A322459 Sum of n-th powers of the roots of x^3 + 7*x^2 + 14*x + 7.

3, -7, 21, -70, 245, -882, 3234, -12005, 44933, -169099, 638666, -2417807, 9167018, -34790490, 132119827, -501941055, 1907443237, -7249766678, 27557748813, -104759610858, 398257159370, -1514069805269, 5756205681709, -21884262613787, 83201447389466, -316323894905207

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

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

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-7,-14,-7).

Crossrefs

Similar sequences with (h,k) values: A275831 (0,0), A215575 (0,2).

Programs

Mathematica

LinearRecurrence[{-7, -14, -7},{3, -7, 21}, 50] (* Amiram Eldar, Dec 09 2018 *)
CoefficientList[Series[(3+14*x+14*x^2)/(1+7*x+14*x^2+7*x^3), {x, 0, 25}], x] (* G. C. Greubel, Dec 16 2018 *)

PARI

Vec((3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3) + O(x^40)) \\ Colin Barker, Dec 09 2018

PARI

polsym(x^3 + 7*x^2 + 14*x + 7, 25) \\ Joerg Arndt, Dec 17 2018

Formula

a(n) = (sqrt(7))^n*( (A/(2*B*C))^n + (B/(2*C*A))^n + (C/(2*A*B))^n ).

a(n) = -7*a(n-1) - 14*a(n-2) - 7*a(n-3) for n>2.

G.f.: (3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3). - Colin Barker, Dec 09 2018