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A322460 Sum of n-th powers of the roots of x^3 + 95*x^2 - 88*x - 1.

%I A322460 #30 Dec 17 2018 12:02:26
%S A322460 3,-95,9201,-882452,84642533,-8118687210,778722945402,-74693039645137,
%T A322460 7164358266796181,-687186244111463849,65913082025027484446,
%U A322460 -6322208017501153044901,606409425694567846432994,-58165183833442021851601272,5579050171430096545235179411
%N A322460 Sum of n-th powers of the roots of x^3 + 95*x^2 - 88*x - 1.
%C A322460 Let A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).
%C A322460 In general, for integer h, k let
%C A322460   X = A^(h+k)/(B^h*C^k),
%C A322460 Y = B^(h+k)/(C^h*A^k),
%C A322460 Z = C^(h+k)/(A^h*B^k).
%C A322460 then X, Y, Z are the roots of a monic equation
%C A322460     t^3 + a*t^2 + b*t + c = 0
%C A322460 where a, b, c are integers and c = 1 or -1.
%C A322460 Then X^n + Y^n + Z^n , n = 0, 1, 2, ... is an integer sequence.
%C A322460 This sequence has (h,k) = (1,3).
%H A322460 Colin Barker, <a href="/A322460/b322460.txt">Table of n, a(n) for n = 0..500</a>
%H A322460 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-95,88,1).
%F A322460 a(n) = (A^4/(B*C^3))^n + (B^4/(C*A^3))^n +  (C^4/(A*B^3))^n.
%F A322460 a(n) = -95*a(n-1) + 88*a(n-2) + a(n-3) for n>2.
%F A322460 G.f.: (3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3). - _Colin Barker_, Dec 09 2018
%p A322460 seq(coeff(series((3+190*x-88*x^2)/(1+95*x-88*x^2-x^3),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Dec 11 2018
%t A322460 LinearRecurrence[{-95, 88, 1}, {3, -95, 9201}, 50] (* _Amiram Eldar_, Dec 09 2018 *)
%o A322460 (PARI) Vec((3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3) + O(x^15)) \\ _Colin Barker_, Dec 09 2018
%o A322460 (PARI) polsym(x^3 + 95*x^2 - 88*x - 1, 25)  \\ _Joerg Arndt_, Dec 17 2018
%Y A322460 Similar sequences with (h,k) values: A215076 (0,1), A274220 (1,0), A274663 (1,1), A248417 (1,2), A215560 (2,1).
%K A322460 sign,easy
%O A322460 0,1
%A A322460 _Kai Wang_, Dec 09 2018