A322460 Sum of n-th powers of the roots of x^3 + 95*x^2 - 88*x - 1.

3, -95, 9201, -882452, 84642533, -8118687210, 778722945402, -74693039645137, 7164358266796181, -687186244111463849, 65913082025027484446, -6322208017501153044901, 606409425694567846432994, -58165183833442021851601272, 5579050171430096545235179411

Offset: 0

Author

Kai Wang, Dec 09 2018

Keywords

• sign
• easy

Comments

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

In general, for integer h, k let

X = A^(h+k)/(B^h*C^k),

Y = B^(h+k)/(C^h*A^k),

Z = C^(h+k)/(A^h*B^k).

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

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

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

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

This sequence has (h,k) = (1,3).

Links

• Colin Barker, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (-95,88,1).

Crossrefs

Similar sequences with (h,k) values: A215076 (0,1), A274220 (1,0), A274663 (1,1), A248417 (1,2), A215560 (2,1).

Programs

Maple

seq(coeff(series((3+190*x-88*x^2)/(1+95*x-88*x^2-x^3),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 11 2018

Mathematica

LinearRecurrence[{-95, 88, 1}, {3, -95, 9201}, 50] (* Amiram Eldar, Dec 09 2018 *)

PARI

Vec((3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3) + O(x^15)) \\ Colin Barker, Dec 09 2018

PARI

polsym(x^3 + 95*x^2 - 88*x - 1, 25)  \\ Joerg Arndt, Dec 17 2018

Formula

a(n) = (A^4/(B*C^3))^n + (B^4/(C*A^3))^n + (C^4/(A*B^3))^n.

a(n) = -95*a(n-1) + 88*a(n-2) + a(n-3) for n>2.

G.f.: (3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3). - Colin Barker, Dec 09 2018