A321175 a(n) = -a(n-1) + 2*a(n-2) + a(n-3), a(0) = -1, a(1) = -2, a(2) = 3.

-1, -2, 3, -8, 12, -25, 41, -79, 136, -253, 446, -816, 1455, -2641, 4735, -8562, 15391, -27780, 50000, -90169, 162389, -292727, 527336, -950401, 1712346, -3085812, 5560103, -10019381, 18053775, -32532434, 58620603

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) = (1, -2, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).

A094648: (a, b, c) = (1, -2, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).

A321461 : (a, b, c) = (1, -2, -1), (u, v, w) = (2*cos(4k), 2*cos(8k), 2*cos(2k)).

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-1,2,1).

Formula

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

Mathematica

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

Prog

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

Crossrefs

Cf. A321461, A094648.

Sequence in context: A194452 A242516 A282281 * A079980 A025080 A024468

Adjacent sequences: A321172 A321173 A321174 * A321176 A321177 A321178

Keyword

sign,easy

Author

Kai Wang, Jan 10 2019

Status

approved