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%I #40 Jan 16 2019 11:14:38
%S 1,-1,-1,-6,-23,-94,-380,-1539,-6231,-25229,-102150,-413597,-1674620,
%T -6780398,-27453271,-111156025,-450061557,-1822262042,-7378188379,
%U -29873674862,-120956040144,-489741008259,-1982920860215,-8028682653825,-32507472410594,-131620068707297,-532918778418092,-2157744082494058,-8736527429861839,-35373477397979841,-143224285995880937
%N a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), a(0) = 1, a(1) = -1, a(2) = -1 .
%C In general, let {X,Y,Z} be the roots of the cubic equation x^3 + ax^2 + xt + 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 Let X = (sin(4k)*sin(8k))/(sin(2k)*sin(2k)),
%C Y = (sin(8k)*sin(2k))/(sin(4k)*sin(4k)),
%C Z = (sin(2k)*sin(4k))/(sin(8k)*sin(8k)).
%C Then {X,Y,Z} are the roots of the cubic equation x^3 - 3*x^2 - 4*x - 1 = 0.
%C This sequence: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k))).
%C A122600: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k))).
%C A321703: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k))).
%H Colin Barker, <a href="/A321715/b321715.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 (3,4,1).
%F G.f.: (1 - 4*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3). - _Colin Barker_, Jan 15 2019
%o (PARI) Vec((1 - 4*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3) + O(x^30)) \\ _Colin Barker_, Jan 15 2019
%Y Cf. A321703, A122600.
%K sign,easy
%O 0,4
%A _Kai Wang_, Jan 14 2019
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