cp's OEIS Frontend

A091030 Partial sums of powers of 13 (A001022).

%I A091030 #50 Jan 19 2024 11:05:54
%S A091030 1,14,183,2380,30941,402234,5229043,67977560,883708281,11488207654,
%T A091030 149346699503,1941507093540,25239592216021,328114698808274,
%U A091030 4265491084507563,55451384098598320,720867993281778161
%N A091030 Partial sums of powers of 13 (A001022).
%C A091030 13^a(n) is highest power of 13 dividing (13^n)!.
%C A091030 For analogs with primes 2, 3, 5, 7 and 11 see A000225, A003462, A003463, A023000 and A016123 respectively.
%C A091030 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
%C A091030 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
%H A091030 Delbert L. Johnson, <a href="/A091030/b091030.txt">Table of n, a(n) for n = 1..898</a>
%H A091030 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-13).
%F A091030 G.f.: x/((1-13*x)*(1-x)) = (1/(1-13*x) - 1/(1-x))/12.
%F A091030 a(n) = Sum_{k=0..n-1} 13^k = (13^n-1)/12.
%F A091030 a(n) = 13*a(n-1)+1 for n>1, a(1)=1. - _Vincenzo Librandi_, Feb 05 2011
%F A091030 a(n) = Sum_{k=0...n-1} 12^k*binomial(n,n-1-k). - _Bruno Berselli_, Nov 12 2015
%F A091030 E.g.f.: exp(x)*(exp(12*x) - 1)/12. - _Stefano Spezia_, Mar 11 2023
%e A091030 For n=6, a(6) = 1*6 + 12*15 + 144*20 + 1728*15 + 20736*6 + 248832*1 = 402234. - _Bruno Berselli_, Nov 12 2015
%p A091030 a:=n->sum(13^(n-j),j=1..n): seq(a(n), n=1..17); # _Zerinvary Lajos_, Jan 04 2007
%t A091030 Table[13^n, {n, 0, 16}] // Accumulate (* _Jean-François Alcover_, Jul 05 2013 *)
%t A091030 LinearRecurrence[{14,-13},{1,14},20] (* _Harvey P. Dale_, Jan 19 2024 *)
%o A091030 (Sage) [gaussian_binomial(n,1,13) for n in range(1,18)] # _Zerinvary Lajos_, May 28 2009
%o A091030 (Sage) [(13^n-1)/12 for n in (1..30)] # _Bruno Berselli_, Nov 12 2015
%o A091030 (Maxima) A091030(n):=(13^n-1)/12$ makelist(A091030(n),n,1,30); /* _Martin Ettl_, Nov 05 2012 */
%o A091030 (PARI) a(n)=([0,1; -13,14]^(n-1)*[1;14])[1,1] \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A091030 Cf. A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125.
%Y A091030 Cf. A001021, A135278.
%K A091030 nonn,easy
%O A091030 1,2
%A A091030 _Wolfdieter Lang_, Jan 23 2004