A322461 - OEIS

%I #32 Jan 13 2019 07:46:33

%S 3,-8,54,-389,2834,-20673,150825,-1100401,8028410,-58574450,427353149,

%T -3117924532,22748056061,-165967472679,1210881576595,-8834467193304,

%U 64455362190778,-470259679983109,3430966161717678,-25031975531635101,182630713764509309

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

%C Let A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).

%C For integers h, k let

%C    X = 2*sqrt(7)*A^(h+k+1)/(B^h*C^k),

%C    Y = 2*sqrt(7)*B^(h+k+1)/(C^h*A^k),

%C    Z = 2*sqrt(7)*C^(h+k+1)/(A^h*B^k).

%C then X, Y, Z are the roots of a monic equation

%C     t^3 + a*t^2 + b*t + c = 0

%C where a, b, c are integers and c = 1 or -1.

%C Then X^n + Y^n + Z^n, n = 0, 1, 2, ... is an integer sequence.

%C This sequence has (h,k) = (0,1).

%H Colin Barker, <a href="/A322461/b322461.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-8,-5,1).

%F 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 a(n) = -8*a(n-2) - 5*a(n-2) + a(n-3) for n > 2.

%F G.f.: (3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3). - _Colin Barker_, Dec 09 2018

%t LinearRecurrence[{-8, -5, 1}, {3, -8, 54}, 50] (* _Amiram Eldar_, Dec 09 2018 *)

%o (PARI) Vec((3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3) + O(x^25)) \\ _Colin Barker_, Dec 09 2018

%o (PARI) polsym(x^3 + 8*x^2 + 5*x - 1, 25) \\ _Joerg Arndt_, Dec 17 2018

%Y Similar sequences with (h,k) values: A094648 (0,0), A274075 (1,0).

%K sign,easy

%O 0,1

%A _Kai Wang_, Dec 09 2018