A321461 a(n) = -a(n-1) + 2*a(n-2) + a(n-3), a(0) = -1, a(1) = -2, a(2) = -4.

-1, -2, -4, -1, -9, 3, -22, 19, -60, 76, -177, 269, -547, 908, -1733, 3002, -5560, 9831, -17949, 32051, -58118, 104271, -188456, 338880, -611521, 1100825, -1984987, 3575116, -6444265, 11609510, -20922924

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

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

A321175: (a, b, c) = (1, -2, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).

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).

Crossrefs

Cf. A321175, A094648.

Programs

PARI

Vec(-(1 + 3*x + 4*x^2) / (1 + x - 2*x^2 - x^3) + O(x^40)) \\ Colin Barker, Feb 19 2019

Formula

G.f.: -(1 + 3*x + 4*x^2) / (1 + x - 2*x^2 - x^3). - Colin Barker, Feb 19 2019

Extensions

Title corrected by Colin Barker, Jan 12 2019