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

Author

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

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

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

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

Cf. A001023, A135278.

Programs

Mathematica

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

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

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

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