%I #33 Apr 25 2024 09:07:18

%S 1,15,211,2955,41371,579195,8108731,113522235,1589311291,22250358075,

%T 311505013051,4361070182715,61054982558011,854769755812155,

%U 11966776581370171,167534872139182395,2345488209948553531

%N Generalized repunits in base 14.

%C Primes are given in A006032.

%C 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

%H Harvey P. Dale, <a href="/A135519/b135519.txt">Table of n, a(n) for n = 1..873</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (15,-14).

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

%F a(n) = 14*a(n-1) + 1 for n>1, a(1)=1. - _Vincenzo Librandi_, Aug 03 2010

%F a(n) = Sum_{i=0..n-1} 13^i*binomial(n,n-1-i). - _Bruno Berselli_, Nov 12 2015

%F From _G. C. Greubel_, Oct 17 2016: (Start)

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

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

%e a(4) = 2955 because (14^4-1)/13 = 38416/13 = 2955.

%e For n=6, a(6) = 1*6 + 13*15 + 169*20 + 2197*15 + 28561*6 + 371293*1 = 579195. - _Bruno Berselli_, Nov 12 2015

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

%o (Sage) [gaussian_binomial(n,1,14) for n in range(1,15)] # _Zerinvary Lajos_, May 28 2009

%o (Sage) [(14^n-1)/13 for n in (1..30)] # _Bruno Berselli_, Nov 12 2015

%o (Maxima) A135519(n):=(14^n-1)/13$ makelist(A135519(n),n,1,30); /* _Martin Ettl_, Nov 05 2012 */

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

%K nonn,easy

%O 1,2

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