A017906 Expansion of 1/(1 - x^12 - x^13 - ...).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 18, 22, 27, 33, 40, 48, 57, 67, 78, 90, 103, 118, 136, 158, 185, 218, 258, 306, 363, 430, 508, 598, 701, 819, 955, 1113, 1298, 1516, 1774

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) = number of compositions of n in which each part is >=12. - Milan Janjic, Jun 28 2010

a(n+23) equals the number of binary words of length n having at least 11 zeros between every two successive ones. - Milan Janjic, Feb 09 2015

Links

• Vincenzo Librandi, 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, 1).

Programs

Maple

a:= n-> (Matrix(12, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [1, 0$10, 1][i], 0)))^n)[12, 12]: seq(a(n), n=0..60); # Alois P. Heinz, Aug 04 2008

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 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^12), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 01 2013 *)

PARI

Vec((x-1)/(x-1+x^12)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

G.f.: (x-1)/(x-1+x^12). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 12*k, and 11 divides n-k, define c(n,k) = binomial(k,(n-k)/11), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+12) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011