|
%I #28 Jun 13 2015 00:52:40
%S 1,3,5,7,9,11,13,15,17,19,25,35,49,67,89,115,145,179,217,267,337,435,
%T 569,747,977,1267,1625,2059,2593,3267,4137,5275,6769,8723,11257,14507,
%U 18625,23811,30345,38619,49169,62707,80153,102667,131681,168931,216553,277243
%N Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=8.
%C a(n) is also the number of length n ternary words with at least 8 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>=17, 3*a(n-17) equals the number of 3-colored compositions of n with all parts >=9, such that no adjacent parts have the same color. - _Milan Janjic_, Nov 27 2011
%H Alois P. Heinz, <a href="/A143451/b143451.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,2).
%F G.f.: 1/(x^8*(1-x-2*x^9)).
%F a(n) = 2n+1 if n<=9, else a(n) = a(n-1) + 2a(n-9). - _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(8): seq(a(n), n=0..62);
%t Series[1/(1-x-2*x^9), {x, 0, 62}] // CoefficientList[#, x]& // Drop[#, 8]& (* _Jean-François Alcover_, Feb 13 2014 *)
%o (PARI) Vec(1/(x^8*(1-x-2*x^9))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012
%Y 8th column of A143453.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Aug 16 2008
|
