A203848 a(n) = sigma(n)*Fibonacci(n), where sigma(n) = A000203(n), the sum of divisors of n.

1, 3, 8, 21, 30, 96, 104, 315, 442, 990, 1068, 4032, 3262, 9048, 14640, 30597, 28746, 100776, 83620, 284130, 350272, 637596, 687768, 2782080, 2325775, 5098506, 7856720, 17797416, 15426870, 59906880, 43080608, 137233467, 169179744, 307955898, 442918320, 1358662032

Offset

1,2

Comments

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

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

Example

G.f.: A(x) = x + 3*x^2 + 8*x^3 + 21*x^4 + 30*x^5 + 96*x^6 + 104*x^7 +...
where A(x) = x/(1-x-x^2) + 2*1*x^2/(1-3*x^2+x^4) + 3*2*x^3/(1-4*x^3-x^6) + 4*3*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 6*8*x^6/(1-18*x^6+x^12) +...+ n*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Mathematica

Table[DivisorSigma[1, n] Fibonacci[n], {n, 40}] (* Wesley Ivan Hurt, Aug 10 2016 *)

Prog

(PARI) {a(n)=sigma(n)*fibonacci(n)}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1, n, m*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
(Magma) [DivisorSigma(1, n)*Fibonacci(n): n in [1..40]]; // Vincenzo Librandi, Aug 12 2016

Crossrefs

Cf. A203847, A203849, A203838, A000203 (sigma), A000204 (Lucas), A000045.

Sequence in context: A101643 A334136 A046815 * A160404 A103736 A172243

Adjacent sequences: A203845 A203846 A203847 * A203849 A203850 A203851

Keyword

nonn,easy

Author

Paul D. Hanna, Jan 12 2012

Status

approved