A017909 Expansion of 1/(1 - x^15 - x^16 - ...).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 21, 25, 30, 36, 43, 51, 60, 70, 81, 93, 106, 120, 135, 151, 169, 190, 215, 245, 281, 324, 375, 435, 505, 586, 679

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n+29) equals the number of binary words of length n having at least 14 zeros between every two successive ones. - Milan Janjic, Feb 09 2015

Number of compositions of n into parts >= 15. - Ilya Gutkovskiy, May 23 2017

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Programs

Maple

a:= n -> (Matrix(15, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$13, 1][i] else 0 fi)^n)[15,15]: seq(a(n), n=0..80); # Alois P. Heinz, Aug 04 2008

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
CoefficientList[Series[(x-1)/(x-1+x^15),{x,0,100}],x] (* Harvey P. Dale, Sep 04 2020 *)

PARI

Vec((x-1)/(x-1+x^15)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

G.f.: (x-1)/(x-1+x^15). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 15*k, and 14 divides n-k, define c(n,k) = binomial(k,(n-k)/14), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+15) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011