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%I #18 Jan 02 2023 12:30:48
%S 1,0,0,1,5,24,116,561,2713,13120,63448,306833,1483837,7175800,
%T 34701996,167818017,811563889,3924703424,18979771248,91785716705,
%U 443873515701,2146561633048,10380720366244,50200913713873,242770409836169,1174031855833216,5677589783043784
%N Expansion of (-x^2+5*x-1)/(x^3-x^2+5*x-1).
%C Peter Lawrence (see links) has posted a challenge to find a 3x3 integer matrix with "smallish" elements whose powers generate a sequence that is not in the OEIS. This sequence is one of the solutions found.
%H Alois P. Heinz, <a href="/A200739/b200739.txt">Table of n, a(n) for n = 0..450</a>
%H Peter Lawrence et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2011-November/008535.html">sequence challenge</a> and follow-up messages on the SeqFan list, Nov 21 2011
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1,1)
%F G.f.: (-x^2+5*x-1)/(x^3-x^2+5*x-1).
%F Term (1,1) in the 3x3 matrix [0,1,0; 0,0,1; 1,-1,5]^n.
%p a:= n-> (<<0|1|0>, <0|0|1>, <1|-1|5>>^n)[1, 1]:
%p seq(a(n), n=0..30);
%t CoefficientList[Series[(-x^2 + 5 x - 1)/(x^3 - x^2 + 5 x - 1), {x, 0, 30}], x] (* or *) LinearRecurrence[{5,-1,1},{1,0,0},30] (* _Harvey P. Dale_, Nov 26 2017 *)
%Y Cf. A200676.
%K nonn,easy
%O 0,5
%A _Alois P. Heinz_, Nov 21 2011
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