cp's OEIS Frontend

A200715 Expansion of (-3*x^2 + x - 1)/(x^3 - 3*x^2 + x - 1).

%I A200715 #68 Apr 28 2025 15:19:59
%S A200715 1,0,0,1,1,-2,-4,3,13,0,-36,-23,85,118,-160,-429,169,1296,360,-3359,
%T A200715 -3143,7294,13364,-11661,-44459,3888,125604,69481,-303443,-386282,
%U A200715 593528,1448931,-717935,-4471200,-868464,11827201,9961393,-26388674,-44445652,44681763
%N A200715 Expansion of (-3*x^2 + x - 1)/(x^3 - 3*x^2 + x - 1).
%C A200715 _Peter A. 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 A200715 |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 A200715 Alois P. Heinz, <a href="/A200715/b200715.txt">Table of n, a(n) for n = 0..1000</a>
%H A200715 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 A200715 Peter Lawrence et al., <a href="https://web.archive.org/web/20220127112002/http://list.seqfan.eu/pipermail/seqfan/2011-November/008535.html">sequence challenge</a> and follow-up messages on the SeqFan list, Nov 21 2011
%H A200715 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,-3,1)
%F A200715 G.f.: (-3*x^2 + x - 1)/(x^3 - 3*x^2 + x - 1).
%F A200715 Term (1,1) in the 3 X 3 matrix [0,1,0; 0,0,1; 1,-3,1]^n.
%F A200715 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 A200715 a:= n-> (<<0|1|0>, <0|0|1>, <1|-3|1>>^n)[1, 1]:
%p A200715 seq(a(n), n=0..50);
%t A200715 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 A200715 (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 A200715 Cf. A200676, A200739.
%K A200715 sign,easy
%O A200715 0,6
%A A200715 _Alois P. Heinz_, Nov 21 2011