%I #8 Sep 13 2024 15:46:41
%S 1,-1,-14,44,136,-976,-224,16064,-28544,-199936,856576,1485824,
%T -16676864,9580544,247668736,-648626176,-2665447424,15708913664,
%U 11229331456,-273801281536,367933259776,3644953985024,-13176840126464,-31965583507456,274760609038336
%N a(n) = 4^n cos(nC), where C is the angle opposite side AB in a triangle ABC having sidelengths |BC|=2, |CA| = 3, |AB| = 4; ABC is the smallest integer-sided scalene triangle.
%C 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.
%H Marc Renault, <a href="https://www.jstor.org/stable/10.4169/math.mag.86.5.372">The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m</a>, Math. Mag. 86 (2013) pp. 372-380.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-16).
%F a(n) = 4^n cos (nB), where C is angle opposite side AB in a triangle ABC having sidelengths |BC|=2, |CA|=3, |AB|=4 (the smallest integer-sided scalene triangle).
%F a(n) = -2 a(n-1) - 16 a(n-2), where a(0) = 1, a(1) = -1.
%t (* Program 1 *)
%t A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2 b c)];
%t Table[TrigExpand[4^n Cos[n A[4, 2, 3]]], {n, 0, 30}]
%t (* Program 2 *)
%t LinearRecurrence[{-2, -16}, {1, -1}, 30]
%Y Cf. A375880.
%K sign,easy
%O 0,3
%A _Clark Kimberling_, Sep 11 2024