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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
1, 17, -446, -32572, -650744, 11228432, 1048128544, 24138456128, -252576120704, -33305367179008, -873744536485376, 4397381750801408, 1044225384888272896, 31000744173380636672, -15261492230649798656, -32263652769383865106432, -1081336426114866019794944
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.
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
Cf. A375880.
Sequence in context: A370543 A301642 A035022 * A012117 A271438 A371488
KEYWORD
sign,easy
AUTHOR
Clark Kimberling, Sep 22 2024
STATUS
approved