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%I #63 Jan 02 2023 12:30:48
%S 1,0,0,1,1,-2,-4,3,13,0,-36,-23,85,118,-160,-429,169,1296,360,-3359,
%T -3143,7294,13364,-11661,-44459,3888,125604,69481,-303443,-386282,
%U 593528,1448931,-717935,-4471200,-868464,11827201,9961393,-26388674,-44445652,44681763
%N Expansion of (-3*x^2 + x - 1)/(x^3 - 3*x^2 + x - 1).
%C Peter Lawrence (see links) has posted a challenge to find a 3 X 3 integer matrix with "smallish" elements whose powers generate a sequence that is not in the OEIS. This sequence is one of the solutions found.
%C |a(n)| is a prime number for n in {5, 7, 8, 11, 19, 27, 108, 276, 371, 608, ...} with values {2, 3, 13, 23, 3359, 69481, 167527749243856707416101, ...}.
%H Alois P. Heinz, <a href="/A200715/b200715.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%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 (1,-3,1)
%F G.f.: (-3*x^2 + x - 1)/(x^3 - 3*x^2 + x - 1).
%F Term (1,1) in the 3 X 3 matrix [0,1,0; 0,0,1; 1,-3,1]^n.
%F a(n) = a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=a(2)=0. - _Harvey P. Dale_, Nov 22 2011
%p a:= n-> (<<0|1|0>, <0|0|1>, <1|-3|1>>^n)[1, 1]:
%p seq(a(n), n=0..50);
%t CoefficientList[Series[(-3x^2+x-1)/(x^3-3x^2+x-1),{x,0,40}],x] (* or *) LinearRecurrence[{1,-3,1},{1,0,0},40] (* _Harvey P. Dale_, Nov 22 2011 *)
%o (PARI) Vec((-3*x^2+x-1)/(x^3-3*x^2+x-1)+O(x^99)) \\ _Charles R Greathouse IV_, Nov 22 2011
%Y Cf. A200676, A200739.
%K sign,easy
%O 0,6
%A _Alois P. Heinz_, Nov 21 2011
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