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 A205966 a(n) = Fibonacci(n)*A004016(n) for n>=1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + x*y + y^2 = n. 7

%I #13 Mar 05 2017 17:05:22

%S 1,6,0,12,18,0,0,156,0,204,0,0,864,2796,0,0,5922,0,0,50172,0,131352,0,

%T 0,0,450150,0,1178508,3813732,0,0,16155228,0,0,0,0,89582112,289893804,

%U 0,758951832,0,0,0,5201933244,0,0,0,0,28845161856,140017356882,0,0

%N a(n) = Fibonacci(n)*A004016(n) for n>=1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + x*y + y^2 = n.

%C Compare g.f. to the Lambert series of A004016: 1 + 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - x^n).

%H G. C. Greubel, <a href="/A205966/b205966.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 + 6*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

%e G.f.: A(x) = 1 + 6*x + 12*x^3 + 18*x^4 + 156*x^7 + 204*x^9 + 864*x^12 +...

%e where A(x) = 1 + 1*6*x + 2*6*x^3 + 3*6*x^4 + 13*12*x^7 + 34*6*x^9 + 144*6*x^12 +...+ Fibonacci(n)*A004016(n)*x^n +...

%e The g.f. is also given by the identity:

%e A(x) = 1 + 6*( 1*x/(1-x-x^2) - 1*x^2/(1-3*x^2+x^4) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1-11*x^5-x^10) + 13*x^7/(1-29*x^7-x^14) - 21*x^8/(1-47*x^8-x^16) +...).

%e The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

%t A004016[n_] := SeriesCoefficient[(QPochhammer[q]^3 + 9 q QPochhammer[q^9]^3)/QPochhammer[q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n]*b[n], {n,1,50}]] (* _G. C. Greubel_, Mar 05 2017 *)

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {a(n)=polcoeff(1 + 6*sum(m=1,n,kronecker(m,3)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}

%o for(n=0,60,print1(a(n),", "))

%Y Cf. A004016, A205965, A205967, A203847, A000204 (Lucas).

%Y Cf. A209446 (Pell variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 03 2012

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Last modified September 12 11:37 EDT 2024. Contains 375851 sequences. (Running on oeis4.)