A205967 a(n) = Fibonacci(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

1, 12, 36, 24, 252, 360, 288, 1248, 3780, 408, 11880, 12816, 12096, 39144, 108576, 43920, 367164, 344952, 93024, 1003440, 3409560, 1050816, 7651152, 8253216, 8346240, 27909300, 61182072, 2357016, 213568992, 185122440, 179720640, 516967296, 1646801604, 507539232

Offset: 0

Author

Paul D. Hanna, Feb 04 2012

Keywords

• nonn

Comments

Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).

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

Examples

			G.f.: A(x) = 1 + 12*x + 36*x^2 + 24*x^3 + 252*x^4 + 360*x^5 + 288*x^6 +...
where A(x) = 1 + 1*12*x + 1*36*x^2 + 2*12*x^3 + 3*84*x^4 + 5*72*x^5 + 8*36*x^6 +...+ Fibonacci(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 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, ...].
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500

Crossrefs

Cf. A008653, A205966, A205968, A203847, A000204 (Lucas).

Cf. A209447 (Pell variant).

Programs

Mathematica

terms = 34; s = 1 + 12*Sum[Fibonacci[n]*KroneckerSymbol[n, 3]^2*n*(x^n/(1 - LucasL[n]*x^n + (-1)^n*x^(2*n))), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
b[n_] := If[n < 1, Boole[n == 0], 12 Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Table[If[n == 0, 1, b[n]*Fibonacci[n]], {n, 0, 50}] (* G. C. Greubel, Jul 17 2018 *)

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 12*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=0,50,print1(a(n),", "))

Formula

G.f.: 1 + 12*Sum_{n>=1} Fibonacci(n)*Chi(n,3)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).