A322504 a(n) = -4*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 1, a(1) = -2, a(2) = 4.

1, -2, 4, -9, 22, -57, 153, -419, 1160, -3230, 9021, -25234, 70643, -197849, 554233, -1552742, 4350420, -12189221, 34152882, -95693445, 268125913, -751270435, 2105010556, -5898105006, 16526117921, -46305146110, 129744125671, -363534946433, 1018602262609, -2854060085466, 7996898607604

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) = (4, 3, -1), (u, v, w) = (1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k))).

A215404: (a, b, c) = (4, 3, -1), (u, v, w) = (1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k))).

A136776: (a, b, c) = (4, 3, -1), (u, v, w) = (1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k))).

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

Links

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

Crossrefs

Cf. A215404, A136776.

Programs

Magma

I:=[1,-2,4]; [n le 3 select I[n] else -4*Self(n-1) - 3*Self(n-2) + Self(n-3): n in [1..31]]; // Vincenzo Librandi, Jan 13 2019

Mathematica

LinearRecurrence[{-4,-3,1},{1,-2,4},50] (* Stefano Spezia, Jan 11 2019 *)
RecurrenceTable[{a[0]==1, a[1]==-2, a[2]==4, a[n]==-4 a[n-1]-3 a[n-2]+a[n-3]}, a, {n, 30}] (* Vincenzo Librandi, Jan 13 2019 *)

PARI

Vec((1 + 2*x - x^2) / (1 + 4*x + 3*x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 11 2019

Formula

G.f.: (1 + 2*x - x^2) / (1 + 4*x + 3*x^2 - x^3). - Colin Barker, Jan 11 2019