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%I #32 Aug 15 2021 11:51:31
%S 1,3,5,7,9,11,13,19,29,43,61,83,109,147,205,291,413,579,797,1091,1501,
%T 2083,2909,4067,5661,7843,10845,15011,20829,28963,40285,55971,77661,
%U 107683,149341,207267,287837,399779,555101,770467,1069149,1483683,2059357,2858915
%N Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=5.
%C 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
%C 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
%H Alois P. Heinz, <a href="/A143448/b143448.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,2).
%F 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
%F a(n) = 2n+1 if n<=6, else a(n) = a(n-1) + 2a(n-6). - _Milan Janjic_, Mar 09 2015
%p 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);
%t Series[1/(1-x-2*x^6), {x, 0, 56}] // CoefficientList[#, x]& // Drop[#, 5]& (* _Jean-François Alcover_, Feb 13 2014 *)
%t LinearRecurrence[{1,0,0,0,0,2},{1,3,5,7,9,11},50] (* _Harvey P. Dale_, Aug 15 2021 *)
%Y 5th column of A143453.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Aug 16 2008
%E Generating function corrected by _Harvey P. Dale_, Aug 15 2021
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