A375892 a(n) = (32)^n*cos (nC - nA), 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, 8, -896, -22528, 557056, 31981568, -58720256, -33688649728, -478888853504, 26834955665408, 919741476634624, -12763130975223808, -1146025367677435904, -5266959764209795072, 1089258620274337644544, 22821504722940232466432, -750256751593878028550144

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..16.

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 (16,-1024).

Formula

a(n) = (32)^n*cos (nC - nA), 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) = 16 a(n-1) - 1024 a(n-2), where a(0) = 1, a(1) = 8.

From Stefano Spezia, Sep 23 2024: (Start)

G.f.: (1 - 8*x)/(1 - 16*x + 1024*x^2).

E.g.f.: exp(8*x)*cos(8*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[(32)^n  Cos[n (A[c, a, b] - A[a, b, c])]], {n, 0, 18}]
(* Program 2 *)
LinearRecurrence[{16, -1024}, {1, 8}, 30]

Crossrefs

Cf. A375880.

Sequence in context: A175411 A027725 A265239 * A290023 A112578 A112579

Adjacent sequences: A375889 A375890 A375891 * A375893 A375894 A375895

Keyword

sign,easy

Author

Clark Kimberling, Sep 22 2024

Status

approved