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
(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
KEYWORD
nonn,easy
AUTHOR
Julien Peter Benney (jpbenney(AT)gmail.com), Feb 19 2008
STATUS
approved