A203849 a(n) = sigma_2(n)*Fibonacci(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

1, 5, 20, 63, 130, 400, 650, 1785, 3094, 7150, 10858, 30240, 39610, 94250, 158600, 336567, 463130, 1175720, 1513522, 3693690, 5473000, 10803710, 15188210, 39412800, 48841275, 103184050, 161062760, 333701550, 432980818, 1081652000, 1295110778, 2973391785, 4299985160

Offset

1,2

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

Links

G. C. Greubel, Table of n, a(n) for n = 1..2500

Formula

G.f.: Sum_{n>=1} n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).

Example

G.f.: A(x) = x + 5*x^2 + 20*x^3 + 63*x^4 + 130*x^5 + 400*x^6 + 650*x^7 +...
where A(x) = x/(1-x-x^2) + 2^2*1*x^2/(1-3*x^2+x^4) + 3^2*2*x^3/(1-4*x^3-x^6) + 4^2*3*x^4/(1-7*x^4+x^8) + 5^2*5*x^5/(1-11*x^5-x^10) + 6^2*8*x^6/(1-18*x^6+x^12) +...+ n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Mathematica

Table[DivisorSigma[2, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)

Prog

(PARI) {a(n)=sigma(n, 2)*fibonacci(n)}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1, n, m^2*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}

Crossrefs

Cf. A203847, A203848, A203838, A001157 (sigma_2), A000204 (Lucas), A000045.

Sequence in context: A270631 A032340 A011854 * A271260 A270734 A271411

Adjacent sequences: A203846 A203847 A203848 * A203850 A203851 A203852

Keyword

nonn

Author

Paul D. Hanna, Jan 12 2012

Status

approved