A143450 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=7.

1, 3, 5, 7, 9, 11, 13, 15, 17, 23, 33, 47, 65, 87, 113, 143, 177, 223, 289, 383, 513, 687, 913, 1199, 1553, 1999, 2577, 3343, 4369, 5743, 7569, 9967, 13073, 17071, 22225, 28911, 37649, 49135, 64273, 84207, 110353, 144495, 188945, 246767, 322065, 420335, 548881

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 7 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>=15, 3*a(n-15) equals the number of 3-colored compositions of n with all parts >=8, 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 (1,0,0,0,0,0,0,2).

Crossrefs

7th 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(7): seq(a(n), n=0..61);

Mathematica

Series[1/(1-x-2*x^8), {x, 0, 61}] // CoefficientList[#, x]& // Drop[#, 7]& (* Jean-François Alcover, Feb 13 2014 *)

Formula

G.f.: 1/(x^7*(1-x-2*x^8)).

a(n) = 2n+1 if n<=8, else a(n) = a(n-1) + 2a(n-8). - Milan Janjic, Mar 09 2015