A209445 a(n) = Pell(n)*A001227(n) for n >= 1, where A001227(n) is the number of odd divisors of n.

1, 2, 10, 12, 58, 140, 338, 408, 2955, 4756, 11482, 27720, 66922, 161564, 780100, 470832, 2273378, 8232630, 13250218, 31988856, 154455860, 186444716, 450117362, 1086679440, 3935214363, 6333631924, 30581480180, 36915112104, 89120964298, 430314081400, 519435045698

Offset: 1

Author

Paul D. Hanna, Mar 09 2012

Keywords

• nonn

Comments

Compare g.f. to the Lambert series of A001227: Sum_{n>=1} x^(2*n-1)/(1 - x^(2*n-1)).

Examples

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 12*x^4 + 58*x^5 + 140*x^6 + 338*x^7 + ...
where A(x) = 1*1*x + 2*1*x^2 + 5*2*x^3 + 12*1*x^4 + 29*2*x^5 + 70*2*x^6 + 169*2*x^7 + 408*1*x^8 + ... + Pell(n)*A001227(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1*x/(1-2*x-x^2) + 5*x^3/(1-14*x^3-x^6) + 29*x^5/(1-82*x^5-x^10) + 169*x^7/(1-478*x^7-x^14) + 985*x^9/(1-2786*x^9-x^18) + 5741*x^11/(1-16238*x^11-x^22) + ...
which involves odd-indexed Pell and A002203 numbers.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001227, A205965, A209444, A209446, A204270.

Programs

Mathematica

A001227[n_]:= Sum[Mod[d, 2], {d, Divisors[n]}]; Table[Fibonacci[n, 2]*A001227[n], {n, 1, 1000}] (* 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(sum(m=1,n,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))),n)}
for(n=1,40,print1(a(n),", "))

Formula

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