A017908 Expansion of 1/(1 - x^14 - x^15 - ...).

1, 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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 20, 24, 29, 35, 42, 50, 59, 69, 80, 92, 105, 119, 134, 151, 171, 195, 224, 259, 301, 351, 410, 479, 559, 651, 756, 875, 1009

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) = number of compositions of n in which each part is >=14. - Milan Janjic, Jun 28 2010

a(n+27) equals the number of binary words of length n having at least 13 zeros between every two successive ones. - Milan Janjic, Feb 09 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Programs

Maple

a:= n-> (Matrix(14, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$12, 1][i] else 0 fi)^n)[14, 14]: seq(a(n), n=0..62); # Alois P. Heinz, Aug 04 2008

Mathematica

LinearRecurrence[{1, 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}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)

PARI

Vec((x-1)/(x-1+x^14)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

G.f.: (x-1)/(x-1+x^14). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 14*k, and 13 divides n-k, define c(n,k) = binomial(k,(n-k)/13), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+14) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011