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A321703 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), a(0) = 1, a(1) = 1, a(2) = 5.

%I A321703 #36 Jan 20 2019 08:39:36
%S A321703 1,1,5,20,81,328,1328,5377,21771,88149,356908,1445091,5851054,
%T A321703 23690434,95920609,388374617,1572496721,6366909240,25779089221,
%U A321703 104377401344,422615470156,1711135105065,6928244597163,28051889681905,113579782539432,459875150943079,1861996472668870,7539069804318358,30525070454573633,123593487053663201,500419812783602493
%N A321703 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), a(0) = 1, a(1) = 1, a(2) = 5.
%C A321703 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 A321703 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 A321703 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 A321703 Let k = Pi/7.
%C A321703 Let X = (sin(4k)*sin(8k))/(sin(2k)*sin(2k)),
%C A321703     Y = (sin(8k)*sin(2k))/(sin(4k)*sin(4k)),
%C A321703     Z = (sin(2k)*sin(4k))/(sin(8k)*sin(8k)).
%C A321703 Then {X,Y,Z} are the roots of the cubic equation x^3 - 3*x^2 - 4*x - 1 = 0.
%C A321703 This sequence: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k))).
%C A321703 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 A321715: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k))).
%H A321703 Colin Barker, <a href="/A321703/b321703.txt">Table of n, a(n) for n = 0..1000</a>
%H A321703 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,4,1).
%F A321703 G.f.: (-1 + 2*x + 2*x^2)/(-1 + 3*x + 4*x^2 + x^3). - _Stefano Spezia_, Jan 14 2019
%t A321703 CoefficientList[Series[(-1 + 2 x + 2 x^2)/(-1 + 3 x + 4 x^2 + x^3), {x, 0, 50}], x] (* _Stefano Spezia_, Jan 14 2019 *)
%o A321703 (PARI) Vec((1 - 2*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3) + O(x^30)) \\ _Colin Barker_, Jan 15 2019
%Y A321703 Cf. A321715, A122600.
%K A321703 nonn,easy
%O A321703 0,3
%A A321703 _Kai Wang_, Jan 14 2019