A143455 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=4.

1, 4, 7, 10, 13, 16, 28, 49, 79, 118, 166, 250, 397, 634, 988, 1486, 2236, 3427, 5329, 8293, 12751, 19459, 29740, 45727, 70606, 108859, 167236, 256456, 393637, 605455, 932032, 1433740, 2203108, 3384019, 5200384, 7996480, 12297700, 18907024

Offset: 0

Author

Alois P. Heinz, Aug 16 2008

Keywords

• nonn
• easy

Comments

a(n) is also the number of length n quaternary words with at least 4 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>=9, 4*a(n-9) equals the number of 4-colored compositions of n with all parts >=5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,3).

Crossrefs

4th column of A143461.

Programs

Maple

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(4^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 3 else 0 fi)^(n+k))[1,1], n) fi end(4): seq(a(n), n=0..50);

Mathematica

Series[1/(1-x-3*x^5), {x, 0, 50}] // CoefficientList[#, x]& // Drop[#, 4]& (* Jean-François Alcover, Feb 13 2014 *)

Formula

G.f.: 1/(x^4*(1-x-3*x^5)).