A205976 a(n) = Fibonacci(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2.

1, 4, 12, 32, 84, 120, 384, 52, 1260, 1768, 3960, 4272, 16128, 13048, 4524, 58560, 122388, 114984, 403104, 334480, 1136520, 175136, 2550384, 2751072, 11128320, 9303100, 20394024, 31426880, 8898708, 61707480, 239627520, 172322432, 548933868, 676718976, 1231823592

Offset

0,2

Comments

Compare g.f. to the Lambert series of A028594:

1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).

Here Chi(n,7) = principal Dirichlet character of n modulo 7.

Links

Table of n, a(n) for n=0..34.

Formula

G.f.: 1 + 4*Sum_{n>=1} Fibonacci(n)*Chi(n,7)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

Example

G.f.: A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + 84*x^4 + 120*x^5 + 384*x^6 + 52*x^7 +...
where A(x) = 1 + 1*4*x + 1*12*x^2 + 2*16*x^3 + 3*28*x^4 + 5*24*x^5 + 8*48*x^6 + 13*4*x^7 + 21*60*x^8 + 34*52*x^9 +...+ Fibonacci(n)*A028594(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) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1-18*x^6+x^12) + 0*13*7*x^7/(1+29*x^7-x^14) +...).
The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].

Prog

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 4*sum(m=1, n, fibonacci(m)*kronecker(m, 7)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A028594, A205975, A203847, A000204 (Lucas).

Cf. A209456 (Pell variant).

Sequence in context: A038592 A048776 A135248 * A291038 A271898 A276178

Adjacent sequences: A205973 A205974 A205975 * A205977 A205978 A205979

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2012

Status

approved