A143448 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=5.

1, 3, 5, 7, 9, 11, 13, 19, 29, 43, 61, 83, 109, 147, 205, 291, 413, 579, 797, 1091, 1501, 2083, 2909, 4067, 5661, 7843, 10845, 15011, 20829, 28963, 40285, 55971, 77661, 107683, 149341, 207267, 287837, 399779, 555101, 770467, 1069149, 1483683, 2059357, 2858915

Offset: 0

Author

Alois P. Heinz, Aug 16 2008

Keywords

• nonn
• easy

Comments

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>=11, 3*a(n-11) equals the number of 3-colored compositions of n with all parts >=6, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

a(n) equals the number of ternary words of length n having at least 5 zeros between every two successive nonzero letters. - Milan Janjic, Mar 09 2015

Links

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

Crossrefs

5th 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(5): seq(a(n), n=0..56);

Mathematica

Series[1/(1-x-2*x^6), {x, 0, 56}] // CoefficientList[#, x]& // Drop[#, 5]& (* Jean-François Alcover, Feb 13 2014 *)
LinearRecurrence[{1,0,0,0,0,2},{1,3,5,7,9,11},50] (* Harvey P. Dale, Aug 15 2021 *)

Formula

G.f.: (-1 - 2 x - 2 x^2 - 2 x^3 - 2 x^4 - 2 x^5)/(-1 + x + 2 x^6) - Harvey P. Dale, Aug 15 2021

a(n) = 2n+1 if n<=6, else a(n) = a(n-1) + 2a(n-6). - Milan Janjic, Mar 09 2015

Extensions

Generating function corrected by Harvey P. Dale, Aug 15 2021