A375895 a(n) = (32)^n*sin (nC - nA)/(6 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, 44, 912, -4928, -1150720, -45585408, -827420672, 10272948224, 1299288489984, 46649194577920, 722093147684864, -15996676749656064, -1443277160214167552, -47123598057775562752, -595522502482817187840, 22051574301918219993088, 1580084311826806480044032

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..17.

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 (44,-1024).

Formula

a(n) = (32)^n*cos (nC - nA)/(6 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 23 2024: (Start)

G.f.: x/(1 - 44*x+ 1024*x^2).

E.g.f.: exp(22*x)*sin(6*sqrt(15)*x)/(6*sqrt(15)). (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[(32)^n  Sin[n (A[c, a, b] - A[a, b, c])]/(6 Sqrt[15])], {n, 0, 22}]
(* Program 2 *)
LinearRecurrence[{44, -1024}, {0, 1}, 30]

Crossrefs

Cf. A375880.

Sequence in context: A183750 A133349 A010838 * A191374 A299466 A010960

Adjacent sequences: A375892 A375893 A375894 * A375896 A375897 A375898

Keyword

sign,easy

Author

Clark Kimberling, Sep 22 2024

Status

approved