A322461 Sum of n-th powers of the roots of x^3 + 8*x^2 + 5*x - 1.

3, -8, 54, -389, 2834, -20673, 150825, -1100401, 8028410, -58574450, 427353149, -3117924532, 22748056061, -165967472679, 1210881576595, -8834467193304, 64455362190778, -470259679983109, 3430966161717678, -25031975531635101, 182630713764509309

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).

For integers h, k let

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

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

Z = 2*sqrt(7)*C^(h+k+1)/(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) = (0,1).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-8,-5,1).

Crossrefs

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

Programs

Mathematica

LinearRecurrence[{-8, -5, 1}, {3, -8, 54}, 50] (* Amiram Eldar, Dec 09 2018 *)

PARI

Vec((3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3) + O(x^25)) \\ Colin Barker, Dec 09 2018

PARI

polsym(x^3 + 8*x^2 + 5*x - 1, 25) \\ Joerg Arndt, Dec 17 2018

Formula

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).

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

G.f.: (3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3). - Colin Barker, Dec 09 2018