OFFSET
0,2
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).
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
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Aug 16 2008
EXTENSIONS
Generating function corrected by Harvey P. Dale, Aug 15 2021
STATUS
approved