A091030 Partial sums of powers of 13 (A001022).

1, 14, 183, 2380, 30941, 402234, 5229043, 67977560, 883708281, 11488207654, 149346699503, 1941507093540, 25239592216021, 328114698808274, 4265491084507563, 55451384098598320, 720867993281778161

Offset: 1

Author

Wolfdieter Lang, Jan 23 2004

Keywords

• nonn
• easy

Comments

13^a(n) is highest power of 13 dividing (13^n)!.

For analogs with primes 2, 3, 5, 7 and 11 see A000225, A003462, A003463, A023000 and A016123 respectively.

Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1,A[i,i]:=13, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=14, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n)*charpoly(A,1). - Milan Janjic, Feb 21 2010

Examples

			For n=6, a(6) = 1*6 + 12*15 + 144*20 + 1728*15 + 20736*6 + 248832*1 = 402234. - _Bruno Berselli_, Nov 12 2015
		

Links

• Delbert L. Johnson, Table of n, a(n) for n = 1..898
• Index entries for linear recurrences with constant coefficients, signature (14,-13).

Crossrefs

Cf. A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125.

Cf. A001021, A135278.

Programs

Maple

a:=n->sum(13^(n-j),j=1..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 04 2007

Mathematica

Table[13^n, {n, 0, 16}] // Accumulate (* Jean-François Alcover, Jul 05 2013 *)
LinearRecurrence[{14,-13},{1,14},20] (* Harvey P. Dale, Jan 19 2024 *)

Maxima

A091030(n):=(13^n-1)/12$ makelist(A091030(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

PARI

a(n)=([0,1; -13,14]^(n-1)*[1;14])[1,1] \\ Charles R Greathouse IV, Sep 24 2015

Sage

[gaussian_binomial(n,1,13) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

Sage

[(13^n-1)/12 for n in (1..30)] # Bruno Berselli, Nov 12 2015

Formula

G.f.: x/((1-13*x)*(1-x)) = (1/(1-13*x) - 1/(1-x))/12.

a(n) = Sum_{k=0..n-1} 13^k = (13^n-1)/12.

a(n) = 13*a(n-1)+1 for n>1, a(1)=1. - Vincenzo Librandi, Feb 05 2011

a(n) = Sum_{k=0...n-1} 12^k*binomial(n,n-1-k). - Bruno Berselli, Nov 12 2015

E.g.f.: exp(x)*(exp(12*x) - 1)/12. - Stefano Spezia, Mar 11 2023