A375882 a(n) = 4^n cos(nC), where C is the angle opposite side AB in a triangle ABC having sidelengths |BC|=2, |CA| = 3, |AB| = 4; ABC is the smallest integer-sided scalene triangle.

1, -1, -14, 44, 136, -976, -224, 16064, -28544, -199936, 856576, 1485824, -16676864, 9580544, 247668736, -648626176, -2665447424, 15708913664, 11229331456, -273801281536, 367933259776, 3644953985024, -13176840126464, -31965583507456, 274760609038336

Offset

0,3

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..24.

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 (-2,-16).

Formula

a(n) = 4^n cos (nB), where C is angle opposite side AB in a triangle ABC having sidelengths |BC|=2, |CA|=3, |AB|=4 (the smallest integer-sided scalene triangle).

a(n) = -2 a(n-1) - 16 a(n-2), where a(0) = 1, a(1) = -1.

Mathematica

(* Program 1 *)
A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2  b  c)];
Table[TrigExpand[4^n  Cos[n A[4, 2, 3]]], {n, 0, 30}]
(* Program 2 *)
LinearRecurrence[{-2, -16}, {1, -1}, 30]

Crossrefs

Cf. A375880.

Sequence in context: A009942 A031130 A215199 * A216258 A064348 A206215

Adjacent sequences: A375879 A375880 A375881 * A375883 A375884 A375885

Keyword

sign,easy

Author

Clark Kimberling, Sep 11 2024

Status

approved