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A197476
Decimal expansion of least x>0 having cos(x) = cos(2*x)^2.
53
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, 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, 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
OFFSET
1,3
COMMENTS
The Mathematica program includes a graph. Guide for least x>0 satisfying cos(b*x) = cos(c*x)^2, for selected b and c:
b.....c......x
1.....2.......A197476
1.....3.......A197477
1.....4.......A197478
2.....1.......A000796, Pi
2.....3.......A197479
2.....4.......A197480
3.....1.......A019669, Pi/2
3.....2.......A197482
3.....4.......A197483
4.....1.......A168229, arctan(sqrt(7))
4.....2.......A019669, Pi/2
4.....3.......A019679
4.....6.......A197485
4.....8.......A197486
6.....2.......A003881
6.....3.......A019670, Pi/3, tangency point
6.....4.......A197488
6.....8.......A197489
1.....4*Pi....A197334
1.....3*Pi....A197335
1.....2*Pi....A197490
1.....3*Pi/2..A197491
1.....Pi......A197492
1.....Pi/2....A197493
1.....Pi/3....A197494
1.....Pi/4....A197495
1.....2*Pi/3..A197506
2.....3*Pi....A197507
2.....3*Pi/2..A197508
2.....2*Pi....A197509
2.....Pi......A197510
2.....Pi/2....A197511
2.....Pi/3....A197512
2.....Pi/4....A197513
2.....Pi/6....A197514
Pi....1.......A197515
Pi....2.......A197516
Pi....1/2.....A197517
2*Pi..1.......A197518
2*Pi..2.......A197519
2*Pi..3.......A197520
Pi/2..Pi/3....A197521
Pi/2..Pi/6....3
Pi/3..1.......A197582
Pi/3..2.......A197583
Pi/3..1/3.....A197584
See A197133 for a guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected b and c.
EXAMPLE
1.137743932905455557789449860055008349584...
MATHEMATICA
b = 1; c = 2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.1, 1.3}, WorkingPrecision -> 200]
RealDigits[t] (* A197476 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]
(* or *)
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 *)
CROSSREFS
Cf. A197133.
Sequence in context: A354966 A021730 A153844 * A246848 A019634 A123879
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 15 2011
EXTENSIONS
Edited by Georg Fischer, Jul 28 2021
STATUS
approved