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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

LINKS

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

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).

EXAMPLE

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, ...].

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 *)

PROG

(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), ", "))

CROSSREFS

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

Sequence in context: A198311 A060930 A169725 * A219302 A338261 A101523

Adjacent sequences:  A209444 A209445 A209446 * A209448 A209449 A209450

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 10 2012

STATUS

approved

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Last modified July 25 13:05 EDT 2021. Contains 346290 sequences. (Running on oeis4.)