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%I #14 Oct 11 2023 19:19:52
%S 1,4,7,10,13,16,19,31,52,82,121,169,226,319,475,721,1084,1591,2269,
%T 3226,4651,6814,10066,14839,21646,31324,45277,65719,95917,140434,
%U 205372,299344,435175,632332,920083,1341385,1957501,2855533,4161058,6058054
%N Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=5.
%C a(n) is also the number of length n quaternary words with at least 5 0-digits between any other digits.
%H Vincenzo Librandi, <a href="/A143456/b143456.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 (1,0,0,0,0,3).
%F G.f.: 1/(x^5*(1-x-3*x^6)).
%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(5): seq(a(n), n=0..52);
%t Series[1/(1-x-3*x^6), {x, 0, 52}] // CoefficientList[#, x]& // Drop[#, 5]& (* _Jean-François Alcover_, Feb 13 2014 *)
%Y 5th column of A143461.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Aug 16 2008
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