A034661 Sum of n-th powers of divisors of 18.

6, 39, 455, 6813, 112931, 1956669, 34591115, 617285253, 11064693731, 198756808749, 3574014537275, 64300154115093, 1157115988280531, 20825519793796029, 374836322743499435, 6746846977808919333

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• T. D. Noe, Table of n, a(n) for n=0..200
• Index entries for linear recurrences with constant coefficients, signature (39,-533,3285,-9594,12636,-5832).

Programs

Magma

[DivisorSigma(n, 18): n in [0..20]]; // Bruno Berselli, Apr 05 2013 (improved MAGMA code by Vincenzo Librandi, Apr 17 2014)

Mathematica

Join[{6}, LinearRecurrence[{39, -533, 3285, -9594, 12636, -5832}, {39, 455, 6813, 112931, 1956669, 34591115}, 15]] (* Bruno Berselli, Apr 05 2013 *)
Table[(2^n + 1) (3^n + 9^n + 1), {n, 0, 15}] (* Bruno Berselli, Apr 05 2013 *)
Total[#^Range[0, 20]&/@Divisors[18]] (* Vincenzo Librandi, Apr 17 2014 *)

Sage

[sigma(18,n)for n in range(0,16)] # [Zerinvary Lajos, Jun 04 2009]

Formula

From Philippe Deléham, Apr 04 2013: (Start)

G.f.: 1/(1-x) + 1/(1-2*x) + 1/(1-3*x) + 1/(1-6*x) + 1/(1-9*x) + 1/(1-18*x).

a(n) = A000051(n)*A034513(n) = (2^n+1)*(3^n+9^n+1).

a(n) = 39*a(n-1) -533*a(n-2) +3285*a(n-3) -9594*a(n-4) +12636*a(n-5) -5832*a(n-6) with n>6, a(0)=6, a(1)=39, a(2)=455, a(3)=6813, a(4)= 112931, a(5)=1956669, a(6)=34591115. (End)