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A143449 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=6. 2


%S 1,3,5,7,9,11,13,15,21,31,45,63,85,111,141,183,245,335,461,631,853,

%T 1135,1501,1991,2661,3583,4845,6551,8821,11823,15805,21127,28293,

%U 37983,51085,68727,92373,123983,166237,222823,298789,400959,538413,723159,971125

%N Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=6.

%C a(n) is also the number of length n ternary words with at least 6 0-digits between any other digits.

%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>=13, 3*a(n-13) equals the number of 3-colored compositions of n with all parts >=7, such that no adjacent parts have the same color. - _Milan Janjic_, Nov 27 2011

%H Alois P. Heinz, <a href="/A143449/b143449.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,2).

%F G.f.: 1/(x^6*(1-x-2*x^7)).

%F a(n) = 2n+1 if n<=7, else a(n) = a(n-1) + 2a(n-7). - _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(6): seq(a(n), n=0..58);

%t Series[1/(1-x-2*x^7), {x, 0, 58}] // CoefficientList[#, x]& // Drop[#, 6]& (* _Jean-Fran├žois Alcover_, Feb 13 2014 *)

%Y 6th column of A143453.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Aug 16 2008

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