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A376285 a(n) = 20^n * cos(n*A), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.

%I A376285 #15 Mar 30 2025 20:24:46
%S A376285 1,16,112,-2816,-134912,-3190784,-48140288,-264175616,10802495488,
%T A376285 451350102016,10122205069312,143370521411584,538974657445888,
%U A376285 -40101019526365184,-1498822487822041088,-31921911799759241216,-421972182463479283712,-734345118927640592384
%N A376285 a(n) = 20^n * cos(n*A), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.
%C A376285 If a prime p divides a term, then the indices n such that p divides a(n) comprise an arithmetic sequence; e.g., 7 divides a(4*n+2) for n >= 0; 17 divides a(8*n+3) for n >= 0. See the Renault paper in References.
%D A376285 Marc Renault, "The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m", Math. Mag. 86 (2013) 372 - 380.
%H A376285 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (32,-400).
%F A376285 a(n) = 20^n * cos(n*A), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.
%F A376285 a(n) = 32*a(n-1) - 400*a(n-2), where a(0) = 1, a(1) = 16.
%F A376285 From _Stefano Spezia_, Oct 03 2024: (Start)
%F A376285 G.f.: (1 - 16*x)/(1 - 32*x + 400*x^2).
%F A376285 E.g.f.: exp(16*x)*cos(12*x). (End)
%t A376285 (*Program 1*)
%t A376285 A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2  b  c)];
%t A376285 Table[TrigExpand[(20)^n  Cos[n  A[3, 4, 5]]], {n, 0, 30}]
%t A376285 (*Program 2*)
%t A376285 LinearRecurrence[{32, -400}, {1, 16}, 30]
%Y A376285 Cf. A066771, A374880, A139030, A376455.
%K A376285 sign,easy
%O A376285 0,2
%A A376285 _Clark Kimberling_, Oct 03 2024