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A322504
a(n) = -4*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 1, a(1) = -2, a(2) = 4.
1
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
0,2
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) = (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)).
FORMULA
G.f.: (1 + 2*x - x^2) / (1 + 4*x + 3*x^2 - x^3). - Colin Barker, Jan 11 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 *)
PROG
(PARI) Vec((1 + 2*x - x^2) / (1 + 4*x + 3*x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 11 2019
(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
CROSSREFS
Sequence in context: A301362 A130018 A352176 * A099754 A105633 A287709
KEYWORD
sign,easy
AUTHOR
Kai Wang, Jan 10 2019
STATUS
approved