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Sum of n-th powers of the roots of x^3 + 11*x^2 - 4*x - 1.
3

%I #28 Jul 11 2016 15:33:04

%S 3,-11,129,-1460,165655,-187926,2131986,-24186985,274396853,

%T -3112981337,35316195134,-400655674969,4545364223858,-51566312967180,

%U 585010243859443,-6636832570098735,75293632933556677,-854192282305658944,9690652804526376357,-109938656346079219026,1247233638742671255770

%N Sum of n-th powers of the roots of x^3 + 11*x^2 - 4*x - 1.

%C This is the other half for A274663.

%C a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3 + 11*x^2 - 3*x - 1.

%C x1 = (cos(2*Pi/7)*cos(4*Pi/7))/(cos(Pi/7))^2,

%C x2 = -(cos(4*Pi/7)*cos(Pi/7))/(cos(2*Pi/7))^2,

%C x3 = -(cos(Pi/7) *cos(2*Pi/7))(cos(4*Pi/7))^2.

%H Colin Barker, <a href="/A274664/b274664.txt">Table of n, a(n) for n = 0..900</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-11,4,1).

%F a(0) = 3, a(1) = -11, a(2) = 129; thereafter a(n) = -11*a(n-1) + 4*a(n-2) + a(n-3).

%F a(n) = ((cos(2*Pi/7)*cos(4*Pi/7))/(cos(Pi/7))^2)^n +(-(cos(4*Pi/7)*cos(Pi/7))/(cos(2*Pi/7))^2)^n +(-(cos(Pi/7)*cos(2*Pi/7))/(cos(4*Pi/7))^2)^n.

%F G.f.: (3+22*x-4*x^2+149090*x^4+1639990*x^5-596360*x^6-149090*x^7) / (1+11*x-4*x^2-x^3). - _Colin Barker_, Jul 03 2016

%o (PARI) Vec((3+22*x-4*x^2+149090*x^4+1639990*x^5-596360*x^6-149090*x^7) / (1+11*x-4*x^2-x^3) + O(x^20)) \\ _Colin Barker_, Jul 03 2016

%o (PARI) first(n)=my(x='x); polsym(x^3+11*x^2-4*x-1,n) \\ _Charles R Greathouse IV_, Jul 10 2016

%Y Cf. A274663.

%K sign,easy

%O 0,1

%A _Kai Wang_, Jul 01 2016