A375891 - OEIS

A375891

a(n) = (32)^n*cos (nB - nC), 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, 17, -446, -32572, -650744, 11228432, 1048128544, 24138456128, -252576120704, -33305367179008, -873744536485376, 4397381750801408, 1044225384888272896, 31000744173380636672, -15261492230649798656, -32263652769383865106432, -1081336426114866019794944

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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 (34,-1024).

FORMULA

a(n) = (32)^n*cos (nB - nC), 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) = 34 a(n-1) - 1024 a(n-2), where a(0) = 1, a(1) = 17.

From Stefano Spezia, Sep 23 2024: (Start)

G.f.: (1 - 17*x)/(1 - 34*x + 1024*x^2).

E.g.f.: exp(17*x)*cos(7*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[b, c, a] - A[c, a, b])]], {n, 0, 18}]

(* Program 2 *)

LinearRecurrence[{34, -1024}, {1, 17}, 30]

CROSSREFS