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


%S 1,3,5,7,9,11,13,15,17,23,33,47,65,87,113,143,177,223,289,383,513,687,

%T 913,1199,1553,1999,2577,3343,4369,5743,7569,9967,13073,17071,22225,

%U 28911,37649,49135,64273,84207,110353,144495,188945,246767,322065,420335,548881

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

%C a(n) is also the number of length n ternary words with at least 7 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>=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

%H Alois P. Heinz, <a href="/A143450/b143450.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,2).

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

%F a(n) = 2n+1 if n<=8, else a(n) = a(n-1) + 2a(n-8). - _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(7): seq(a(n), n=0..61);

%t Series[1/(1-x-2*x^8), {x, 0, 61}] // CoefficientList[#, x]& // Drop[#, 7]& (* _Jean-Fran├žois Alcover_, Feb 13 2014 *)

%Y 7th column of A143453.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Aug 16 2008

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