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.

1, 4, 4, 8, 60, 120, 32, 416, 1092, 136, 1320, 4272, 2880, 13048, 12064, 14640, 114492, 114984, 10336, 334480, 811800, 350272, 850128, 2751072, 2411136, 9303100, 6798008, 785672, 50849760, 61707480, 19968960, 172322432, 531507396, 169179744, 410607864

Offset

0,2

Comments

Compare g.f. to the Lambert series of A034896:

1 + 4*Sum_{n>=1} Chi(n,3)*n*x^n/(1 - (-x)^n).

Here Chi(n,3) = principal Dirichlet character of n modulo 3.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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)).

Example

G.f.: A(x) = 1 + 4*x + 4*x^2 + 8*x^3 + 60*x^4 + 120*x^5 + 32*x^6 + ...
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 + ...
The g.f. is also given by the identity:
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) + ...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0,...].

Mathematica

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 *)

Prog

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{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)}
for(n=0, 61, print1(a(n), ", "))

Crossrefs

Cf. A034896, A205882, A205967, A205970, A205972, A203847, A000204 (Lucas).

Cf. A209451 (Pell variant).

Sequence in context: A102369 A298569 A281717 * A118016 A201989 A071775

Adjacent sequences: A205968 A205969 A205970 * A205972 A205973 A205974

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2012

Status

approved