A274592 Sum of n-th powers of the roots of x^3 -31* x^2 - 25*x - 1.

3, 31, 1011, 32119, 1020995, 32454831, 1031656755, 32793751175, 1042430160131, 33136210400191, 1053316070160371, 33482245865136407, 1064315659783638083, 33831894915991351119, 1075430116136187973171, 34185195288781394584359, 1086660638750543922795523

Offset: 0

Author

Kai Wang, Jun 29 2016

Keywords

• nonn
• easy

Comments

This is one side of a two sided sequence (see A248417).

a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial

x^3 -31* x^2 - 25*x - 1.

x1 = (tan(Pi/7))^2/(tan(2*Pi/7)*tan(4*Pi/7)),

x2 = (tan(2*Pi/7))^2/(tan(4*Pi/7)*tan(Pi/7)),

x3 = (tan(4*Pi/7))^2/(tan(Pi/7)*tan(2*Pi/7)).

Links

• Colin Barker, Table of n, a(n) for n = 0..600
• Index entries for linear recurrences with constant coefficients, signature (31,25,1).

Crossrefs

Cf. A248417, A274032, A274075.

Programs

Mathematica

LinearRecurrence[{31,25,1},{3,31,1011},20] (* Harvey P. Dale, Feb 02 2022 *)

PARI

Vec((3-62*x-25*x^2)/(1-31*x-25*x^2-x^3) + O(x^20)) \\ Colin Barker, Jun 30 2016

PARI

polsym(x^3 -31* x^2 - 25*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

Formula

a(n) = ((tan(Pi/7))^2/(tan(2*Pi/7)*tan(4*Pi/7)))^n + ((tan(2*Pi/7))^2/(tan(4*Pi/7)*tan(Pi/7)))^n + ((tan(4*Pi/7))^2/(tan(Pi/7)*tan(2*Pi/7)))^n.

a(n) = 31*a(n-1) + 25*a(n-2) + a(n-3).

G.f.: (3-62*x-25*x^2) / (1-31*x-25*x^2-x^3). - Colin Barker, Jun 30 2016