A197017 Decimal expansion of the radius of the circle tangent to the curve y=cos(2x) and to the positive x and y axes.

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

Offset

0,1

Comments

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

x=0.556627409764774263651183045638839616840052780212...

y=0.441743828977740325730277185387438343947805907493...

slope=-0.5283257380737094443139057566841614427843590...

(The Mathematica program includes a graph.)

Links

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

Example

radius=0.2971056352748227167122144365263161994072960710...

Mathematica

r = .297; c = 2;
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] (* A197017 *)
slope = -Sin[x1] (* slope at tangency point *)

Crossrefs

Cf. A197016, A197018, A197019, A197020.

Sequence in context: A198842 A016596 A201616 * A094044 A357109 A011070

Adjacent sequences: A197014 A197015 A197016 * A197018 A197019 A197020

Keyword

nonn,cons

Author

Clark Kimberling, Oct 08 2011

Status

approved