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%I #24 Jul 28 2021 13:43:05
%S 1,1,3,7,7,4,3,9,3,2,9,0,5,4,5,5,5,5,7,7,8,9,4,4,9,8,6,0,0,5,5,0,0,8,
%T 3,4,9,5,8,4,8,0,4,2,9,0,3,4,9,5,7,5,2,7,2,0,1,5,1,8,2,5,2,6,7,3,6,0,
%U 9,8,3,4,7,3,4,7,2,7,2,1,7,7,9,8,8,0,3,2,8,0,5,1,7,6,4,4,7,2,7
%N Decimal expansion of least x>0 having cos(x) = cos(2*x)^2.
%C The Mathematica program includes a graph. Guide for least x>0 satisfying cos(b*x) = cos(c*x)^2, for selected b and c:
%C b.....c......x
%C 1.....2.......A197476
%C 1.....3.......A197477
%C 1.....4.......A197478
%C 2.....1.......A000796, Pi
%C 2.....3.......A197479
%C 2.....4.......A197480
%C 3.....1.......A019669, Pi/2
%C 3.....2.......A197482
%C 3.....4.......A197483
%C 4.....1.......A168229, arctan(sqrt(7))
%C 4.....2.......A019669, Pi/2
%C 4.....3.......A019679
%C 4.....6.......A197485
%C 4.....8.......A197486
%C 6.....2.......A003881
%C 6.....3.......A019670, Pi/3, tangency point
%C 6.....4.......A197488
%C 6.....8.......A197489
%C 1.....4*Pi....A197334
%C 1.....3*Pi....A197335
%C 1.....2*Pi....A197490
%C 1.....3*Pi/2..A197491
%C 1.....Pi......A197492
%C 1.....Pi/2....A197493
%C 1.....Pi/3....A197494
%C 1.....Pi/4....A197495
%C 1.....2*Pi/3..A197506
%C 2.....3*Pi....A197507
%C 2.....3*Pi/2..A197508
%C 2.....2*Pi....A197509
%C 2.....Pi......A197510
%C 2.....Pi/2....A197511
%C 2.....Pi/3....A197512
%C 2.....Pi/4....A197513
%C 2.....Pi/6....A197514
%C Pi....1.......A197515
%C Pi....2.......A197516
%C Pi....1/2.....A197517
%C 2*Pi..1.......A197518
%C 2*Pi..2.......A197519
%C 2*Pi..3.......A197520
%C Pi/2..Pi/3....A197521
%C Pi/2..Pi/6....3
%C Pi/3..1.......A197582
%C Pi/3..2.......A197583
%C Pi/3..1/3.....A197584
%C See A197133 for a guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected b and c.
%e 1.137743932905455557789449860055008349584...
%t b = 1; c = 2; f[x_] := Cos[x]
%t t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.1, 1.3}, WorkingPrecision -> 200]
%t RealDigits[t] (* A197476 *)
%t Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]
%t (* or *)
%t RealDigits[ ArcCos[ ((19 - 3*Sqrt[33])^(1/3) + (19 + 3*Sqrt[33])^(1/3) - 2)/6], 10, 99] // First (* _Jean-François Alcover_, Feb 19 2013 *)
%Y Cf. A197133.
%K nonn,cons
%O 1,3
%A _Clark Kimberling_, Oct 15 2011
%E Edited by _Georg Fischer_, Jul 28 2021
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