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

3, 20, 418, 8543, 174642, 3570145, 72983221, 1491970367, 30499826474, 623497246004, 12745935328713, 260560681614770, 5326550547499821, 108888803019858063, 2225975576006274419, 45504837297851710768, 930239414944110543194, 19016557810138882535211

Offset: 0

Author

Kai Wang, Dec 09 2018

Keywords

• nonn
• easy

Comments

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

In general, for integer h, k let

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

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

Z = (A^h*B^k)/C^(h+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) and its other half is A320918.

Links

• Colin Barker, Table of n, a(n) for n = 0..750
• Index entries for linear recurrences with constant coefficients, signature (20,9,1).

Crossrefs

Similar sequences with (h,k) values: A033304 (0,1), A215076 (1,1), A274032 (1,2).

Programs

Mathematica

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 *)

PARI

Vec((3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3) + O(x^20)) \\ Colin Barker, Dec 09 2018

PARI

polsym(x^3 - 20*x^2 - 9*x - 1, 25) \\ Joerg Arndt, Dec 17 2018

Formula

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

a(n) = 20*a(n-1) + 9*a(n-2) + a(n-3) for n > 2.

G.f.: (3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3). - Colin Barker, Dec 09 2018