A135518 Generalized repunits in base 15.

1, 16, 241, 3616, 54241, 813616, 12204241, 183063616, 2745954241, 41189313616, 617839704241, 9267595563616, 139013933454241, 2085209001813616, 31278135027204241, 469172025408063616, 7037580381120954241, 105563705716814313616, 1583455585752214704241

Offset

1,2

Comments

Primes in this sequence are given in A006033.

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..250

Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.

Index entries for linear recurrences with constant coefficients, signature (16,-15).

Formula

a(n) = (15^n - 1)/14.

a(n) = 15*a(n-1) + 1 with n>1, a(1)=1. - Vincenzo Librandi, Aug 03 2010

G.f.: x/((1-x)*(1-15*x)). - Bruno Berselli, Nov 07 2012

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

a(n) = Sum_{i=0...n-1} 14^i*binomial(n,n-1-i). - Bruno Berselli, Nov 12 2015

E.g.f.: (1/14)*(exp(15*x) - exp(x)). - G. C. Greubel, Oct 17 2016

Example

For n=4, a(4) = 15^3+15^2+15^1+1 = 3375+225+15+1 = 3616.
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

Mathematica

Table[FromDigits[PadRight[{}, n, 1], 15], {n, 20}] (* or *) LinearRecurrence[{16, -15}, {1, 16}, 20] (* Harvey P. Dale, Jul 08 2013 *)

Prog

(Sage) [gaussian_binomial(n, 1, 15) for n in range(1, 15)] # Zerinvary Lajos, May 28 2009
(Sage) [(15^n-1)/14 for n in (1..30)] # Bruno Berselli, Nov 12 2015
(Maxima) A135518(n):=(15^n-1)/14$ makelist(A135518(n), n, 1, 30); /* Martin Ettl, Nov 05 2012 */
(PARI) a(n)=(15^n-1)/14 \\ Charles R Greathouse IV, Sep 24 2015
(Python)
def a(n): return int('1'*n, 15)
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jan 16 2021

Crossrefs

Cf. A000225, A003462, A002450, A003463, A003464.

Cf. A023000, A023001, A002452, A016123, A016125.

Cf. A001023, A135278.

Sequence in context: A173605 A175720 A342031 * A369393 A179092 A231020

Adjacent sequences: A135515 A135516 A135517 * A135519 A135520 A135521

Keyword

nonn,easy

Author

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

Status

approved