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%I A375891 #13 Sep 28 2024 07:38:44
%S A375891 1,17,-446,-32572,-650744,11228432,1048128544,24138456128,
%T A375891 -252576120704,-33305367179008,-873744536485376,4397381750801408,
%U A375891 1044225384888272896,31000744173380636672,-15261492230649798656,-32263652769383865106432,-1081336426114866019794944
%N 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.
%C A375891 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 A375891 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 A375891 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (34,-1024).
%F 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.
%F A375891 a(n) = 34 a(n-1) - 1024 a(n-2), where a(0) = 1, a(1) = 17.
%F A375891 From _Stefano Spezia_, Sep 23 2024: (Start)
%F A375891 G.f.: (1 - 17*x)/(1 - 34*x + 1024*x^2).
%F A375891 E.g.f.: exp(17*x)*cos(7*sqrt(15)*x). (End)
%t A375891 (* Program 1 *)
%t A375891 A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2  b  c)];
%t A375891 {a, b, c} = {2, 3, 4};
%t A375891 Table[TrigExpand[(32)^n  Cos[n (A[b, c, a] - A[c, a, b])]], {n, 0, 18}]
%t A375891 (* Program 2 *)
%t A375891 LinearRecurrence[{34, -1024}, {1, 17}, 30]
%Y A375891 Cf. A375880.
%K A375891 sign,easy,new
%O A375891 0,2
%A A375891 _Clark Kimberling_, Sep 22 2024

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