A274975 Sum of n-th powers of the three roots of x^3-2*x^2-x+1.

3, 2, 6, 11, 26, 57, 129, 289, 650, 1460, 3281, 7372, 16565, 37221, 83635, 187926, 422266, 948823, 2131986, 4790529, 10764221, 24186985, 54347662, 122118088, 274396853, 616564132, 1385407029, 3112981337, 6994805571, 15717185450, 35316195134, 79354770147, 178308549978

Offset: 0

Author

Kai Wang, Jul 14 2016

Keywords

• nonn
• easy

Comments

a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3-2*x^2-x+1.

x1 = 1/(2*cos(Pi/7)),

x2 = 1/(-2*cos(2*Pi/7)),

x3 = 1/(-2*cos(4*Pi/7)).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
• Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Crossrefs

Cf. A096975.

3 followed by terms of A033304.

Programs

Mathematica

CoefficientList[Series[-(x^2 + 4 x - 3)/(x^3 - x^2 - 2 x + 1), {x, 0, 32}], x] (* Michael De Vlieger, Jul 14 2016 *)

PARI

Vec(-(x^2+4*x-3)/(x^3-x^2-2*x+1) + O(x^50)) \\ Colin Barker, Aug 02 2016

Formula

G.f.: -(x^2+4*x-3)/(x^3-x^2-2*x+1). - Alois P. Heinz, Jul 14 2016

a(0)=3, a(1)=2, a(2)=6; thereafter a(n)=2*a(n-1)+a(n-2)-a(n-3).

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

a(n) = A033304(n-1) for n>0.