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A203848 a(n) = sigma(n)*Fibonacci(n), where sigma(n) = A000203(n), the sum of divisors of n. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 09:12 EST 2022. Contains 358585 sequences. (Running on oeis4.)