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
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
Programs
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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 */
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PARI
a(n)=(15^n-1)/14 \\ Charles R Greathouse IV, Sep 24 2015
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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
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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
Comments