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%I #32 Sep 08 2022 08:45:15
%S -1,-2,-1,2,4,2,-3,-6,-3,4,8,4,-5,-10,-5,6,12,6,-7,-14,-7,8,16,8,-9,
%T -18,-9,10,20,10,-11,-22,-11,12,24,12,-13,-26,-13,14,28,14,-15,-30,
%U -15,16,32,16,-17,-34,-17,18,36,18,-19,-38,-19,20,40,20,-21,-42
%N A sequence generated from the Quadrifoil.
%C a(3n - 1) = 2n (unsigned; n = 1, 2, 3...). In A099471, a(3n) = (2n + 1), unsigned. Odifreddi, p. 135: "Since the trefoil has polynomial x^2 - x + 1 and the quadrifoil (or flat knot) is the sum of two trefoils, its polynomial is (x^2 - x + 1)^2 = x^4 - 2x^3 + 3x^2 - 2x + 1."
%C Coefficient of x of the characteristic polynomial of the n X n matrix with 1's along the superdiagonal, main diagonal and subdiagonal. - _John M. Campbell_, Sep 14 2011
%D Piergiorgio Odifreddi, "The Mathematical Century; The 30 Greatest Problems of the Last 100 Years", Princeton University Press, 2000, page 135.
%H Vincenzo Librandi, <a href="/A099470/b099470.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,2,-1).
%F M = the 4 X 4 companion matrix to the Quadrafoil polynomial x^4 - 2x^3 + 3x^2 - 2x + 1: [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 2 -3 2]. a(n) = rightmost term in M^n * [1 0 0 0].
%F O.g.f.: -x/(x^2-x+1)^2. a(n) = 2*a(n-1)-3*a(n-2)+2*a(n-3)-a(n-4) = -A099254(n-1). - _R. J. Mathar_, Apr 06 2008, Apr 23 2009
%e a(7) = -3 since M^7 * [1 0 0 0] = [2 4 2 -3].
%t Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1,#2] + KroneckerDelta[#1,#2+1] + KroneckerDelta[#1,#2-1] &, {n,n}], x], x], {n,75}] (* _John M. Campbell_, Sep 14 2011 *)
%t Table[(3 n Cos[Pi n/3] - Sqrt[3] (3 n + 4) Sin[Pi n/3])/9, {n, 20}] (* _Vladimir Reshetnikov_, Sep 08 2016 *)
%t LinearRecurrence[{2, -3, 2, -1}, {-1, -2, -1, 2}, 90] (* _Vincenzo Librandi_, Sep 09 2016 *)
%o (Magma) I:=[-1,-2,-1,2]; [n le 4 select I[n] else 2*Self(n-1)-3*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..80]]; // _Vincenzo Librandi_, Sep 09 2016
%Y Cf. A099254, A099471.
%K sign,easy
%O 1,2
%A _Gary W. Adamson_, Oct 17 2004
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