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A203847 a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n. 22
1, 2, 4, 9, 10, 32, 26, 84, 102, 220, 178, 864, 466, 1508, 2440, 4935, 3194, 15504, 8362, 40590, 43784, 70844, 57314, 370944, 225075, 485572, 785672, 1906866, 1028458, 6656320, 2692538, 13069854, 14098312, 22811548, 36909860, 134373168, 48315634, 156352676, 252983944 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Related identities:

(1) Sum_{n>=1} n^k*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Fibonacci(n)*x^n for k>=0.

(2) Sum_{n>=1} phi(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Fibonacci(n)*x^n.

(3) Sum_{n>=1} moebius(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = x.

(4) Sum_{n>=1} lambda(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Fibonacci(n^2)*x^(n^2).

LINKS

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

FORMULA

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

EXAMPLE

G.f.: A(x) = x + 2*x^2 + 4*x^3 + 9*x^4 + 10*x^5 + 32*x^6 + 26*x^7 +...

where A(x) = x/(1-x-x^2) + x^2/(1-3*x^2+x^4) + 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) + 5*x^5/(1-11*x^5-x^10) + 8*x^6/(1-18*x^6+x^12) +...+ Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

MATHEMATICA

Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)

PROG

(PARI) {a(n)=sigma(n, 0)*fibonacci(n)}

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

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

(PARI) a(n) = numdiv(n)*fibonacci(n); \\ Michel Marcus, Jul 18 2018

CROSSREFS

Cf. A203848, A203849, A203838, A204060, A000005 (tau), A000204 (Lucas), A000045.

Cf. A205507, A205882, A205963, A205964, A205965, A205966.

Cf. A205967, A205968, A205969, A205970, A205971.

Cf. A205972, A205973, A205974, A205975, A205976.

Cf. A204291, A203801, A203850, A203860, A203861.

Sequence in context: A075347 A005733 A096692 * A172147 A266726 A030194

Adjacent sequences:  A203844 A203845 A203846 * A203848 A203849 A203850

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 11 2012

STATUS

approved

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Last modified August 12 12:00 EDT 2020. Contains 336439 sequences. (Running on oeis4.)