A205508 a(n) = Pell(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

1, 4, 8, 0, 48, 232, 0, 0, 1632, 3940, 19024, 0, 0, 267688, 0, 0, 1883328, 9093512, 10976840, 0, 127955424, 0, 0, 0, 0, 15740857452, 25334527696, 0, 0, 356483857192, 0, 0, 2508054264192, 0, 29236023007504, 0, 85200014758320, 411382062287848, 0, 0, 5788584895037376

Offset

0,2

Comments

Compare to the g.f. of A004018 given by the Lambert series identity:

1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.

Links

Table of n, a(n) for n=0..40.

Formula

G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Pell(2*n+1)*x^(2*n+1) / (1 - A002203(2*n+1)*x^(2*n+1) - x^(4*n+2)), where A002203 is the companion Pell numbers.

Example

 G.f.: A(x) = 1 + 4*x + 8*x^2 + 48*x^4 + 232*x^5 + 1632*x^8 + 3940*x^9 + 19024*x^10 +...
Compare the g.f to the square of the Jacobi theta_3 series:
theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...
The g.f. equals the sum:
A(x) = 1 + 4*x/(1-2*x-x^2) - 4*5*x^3/(1-14*x^3-x^6) + 4*29*x^5/(1-82*x^5-x^10) - 4*169*x^7/(1-478*x^7-x^14) + 4*985*x^9/(1-2786*x^9-x^18) - 4*5741*x^11/(1-16238*x^11-x^22) + 4*33461*x^13/(1-94642*x^13-x^26) - 4*195025*x^15/(1-551614*x^15-x^30) +...
which involves odd-indexed Pell and companion Pell numbers.

Prog

 (PARI) {A004018(n)=polcoeff((1+2*sum(k=1, sqrtint(n+1), x^(k^2), x*O(x^n)))^2, n)}
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=if(n==0, 1, Pell(n)*A004018(n))}
(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=0, n+1, (-1)^m*Pell(2*m+1)*x^(2*m+1)/(1-A002203(2*m+1)*x^(2*m+1)-x^(4*m+2)+x*O(x^n))))^(1/1), n)}

Crossrefs

Cf. A205507, A204384, A204270, A004018, A000129 (Pell), A002203.

Sequence in context: A280652 A104538 A120580 * A013328 A143462 A190966

Adjacent sequences: A205505 A205506 A205507 * A205509 A205510 A205511

Keyword

nonn

Author

Paul D. Hanna, Jan 28 2012

Status

approved