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%I #39 Jan 02 2023 12:30:48
%S 1,0,0,1,5,22,96,419,1829,7984,34852,152137,664113,2899006,12654828,
%T 55241235,241140697,1052634608,4594992184,20058197793,87558647021,
%U 382213633910,1668450426280,7283169876691,31792711738525,138782499488832,605817532105276
%N Expansion of -(3*x^2-5*x+1)/(x^3-3*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="/A200676/b200676.txt">Table of n, a(n) for n = 0..500</a>
%H D. Birmajer, J. B. Gil, M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3 , example 14
%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%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,-3,1)
%F G.f.: -(3*x^2-5*x+1)/(x^3-3*x^2+5*x-1).
%F Term (1,1) in the 3x3 matrix [0,1,0; 0,0,1; 1,-3,5]^n.
%p a:= n-> (<<0|1|0>, <0|0|1>, <1|-3|5>>^n)[1, 1]:
%p seq(a(n), n=0..30);
%t CoefficientList[Series[-(3 x^2 - 5 x + 1)/(x^3 - 3 x^2 + 5 x - 1), {x, 0, 26}], x] (* _Michael De Vlieger_, Sep 04 2018 *)
%t LinearRecurrence[{5,-3,1},{1,0,0},40] (* _Harvey P. Dale_, Aug 18 2021 *)
%Y Cf. A200739.
%K nonn,easy
%O 0,5
%A _Alois P. Heinz_, Nov 21 2011
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