A203847 - OEIS

A203847

a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.

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

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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.