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A322455 Sum of n-th powers of the roots of x^3 - 20*x^2 - 9*x - 1.

%I A322455 #34 Dec 17 2018 12:02:44
%S A322455 3,20,418,8543,174642,3570145,72983221,1491970367,30499826474,
%T A322455 623497246004,12745935328713,260560681614770,5326550547499821,
%U A322455 108888803019858063,2225975576006274419,45504837297851710768,930239414944110543194,19016557810138882535211
%N A322455 Sum of n-th powers of the roots of x^3 - 20*x^2 - 9*x - 1.
%C A322455 Let A = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
%C A322455 In general, for integer h, k let
%C A322455 X = (B^h*C^k)/A^(h+k),
%C A322455 Y = (C^h*A^k)/B^(h+k),
%C A322455 Z = (A^h*B^k)/C^(h+k).
%C A322455 then X, Y, Z are the roots of a monic equation
%C A322455     t^3 + a*t^2 + b*t + c = 0
%C A322455 where a, b, c are integers and c = 1 or -1.
%C A322455 Then X^n + Y^n + Z^n, n = 0, 1, 2, ... is an integer sequence.
%C A322455 This sequence has (h,k) = (1,3) and its other half is A320918.
%H A322455 Colin Barker, <a href="/A322455/b322455.txt">Table of n, a(n) for n = 0..750</a>
%H A322455 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (20,9,1).
%F A322455 a(n) = (B*C^3/A^4)^n + (C*A^3/B^4)^n  + (A*B^3/C^4)^n.
%F A322455 a(n) = 20*a(n-1) + 9*a(n-2) + a(n-3) for n > 2.
%F A322455 G.f.: (3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3). - _Colin Barker_, Dec 09 2018
%t A322455 CoefficientList[Series[(3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3) , {x, 0, 50}], x] (* _Amiram Eldar_, Dec 09 2018 *)
%o A322455 (PARI) Vec((3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3) + O(x^20)) \\ _Colin Barker_, Dec 09 2018
%o A322455 (PARI) polsym(x^3 - 20*x^2 - 9*x - 1, 25) \\ _Joerg Arndt_, Dec 17 2018
%Y A322455 Similar sequences with (h,k) values: A033304 (0,1), A215076 (1,1), A274032 (1,2).
%K A322455 nonn,easy
%O A322455 0,1
%A A322455 _Kai Wang_, Dec 09 2018