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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.
1
0, -1, -14, -132, -952, -4880, -7392, 208832, 3396736, 34189056, 261255680, 1469479936, 3852355584, -40113737728, -808143085568, -8746723983360, -70732978290688, -430471361134592, -1499688445280256, 6554528878690304, 187743464799600640, 2208918658958229504
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) = 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
Cf. A375880.
Sequence in context: A026936 A021114 A328785 * A113976 A022738 A017269
KEYWORD
sign,easy
AUTHOR
Clark Kimberling, Sep 22 2024
STATUS
approved