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%I #16 Mar 14 2024 15:19:32
%S -3,3,-2,-1,9,-30,85,-229,606,-1593,4177,-10942,28653,-75021,196414,
%T -514225,1346265,-3524574,9227461,-24157813,63245982,-165580137,
%U 433494433,-1134903166,2971215069,-7778742045,20365011070,-53316291169,139583862441,-365435296158,956722026037
%N Expansion of -(3+9*x+2*x^2)/((x+1)*(x^2+3*x+1)).
%C Floretion Algebra Multiplication Program, FAMP Code: 2tesforseq[A*B], with A = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = + .5'ij' + .5'ji'; 1vesforseq(n) = (-1)^(n+1)*2, ForType: 1A
%H Colin Barker, <a href="/A131589/b131589.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-4,-1).
%F a(n) + a(n+1) = (-1)^(n+1)*A001906(n) = (-1)^(n+1)*F(2n).
%F From _Colin Barker_, May 01 2019: (Start)
%F a(n) = -(2^(-1-n)*(5*(-1)^n*2^(3+n) + (-3-sqrt(5))^n*(-5+sqrt(5)) - (-3+sqrt(5))^n*(5+sqrt(5)))) / 5.
%F a(n) = -4*a(n-1) - 4*a(n-2) - a(n-3) for n>2. (End)
%t CoefficientList[Series[-(3+9x+2x^2)/((x+1)(x^2+3x+1)),{x,0,40}],x] (* or *) LinearRecurrence[{-4,-4,-1},{-3,3,-2},40] (* _Harvey P. Dale_, Jun 22 2022 *)
%o (PARI) Vec(-(3 + 9*x + 2*x^2) / ((1 + x)*(1 + 3*x + x^2)) + O(x^35)) \\ _Colin Barker_, May 01 2019
%Y Cf. A001906.
%K easy,sign
%O 0,1
%A _Creighton Dement_, Aug 30 2007
%E Definition corrected (by negating prior formula) by _Harvey P. Dale_, Jun 22 2022
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