login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A091030 Partial sums of powers of 13 (A001022). 44
1, 14, 183, 2380, 30941, 402234, 5229043, 67977560, 883708281, 11488207654, 149346699503, 1941507093540, 25239592216021, 328114698808274, 4265491084507563, 55451384098598320, 720867993281778161 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
LINKS
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
EXAMPLE
For n=6, a(6) = 1*6 + 12*15 + 144*20 + 1728*15 + 20736*6 + 248832*1 = 402234. - Bruno Berselli, Nov 12 2015
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 *)
PROG
(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
(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
CROSSREFS
Sequence in context: A170733 A186229 A181237 * A179090 A165152 A263384
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 23 2004
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)