A209446 a(n) = Pell(n)*A004016(n) for n >= 1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + x*y + y^2 = n.

1, 6, 0, 30, 72, 0, 0, 2028, 0, 5910, 0, 0, 83160, 401532, 0, 0, 2824992, 0, 0, 79501308, 0, 463367580, 0, 0, 0, 7870428726, 0, 45872220270, 221490672624, 0, 0, 3116610274188, 0, 0, 0, 0, 127800022137480, 617073093431772, 0, 3596565555708780, 0, 0, 0, 122177355889216668

Offset: 0

Author

Paul D. Hanna, Mar 10 2012

Keywords

• nonn

Comments

Compare g.f. to the Lambert series of A004016: 1 + 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - x^n).

Examples

			G.f.: A(x) = 1 + 6*x + 30*x^3 + 72*x^4 + 2028*x^7 + 5910*x^9 + 83160*x^12 + ...
where A(x) = 1 + 1*6*x + 5*6*x^3 + 12*6*x^4 + 169*12*x^7 + 985*6*x^9 + 13860*6*x^12 + ... + Pell(n)*A004016(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 + 6*( 1*x/(1-2*x-x^2) - 2*x^2/(1-6*x^2+x^4) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1-82*x^5-x^10) + 169*x^7/(1-478*x^7-x^14) + ...).
The values of the symbol Kronecker(n,3) repeat [1, -1, 0, ...].
		

Links

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

Crossrefs

Cf. A004016, A205966, A209445, A209447, A204270, A000129 (Pell), A002203.

Programs

Mathematica

A004016[n_]:= If[n < 1, Boole[n == 0], 6 DivisorSum[n, KroneckerSymbol[#, 3] &]]; Join[{1}, Table[Fibonacci[n, 2]*A004016[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)

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 + 6*sum(m=1,n,kronecker(m,3)*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))

Formula

G.f.: 1 + 6*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).