A321715 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), a(0) = 1, a(1) = -1, a(2) = -1 .

1, -1, -1, -6, -23, -94, -380, -1539, -6231, -25229, -102150, -413597, -1674620, -6780398, -27453271, -111156025, -450061557, -1822262042, -7378188379, -29873674862, -120956040144, -489741008259, -1982920860215, -8028682653825, -32507472410594, -131620068707297, -532918778418092, -2157744082494058, -8736527429861839, -35373477397979841, -143224285995880937

Offset: 0

Author

Kai Wang, Jan 14 2019

Keywords

• sign
• easy

Comments

In general, let {X,Y,Z} be the roots of the cubic equation x^3 + ax^2 + xt + 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.

Let X = (sin(4k)*sin(8k))/(sin(2k)*sin(2k)),

Y = (sin(8k)*sin(2k))/(sin(4k)*sin(4k)),

Z = (sin(2k)*sin(4k))/(sin(8k)*sin(8k)).

Then {X,Y,Z} are the roots of the cubic equation x^3 - 3*x^2 - 4*x - 1 = 0.

This sequence: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k))).

A122600: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k))).

A321703: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k))).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,4,1).

Crossrefs

Cf. A321703, A122600.

Programs

PARI

Vec((1 - 4*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 15 2019

Formula

G.f.: (1 - 4*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3). - Colin Barker, Jan 15 2019