A209451 a(n) = Pell(n)*A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

1, 4, 8, 20, 240, 696, 280, 5408, 21216, 3940, 57072, 275568, 277200, 1873816, 2585024, 4680600, 54616512, 81841608, 10976840, 530008720, 1919331360, 1235646880, 4474673184, 21605633376, 28253665440, 162655527004, 177341693872, 30581480180, 2953208968320

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 modulo 3.

Links

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

Formula

G.f.: 1 + 4*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 + 4*x + 8*x^2 + 20*x^3 + 240*x^4 + 696*x^5 + 280*x^6 + ...
where A(x) = 1 + 1*4*x + 2*4*x^2 + 5*4*x^3 + 12*20*x^4 + 29*24*x^5 + 70*4*x^6 + ... + Pell(n)*A034896(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 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

A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A034896[n], {n, 1, 50}]] (* G. C. Greubel, Dec 24 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 + 4*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, 61, print1(a(n), ", "))

Crossrefs

Cf. A034896, A205971, A205884, A209447, A209450, A209452, A204270, A000129 (Pell), A002203.

Sequence in context: A086912 A168451 A000585 * A102559 A308233 A371001

Adjacent sequences: A209448 A209449 A209450 * A209452 A209453 A209454

Keyword

nonn

Author

Paul D. Hanna, Mar 10 2012

Status

approved