A197018 Decimal expansion of the radius of the circle tangent to the curve y=cos(3x) and to the positive x and y axes.

2, 1, 8, 7, 2, 9, 4, 8, 8, 8, 0, 3, 6, 4, 4, 0, 6, 5, 8, 9, 7, 2, 8, 5, 2, 2, 3, 2, 6, 8, 1, 2, 1, 0, 4, 9, 3, 0, 3, 6, 3, 6, 1, 9, 9, 7, 3, 1, 4, 1, 4, 9, 9, 5, 8, 2, 2, 1, 6, 6, 9, 4, 6, 6, 9, 0, 3, 1, 8, 5, 8, 6, 5, 0, 7, 6, 2, 9, 6, 0, 6, 3, 4, 5, 6, 6, 6, 1, 3, 7, 9, 4, 2, 8, 4, 3, 0, 0, 7

Offset

0,1

Comments

Let (x,y) denote the point of tangency. Then

x=0.4252834568497833490618545391964703664552948...

y=0.2906881405190418936802785128662388404186594...

slope=-0.41257900534470955829852211550705870735...

(The Mathematica program includes a graph.)

Links

Table of n, a(n) for n=0..98.

Example

radius=0.218729488803644065897285223268121049303636199...

Mathematica

r = .219; c = 3;
Show[Plot[Cos[c*x], {x, 0, Pi}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
f[x_] := (x - c*Sin[c*x] Cos[c*x])/(1 - c*Sin[c*x]);
t = x /. FindRoot[Cos[c*x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
x1 = Re[t]      (* x coordinate of tangency point *)
y = Cos[c*x1]   (* y coordinate of tangency point *)
radius = f[x1]
RealDigits[radius] (* A197018 *)
slope = -Sin[x1] (* slope at tangency point *)

Crossrefs

Cf. A197016, A197017, A197019, A197020.

Sequence in context: A280757 A298641 A293415 * A082532 A049250 A060587

Adjacent sequences: A197015 A197016 A197017 * A197019 A197020 A197021

Keyword

nonn,cons

Author

Clark Kimberling, Oct 08 2011

Status

approved