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A197476 Decimal expansion of least x>0 having cos(x)=(cos 2x)^2. 52
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 (list; constant; graph; refs; listen; history; text; internal format)
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.... x=3

...

See A197133 for a guide for least x>0 satisfying sin(b*x)=(sin(c*x))^2 for selected b and c.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=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}]

RealDigits[ ArcCos[ ((19 - 3*Sqrt[33])^(1/3) + (19 + 3*Sqrt[33])^(1/3) - 2)/6], 10, 99] // First (* Jean-Fran├žois Alcover, Feb 19 2013 *)

CROSSREFS

Cf. A197133.

Sequence in context: A219313 A021730 A153844 * A246848 A019634 A123879

Adjacent sequences:  A197473 A197474 A197475 * A197477 A197478 A197479

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 15 2011

STATUS

approved

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Last modified October 14 00:26 EDT 2019. Contains 327991 sequences. (Running on oeis4.)