A143452 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=9.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 27, 37, 51, 69, 91, 117, 147, 181, 219, 261, 315, 389, 491, 629, 811, 1045, 1339, 1701, 2139, 2661, 3291, 4069, 5051, 6309, 7931, 10021, 12699, 16101, 20379, 25701, 32283, 40421, 50523, 63141, 79003, 99045, 124443, 156645

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 9 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>=19, 3*a(n-19) equals the number of 3-colored compositions of n with all parts >=10, 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 (1,0,0,0,0,0,0,0,0,2).

Crossrefs

9th 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(9): seq(a(n), n=0..64);

Mathematica

Series[1/(1-x-2*x^10), {x, 0, 64}] // CoefficientList[#, x]& // Drop[#, 9]& (* Jean-François Alcover, Feb 13 2014 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,2},{1,3,5,7,9,11,13,15,17,19},50] (* Harvey P. Dale, Nov 28 2015 *)

Formula

G.f.: 1/(x^9*(1-x-2*x^10)).

a(n) = 2n+1 if n<=10, else a(n) = a(n-1) + 2a(n-10). - Milan Janjic, Mar 09 2015