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%I #12 Oct 11 2023 19:20:41
%S 1,4,7,10,13,16,19,22,25,28,40,61,91,130,178,235,301,376,460,580,763,
%T 1036,1426,1960,2665,3568,4696,6076,7816,10105,13213,17491,23371,
%U 31366,42070,56158,74386,97834,128149,167788,220261,290374,384472,510682
%N Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=8.
%C a(n) is also the number of length n quaternary words with at least 8 0-digits between any other digits.
%H Vincenzo Librandi, <a href="/A143459/b143459.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 (1,0,0,0,0,0,0,0,3).
%F G.f.: 1/(x^8*(1-x-3*x^9)).
%p a := proc(k::nonnegint) local n,i,j; if k=0 then unapply (4^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 3 else 0 fi)^(n+k))[1,1], n) fi end(8): seq (a(n), n=0..58);
%t Series[1/(1-x-3*x^9), {x, 0, 58}] // CoefficientList[#, x]& // Drop[#, 8]& (* _Jean-François Alcover_, Feb 13 2014 *)
%Y 8th column of A143461.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Aug 16 2008
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