A205968 a(n) = Fibonacci(n)*A008655(n) for n >= 1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x) + theta_2(x)*theta_2(3*x))^4.

1, 24, 216, 1776, 5256, 15120, 63936, 107328, 294840, 823344, 1496880, 2845152, 9334656, 12291216, 28012608, 68251680, 110883528, 188343792, 563167296, 688359840, 1493387280, 3343696512, 5095667232, 8368761024, 24087248640, 28361250600, 57633511824, 128471514096

Offset: 0

Author

Paul D. Hanna, Feb 04 2012

Keywords

• nonn

Comments

Compare g.f. to the Lambert series of A008655:

1 + Sum_{n>=1} 24*n^3*x^n/(1-x^n) + 8*(3*n)^3*x^(3*n)/(1-x^(3*n)).

Examples

			G.f.: A(x) = 1 + 24*x + 216*x^2 + 1776*x^3 + 5256*x^4 + 15120*x^5 + ...
where A(x) = 1 + 1*24*x + 1*216*x^2 + 2*888*x^3 + 3*1752*x^4 + 5*3024*x^5 + ... + Fibonacci(n)*A008655(n)*x^n + ...
		

Links

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

Crossrefs

Cf. A008655, A205967, A205969, A203847, A000204 (Lucas).

Cf. A209448 (Pell variant).

Programs

Mathematica

A008655 := CoefficientList[Series[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, 250}], q]; b := Table[A008655[[n]], {n, 1, 120}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jul 16 2018 *)

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + sum(m=1,n, 24*fibonacci(m)*m^3*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*fibonacci(3*m)*(3*m)^3*x^(3*m)/(1-Lucas(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n)) ),n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: 1 + Sum_{n>=1} 24*Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) + 8*Fibonacci(3*n)*(3*n)^3*x^(3*n)/(1 - Lucas(3*n)*x^(3*n) + (-1)^n*x^(6*n)).