n +...
%e A205964 The g.f. is also given by the identity:
%e A205964 A(x) = 1 + 16*( 1*1*x/(1+x-x^2) + 1*8*x^2/(1-3*x^2+x^4) + 2*27*x^3/(1+4*x^3-x^6) + 3*64*x^4/(1-7*x^4+x^8) + 5*125*x^5/(1+11*x^5-x^10) + 8*216*x^6/(1-18*x^6+x^12) + 13*343*x^7/(1+29*x^7-x^14) +...).
%t A205964 Join[{1}, Table[Fibonacci[n]*SquaresR[8, n], {n,1,30}]] (* _G. C. Greubel_, Mar 05 2017 *)
%o A205964 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205964 {a(n)=polcoeff(1+16*sum(m=1,n,fibonacci(m)*m^3*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%o A205964 for(n=0,31,print1(a(n),", "))
%Y A205964 Cf. A000143, A205507, A205963, A203847, A000204 (Lucas).
%Y A205964 Cf. A209444 (Pell variant).
%K A205964 nonn
%O A205964 0,2
%A A205964 _Paul D. Hanna_, Feb 03 2012