A017907 Expansion of 1/(1 - x^13 - x^14 - ...).

1, 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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 19, 23, 28, 34, 41, 49, 58, 68, 79, 91, 104, 118, 134, 153, 176, 204, 238, 279, 328, 386, 454, 533, 624, 728, 846, 980, 1133, 1309

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) = number of compositions of n in which each part is >= 13. - Milan Janjic, Jun 28 2010

a(n+25) equals the number of binary words of length n having at least 12 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, 1).

Crossrefs

Column k=12 of A141539, k=13 of A220122. - Alois P. Heinz, Dec 09 2012

Programs

Maple

a:= n-> (Matrix(13, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$11, 1][i] else 0 fi)^n)[13,13]: 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, 1}, {1, 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^13),{x,0,70}],x] (* Harvey P. Dale, Feb 07 2015 *)

PARI

Vec((x-1)/(x-1+x^13)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

G.f.: (x-1)/(x-1+x^13). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 13*k, and 12 divides n-k, define c(n,k) = binomial(k,(n-k)/12), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+13) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011

a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=0, a(n)=a(n-1)+a(n-13). - Harvey P. Dale, Feb 07 2015