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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.
1
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.
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
KEYWORD
sign,easy
AUTHOR
Clark Kimberling, Sep 25 2024
STATUS
approved