A321173 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = 3, a(2) = -9.

-1, 3, -9, 20, -46, 103, -232, 521, -1171, 2631, -5912, 13284, -29849, 67070, -150705, 338631, -760897, 1709720, -3841706, 8632235, -19396456, 43583441, -97931103, 220049191, -494446044, 1111010176, -2496417205, 5609398542, -12604204113, 28321389563, -63637584697

Offset

0,2

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(4k), 2*cos(8k), 2*cos(2k)).

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

A321174 : (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).

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).

Formula

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

Mathematica

LinearRecurrence[{-2, 1, 1}, {-1, 3, -9}, 50] (* Stefano Spezia, Jan 11 2019 *)

Prog

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

Crossrefs

Cf. A321174, A033304, A274975.

Sequence in context: A203861 A073801 A065891 * A147328 A364914 A220770

Adjacent sequences: A321170 A321171 A321172 * A321174 A321175 A321176

Keyword

sign,easy

Author

Kai Wang, Jan 10 2019

Status

approved