A143451 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=8.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 25, 35, 49, 67, 89, 115, 145, 179, 217, 267, 337, 435, 569, 747, 977, 1267, 1625, 2059, 2593, 3267, 4137, 5275, 6769, 8723, 11257, 14507, 18625, 23811, 30345, 38619, 49169, 62707, 80153, 102667, 131681, 168931, 216553, 277243

Offset: 0

Author

Alois P. Heinz, Aug 16 2008

Keywords

• nonn
• easy

Comments

a(n) is also the number of length n ternary words with at least 8 0-digits between any other digits.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=17, 3*a(n-17) equals the number of 3-colored compositions of n with all parts >=9, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,2).

Crossrefs

8th column of A143453.

Programs

Maple

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^n,n) else unapply((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 2 else 0 fi)^(n+k))[1,1], n) fi end(8): seq(a(n), n=0..62);

Mathematica

Series[1/(1-x-2*x^9), {x, 0, 62}] // CoefficientList[#, x]& // Drop[#, 8]& (* Jean-François Alcover, Feb 13 2014 *)

PARI

Vec(1/(x^8*(1-x-2*x^9))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

G.f.: 1/(x^8*(1-x-2*x^9)).

a(n) = 2n+1 if n<=9, else a(n) = a(n-1) + 2a(n-9). - Milan Janjic, Mar 09 2015