A209447 a(n) = Pell(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, 72, 60, 1008, 2088, 2520, 16224, 73440, 11820, 513648, 826704, 1164240, 5621448, 23265216, 14041800, 175149504, 245524824, 98791560, 1590026160, 8061191712, 3706940640, 40272058656, 64816900128, 97801149600, 487966581012, 1596075244848, 91744440540

Offset: 0

Author

Paul D. Hanna, Mar 10 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 + 72*x^2 + 60*x^3 + 1008*x^4 + 2088*x^5 + 2520*x^6 +...
where A(x) = 1 + 1*12*x + 2*36*x^2 + 5*12*x^3 + 12*84*x^4 + 29*72*x^5 + 70*36*x^6 +...+ Pell(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 169*7*x^7/(1-478*x^7-x^14) + 408*8*x^8/(1-1154*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..1000

Crossrefs

Cf. A008653, A205967, A209446, A209448, A204270, A000129 (Pell), A002203.

Programs

Mathematica

A008653[n_]:= If[n < 1, Boole[n == 0], 12*Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A008653[n], {n, 1, 1000}]] (* G. C. Greubel, Jan 02 2017 *)

PARI

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 + 12*sum(m=1,n,Pell(m)*kronecker(m,3)^2*m*x^m/(1-A002203(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} Pell(n)*Chi(n,3)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).