A135519 Generalized repunits in base 14.

1, 15, 211, 2955, 41371, 579195, 8108731, 113522235, 1589311291, 22250358075, 311505013051, 4361070182715, 61054982558011, 854769755812155, 11966776581370171, 167534872139182395, 2345488209948553531

Offset: 1

Author

Julien Peter Benney (jpbenney(AT)gmail.com), Feb 19 2008

Keywords

• nonn
• easy

Comments

Primes are given in A006032.

Let A be the Hessenberg matrix of the 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)=det(A). - Milan Janjic, Feb 21 2010

Examples

			a(4) = 2955 because (14^4-1)/13 = 38416/13 = 2955.
For n=6, a(6) = 1*6 + 13*15 + 169*20 + 2197*15 + 28561*6 + 371293*1 = 579195. - _Bruno Berselli_, Nov 12 2015
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..873
• Index entries for linear recurrences with constant coefficients, signature (15,-14).

Crossrefs

Cf. A000225, A001022, A002450, A002452, A003462, A003463, A003464, A016123, A016125, A023000, A023001, A135278.

Programs

Mathematica

Table[FromDigits[PadRight[{}, n, 1], 14], {n, 20}] (* or *) LinearRecurrence[{15, -14}, {1, 15}, 20] (* Harvey P. Dale, Aug 29 2016 *)

Maxima

A135519(n):=(14^n-1)/13$ makelist(A135519(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

Sage

[gaussian_binomial(n,1,14) for n in range(1,15)] # Zerinvary Lajos, May 28 2009

Sage

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

Formula

a(n) = (14^n - 1)/13.

a(n) = 14*a(n-1) + 1 for n>1, a(1)=1. - Vincenzo Librandi, Aug 03 2010

a(n) = Sum_{i=0..n-1} 13^i*binomial(n,n-1-i). - Bruno Berselli, Nov 12 2015

From G. C. Greubel, Oct 17 2016: (Start)

G.f.: x/((1-x)*(1-14*x)).

E.g.f.: (1/13)*(exp(14*x) - exp(x)). (End)