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A322459 Sum of n-th powers of the roots of x^3 + 7*x^2 + 14*x + 7.

%I A322459 #28 Dec 17 2018 12:02:35
%S A322459 3,-7,21,-70,245,-882,3234,-12005,44933,-169099,638666,-2417807,
%T A322459 9167018,-34790490,132119827,-501941055,1907443237,-7249766678,
%U A322459 27557748813,-104759610858,398257159370,-1514069805269,5756205681709,-21884262613787,83201447389466,-316323894905207
%N A322459 Sum of n-th powers of the roots of x^3 + 7*x^2 + 14*x + 7.
%C A322459 Let A = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
%C A322459 In general, for integer h, k let
%C A322459 X = sqrt(7)*A^(h+k-1)/(2*B^h*C^k),
%C A322459 Y = sqrt(7)*B^(h+k-1)/(2*C^h*A^k),
%C A322459 Z = sqrt(7)*C^(h+k-1)/(2*A^h*B^k),
%C A322459 then X, Y, Z are the roots of a monic equation
%C A322459     t^3 + a*t^2 + b*t + c = 0
%C A322459 where a, b, c are integers and c = 1 or -1.
%C A322459 Then X^n + Y^n + Z^n , n = 0, 1, 2, ... is an integer sequence.
%C A322459 This sequence has (h,k) = (1,1).
%H A322459 Colin Barker, <a href="/A322459/b322459.txt">Table of n, a(n) for n = 0..1000</a>
%H A322459 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-7,-14,-7).
%F A322459 a(n) = (sqrt(7))^n*( (A/(2*B*C))^n + (B/(2*C*A))^n + (C/(2*A*B))^n ).
%F A322459 a(n) = -7*a(n-1) - 14*a(n-2) - 7*a(n-3) for n>2.
%F A322459 G.f.: (3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3). - _Colin Barker_, Dec 09 2018
%t A322459 LinearRecurrence[{-7, -14, -7},{3, -7, 21}, 50] (* _Amiram Eldar_, Dec 09 2018 *)
%t A322459 CoefficientList[Series[(3+14*x+14*x^2)/(1+7*x+14*x^2+7*x^3), {x, 0, 25}], x] (* _G. C. Greubel_, Dec 16 2018 *)
%o A322459 (PARI) Vec((3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3) + O(x^40)) \\ _Colin Barker_, Dec 09 2018
%o A322459 (PARI) polsym(x^3 + 7*x^2 + 14*x + 7, 25) \\ _Joerg Arndt_, Dec 17 2018
%Y A322459 Similar sequences with (h,k) values: A275831 (0,0), A215575 (0,2).
%K A322459 sign,easy
%O A322459 0,1
%A A322459 _Kai Wang_, Dec 09 2018