A375896 a(n) = (64)^n*sin (nA - nB)/(5 sqrt(15)), 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.

0, -1, -122, -10788, -816424, -55416080, -3416689056, -189851801152, -9167161367168, -340760709275904, -4024113571740160, 904814009441803264, 126870078341747693568, 11772031375019592445952, 916527986864591725551616, 63598173885399939858677760

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..15.

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)/(5 sqrt(15)), 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) = 0, a(1) = -1.

From Stefano Spezia, Sep 25 2024: (Start)

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

E.g.f.: -exp(61*x)*sin(5*sqrt(105)*x)/(5*sqrt(5)). (End)

Mathematica

(* Program 1 *)
A[a_, b_, c_] := ArcSin[Sqrt[1 - ((b^2 + c^2 - a^2)/(2  b  c))^2]];
{a, b, c} = {2, 3, 4};
Table[TrigExpand[(64)^n  Sin[n (A[a, b, c] - A[b, c, a])]/(5 Sqrt[15])], {n, 0, 18}]
(* Program 2 *)
LinearRecurrence[{122, -4096}, {0, -1}, 30]

Crossrefs

Cf. A375880.

Sequence in context: A281609 A233071 A316526 * A324453 A275502 A275399

Adjacent sequences: A375893 A375894 A375895 * A375897 A375898 A375899

Keyword

sign,easy

Author

Clark Kimberling, Sep 25 2024

Status

approved