A375893 a(n) = (64)^n*cos (nA - nB), where A, B, C are, respectively, the angles opposite sides BC, CA, AB in a triangle ABC having sidelengths |BC| = 2, |CA| = 3, |AB| = 4; ABC is the smallest integer-sided scalene triangle.

1, 61, 3346, 158356, 5614216, 36308176, -18566231264, -2413798503104, -218436134121344, -16762289694089984, -1150284937317953024, -71676423765797694464, -4032956596172983138304, -198434072988396586348544, -7689966686659844600029184

Offset

0,2

Comments

If a prime p divides a term, then the indices n such that p divides a(n) comprise an arithmetic sequence; see the Renault paper in References. For a guide to related sequences, see A375880.

Links

Table of n, a(n) for n=0..14.

Marc Renault, The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m, Math. Mag. 86 (2013) pp. 372-380.

Index entries for linear recurrences with constant coefficients, signature (122,-4096).

Formula

a(n) = (64)^n*cos (nA - nB), where A, B, C are, respectively, the angles opposite sides BC, CA, AB in a triangle ABC having sidelengths |BC| = 2, |CA| = 3, |AB| = 4.

a(n) = 122 a(n-1) - 4096 a(n-2), where a(0) = 1, a(1) = 61.

From Stefano Spezia, Sep 23 2024: (Start)

G.f.: (1 - 61*x)/(1 - 122*x+ 4096*x^2).

E.g.f.: exp(61*x)*cos(5*sqrt(15)*x). (End)

Mathematica

(* Program 1 *)
A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2  b  c)];
{a, b, c} = {2, 3, 4};
Table[TrigExpand[(64)^n  Cos[n (A[a, b, c] - A[b, c, a])]], {n, 0, 18}]
(* Program 2 *)
LinearRecurrence[{122, -4096}, {1, 61}, 30]

Crossrefs

Cf. A375880.

Sequence in context: A191092 A234028 A135647 * A269025 A207231 A207224

Adjacent sequences: A375890 A375891 A375892 * A375894 A375895 A375896

Keyword

sign,easy

Author

Clark Kimberling, Sep 22 2024

Status

approved