%I #46 Jan 17 2021 11:24:10
%S 1,16,241,3616,54241,813616,12204241,183063616,2745954241,41189313616,
%T 617839704241,9267595563616,139013933454241,2085209001813616,
%U 31278135027204241,469172025408063616,7037580381120954241,105563705716814313616,1583455585752214704241
%N Generalized repunits in base 15.
%C Primes in this sequence are given in A006033.
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=15, (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 Partial sums are in A014898. Also, the sequence is related to A014930 by A014930(n) = n*a(n) - Sum_{i=1..n-1}( a(i) ). - _Bruno Berselli_, Nov 06 2012
%H G. C. Greubel, <a href="/A135518/b135518.txt">Table of n, a(n) for n = 1..250</a>
%H Jon Grantham and Hester Graves, <a href="https://arxiv.org/abs/2009.04052">The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits</a>, arXiv:2009.04052 [math.NT], 2020.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-15).
%F a(n) = (15^n - 1)/14.
%F a(n) = 15*a(n-1) + 1 with n>1, a(1)=1. - _Vincenzo Librandi_, Aug 03 2010
%F G.f.: x/((1-x)*(1-15*x)). - _Bruno Berselli_, Nov 07 2012
%F a(1)=1, a(2)=16; for n>2, a(n) = 16*a(n-1) - 15*a(n-2). - _Harvey P. Dale_, Jul 08 2013
%F a(n) = Sum_{i=0...n-1} 14^i*binomial(n,n-1-i). - _Bruno Berselli_, Nov 12 2015
%F E.g.f.: (1/14)*(exp(15*x) - exp(x)). - _G. C. Greubel_, Oct 17 2016
%e For n=4, a(4) = 15^3+15^2+15^1+1 = 3375+225+15+1 = 3616.
%e For n=6, a(6) = 1*6 + 14*15 + 14^2*20 + 14^3*15 + 14^4*6 + 14^5*1 = 813616. - _Bruno Berselli_, Nov 12 2015
%t Table[FromDigits[PadRight[{},n,1],15],{n,20}] (* or *) LinearRecurrence[{16,-15},{1,16},20] (* _Harvey P. Dale_, Jul 08 2013 *)
%o (Sage) [gaussian_binomial(n,1,15) for n in range(1,15)] # _Zerinvary Lajos_, May 28 2009
%o (Sage) [(15^n-1)/14 for n in (1..30)] # _Bruno Berselli_, Nov 12 2015
%o (Maxima) A135518(n):=(15^n-1)/14$ makelist(A135518(n),n,1,30); /* _Martin Ettl_, Nov 05 2012 */
%o (PARI) a(n)=(15^n-1)/14 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Python)
%o def a(n): return int('1'*n, 15)
%o print([a(n) for n in range(1, 20)]) # _Michael S. Branicky_, Jan 16 2021
%Y Cf. A000225, A003462, A002450, A003463, A003464.
%Y Cf. A023000, A023001, A002452, A016123, A016125.
%Y Cf. A001023, A135278.
%K nonn,easy
%O 1,2
%A Julien Peter Benney (jpbenney(AT)gmail.com), Feb 19 2008