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A322461 Sum of n-th powers of the roots of x^3 + 8*x^2 + 5*x - 1.

%I A322461 #32 Jan 13 2019 07:46:33
%S A322461 3,-8,54,-389,2834,-20673,150825,-1100401,8028410,-58574450,427353149,
%T A322461 -3117924532,22748056061,-165967472679,1210881576595,-8834467193304,
%U A322461 64455362190778,-470259679983109,3430966161717678,-25031975531635101,182630713764509309
%N A322461 Sum of n-th powers of the roots of x^3 + 8*x^2 + 5*x - 1.
%C A322461 Let A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).
%C A322461 For integers h, k let
%C A322461    X = 2*sqrt(7)*A^(h+k+1)/(B^h*C^k),
%C A322461    Y = 2*sqrt(7)*B^(h+k+1)/(C^h*A^k),
%C A322461    Z = 2*sqrt(7)*C^(h+k+1)/(A^h*B^k).
%C A322461 then X, Y, Z are the roots of a monic equation
%C A322461     t^3 + a*t^2 + b*t + c = 0
%C A322461 where a, b, c are integers and c = 1 or -1.
%C A322461 Then X^n + Y^n + Z^n, n = 0, 1, 2, ... is an integer sequence.
%C A322461 This sequence has (h,k) = (0,1).
%H A322461 Colin Barker, <a href="/A322461/b322461.txt">Table of n, a(n) for n = 0..1000</a>
%H A322461 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-8,-5,1).
%F A322461 a(n) = (2*sqrt(7)*A^2/C)^n + (2*sqrt(7)*B^2/A)^n + (2*sqrt(7)*C^2/B)^n, where A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).
%F A322461 a(n) = -8*a(n-2) - 5*a(n-2) + a(n-3) for n > 2.
%F A322461 G.f.: (3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3). - _Colin Barker_, Dec 09 2018
%t A322461 LinearRecurrence[{-8, -5, 1}, {3, -8, 54}, 50] (* _Amiram Eldar_, Dec 09 2018 *)
%o A322461 (PARI) Vec((3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3) + O(x^25)) \\ _Colin Barker_, Dec 09 2018
%o A322461 (PARI) polsym(x^3 + 8*x^2 + 5*x - 1, 25) \\ _Joerg Arndt_, Dec 17 2018
%Y A322461 Similar sequences with (h,k) values: A094648 (0,0), A274075 (1,0).
%K A322461 sign,easy
%O A322461 0,1
%A A322461 _Kai Wang_, Dec 09 2018