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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.)