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%I #27 Feb 19 2019 09:20:03
%S -1,-2,3,-8,12,-25,41,-79,136,-253,446,-816,1455,-2641,4735,-8562,
%T 15391,-27780,50000,-90169,162389,-292727,527336,-950401,1712346,
%U -3085812,5560103,-10019381,18053775,-32532434,58620603
%N a(n) = -a(n-1) + 2*a(n-2) + a(n-3), a(0) = -1, a(1) = -2, a(2) = 3.
%C Let {X,Y,Z} be the roots of the cubic equation t^3 + at^2 + bt + c = 0 where {a, b, c} are integers.
%C Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers.
%C Then {p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...} is an integer sequence with the recurrence relation: p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).
%C Let k = Pi/7.
%C This sequence has (a, b, c) = (1, -2, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).
%C A094648: (a, b, c) = (1, -2, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).
%C A321461 : (a, b, c) = (1, -2, -1), (u, v, w) = (2*cos(4k), 2*cos(8k), 2*cos(2k)).
%C X = sin(2k)/sin(8k), Y = sin(4k)/sin(2k), Z = sin(8k)/sin(4k).
%H Colin Barker, <a href="/A321175/b321175.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 (-1,2,1).
%F G.f.: -(1 + 3*x - 3*x^2) / (1 + x - 2*x^2 - x^3). - _Colin Barker_, Jan 11 2019
%t LinearRecurrence[{-1,2,1},{-1,-2,3},50] (* _Stefano Spezia_, Jan 11 2019 *)
%o (PARI) Vec(-(1 + 3*x - 3*x^2) / (1 + x - 2*x^2 - x^3) + O(x^30)) \\ _Colin Barker_, Jan 11 2019
%Y Cf. A321461, A094648.
%K sign,easy
%O 0,2
%A _Kai Wang_, Jan 10 2019
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