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A322458 Sum of n-th powers of the roots of x^3 - 49*x + 49.

%I A322458 #35 Dec 17 2018 12:02:39
%S A322458 3,0,98,-147,4802,-12005,242501,-823543,12470794,-52236156,651422513,
%T A322458 -3170640550,34479274781,-187281090087,1844845851219,-10866257878532,
%U A322458 99574220123994,-622844082757799,5411583422123774,-35398496841207857,295686947739197077,-1999693932903249919
%N A322458 Sum of n-th powers of the roots of x^3 - 49*x + 49.
%C A322458 Let A = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
%C A322458 In general, for integer h, k let
%C A322458   X = 2*sqrt(7)*A^(h+k+1)/(B^h*C^k),
%C A322458 Y = 2*sqrt(7)*B^(h+k+1)/(C^h*A^k),
%C A322458 Z = 2*sqrt(7)*C^(h+k+1)/(A^h*B^k).
%C A322458 then X, Y, Z are the roots of a monic equation
%C A322458     t^3 + a*t^2 + b*t + c = 0
%C A322458 where a, b, c are integers and c = 1 or -1.
%C A322458 Then X^n + Y^n + Z^n , n = 0, 1, 2, ... is an integer sequence.
%C A322458 This sequence has (h,k) = (0,1).
%H A322458 Colin Barker, <a href="/A322458/b322458.txt">Table of n, a(n) for n = 0..1000</a>
%H A322458 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,49,-49).
%F A322458 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 = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
%F A322458 a(n) = 49*a(n-2) - 49 a(n-3) for n>2.
%F A322458 G.f.: (3 - 49*x^2) / (1 - 49*x^2 + 49*x^3). - _Colin Barker_, Dec 09 2018
%t A322458 LinearRecurrence[{0, 49, -49}, {3, 0, 98}, 50] (* _Amiram Eldar_, Dec 09 2018 *)
%o A322458 (PARI) Vec((3 - 49*x^2) / (1 - 49*x^2 + 49*x^3) + O(x^25)) \\ _Colin Barker_, Dec 09 2018
%o A322458 (PARI) polsym(x^3 - 49*x + 49, 25) \\ _Joerg Arndt_, Dec 17 2018
%Y A322458 Similar sequences with (h,k) values: A275830 (0,0).
%K A322458 sign,easy
%O A322458 0,1
%A A322458 _Kai Wang_, Dec 09 2018