A375894 - OEIS

A375894

a(n) = 8^n*sin (nB - nC)/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, -14, -132, -952, -4880, -7392, 208832, 3396736, 34189056, 261255680, 1469479936, 3852355584, -40113737728, -808143085568, -8746723983360, -70732978290688, -430471361134592, -1499688445280256, 6554528878690304, 187743464799600640, 2208918658958229504

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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..21.

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 (14,-64).

FORMULA

a(n) = 8^n*cos (nB - nC)/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) = 14 a(n-1) - 64 a(n-2), where a(0) = 0, a(1) = -1.

From Stefano Spezia, Sep 23 2024: (Start)

G.f.: x/(1 - 14*x+ 64*x^2).

E.g.f.: exp(7*x)*sin(sqrt(15)*x)/(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[8^n Sin[n (A[b, c, a] - A[c, a, b])]/Sqrt[15]], {n, 0, 22}]

(* Program 2 *)

LinearRecurrence[{14, -64}, {0, -1}, 30]

CROSSREFS