A321174 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = -4, a(2) = 5.

-1, -4, 5, -15, 31, -72, 160, -361, 810, -1821, 4091, -9193, 20656, -46414, 104291, -234340, 526557, -1183163, 2658543, -5973692, 13422764, -30160677, 67770426, -152278765, 342167279, -768842897, 1727574308, -3881824234, 8722379879, -19599009684, 44038575013

Offset: 0

Author

Kai Wang, Jan 10 2019

Keywords

• sign
• easy

Comments

Let {X,Y,Z} be the roots of the cubic equation t^3 + at^2 + bt + c = 0 where {a, b, c} are integers.

Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers.

Then {p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...} is an integer sequence with the recurrence relation: p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).

Let k = Pi/7.

This sequence has (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).

A033304, A274975: (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).

A321173 : (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(4k), 2*cos(8k), 2*cos(2k)).

X = sin(2k)/sin(4k), Y = sin(4k)/sin(8k), Z = sin(8k)/sin(2k).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,1,1).

Crossrefs

Cf. A321173, A033304, A274975.

Programs

Mathematica

LinearRecurrence[{-2,1,1},{-1,-4,5},50] (* Stefano Spezia, Jan 11 2019 *)

PARI

Vec(-(1 + 6*x + 2*x^2) / (1 + 2*x - x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 11 2019

Formula

G.f.: -(1 + 6*x + 2*x^2) / (1 + 2*x - x^2 - x^3). - Colin Barker, Jan 11 2019