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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2012
STATUS
approved