A197022 Decimal expansion of the radius of the circle tangent to the curve y=cos(4x) at points (x,y) and (-x,y), where 0<x<1.

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

Offset

0,1

Comments

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

x=0.294083445311344461181635110698988639348667...

y=0.384312064643508105468613486692501669417807...

slope=-3.69281299167871547859350850472131295652...

(The Mathematica program includes a graph.)

Links

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

Example

radius=0.30467536330660745240216843166775819548...

Mathematica

r = .304; c = 4;
Show[Plot[Cos[c*x], {x, -.5, .5}],
ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
t = x /. FindRoot[
   c*Sin[c*x] Cos[c*x] - x == x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
RealDigits[t]  (* x coordinate of tangency point *)
y = Cos[c*t]   (* y coordinate of tangency point *)
radius = Cos[c*t] - t/(c*Sin[c*t])
RealDigits[radius]  (* A197022 *)
slope = -c*Sin[c*t] (* slope at tangency point *)

Crossrefs

Cf. A197020, A196021, A196026, A197027, A197016.

Sequence in context: A188720 A113792 A216105 * A112238 A111493 A190181

Adjacent sequences: A197019 A197020 A197021 * A197023 A197024 A197025

Keyword

nonn,cons

Author

Clark Kimberling, Oct 08 2011

Status

approved