A322458 Sum of n-th powers of the roots of x^3 - 49*x + 49.

3, 0, 98, -147, 4802, -12005, 242501, -823543, 12470794, -52236156, 651422513, -3170640550, 34479274781, -187281090087, 1844845851219, -10866257878532, 99574220123994, -622844082757799, 5411583422123774, -35398496841207857, 295686947739197077, -1999693932903249919

Offset: 0

Author

Kai Wang, Dec 09 2018

Keywords

• sign
• 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 = 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 (0,49,-49).

Crossrefs

Similar sequences with (h,k) values: A275830 (0,0).

Programs

Mathematica

LinearRecurrence[{0, 49, -49}, {3, 0, 98}, 50] (* Amiram Eldar, Dec 09 2018 *)

PARI

Vec((3 - 49*x^2) / (1 - 49*x^2 + 49*x^3) + O(x^25)) \\ Colin Barker, Dec 09 2018

PARI

polsym(x^3 - 49*x + 49, 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 = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).

a(n) = 49*a(n-2) - 49 a(n-3) for n>2.

G.f.: (3 - 49*x^2) / (1 - 49*x^2 + 49*x^3). - Colin Barker, Dec 09 2018