A197025 Decimal expansion of the radius of the circle tangent to the curve y=3/(1+x^2) and to the positive x and y axes.

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

Offset

1,1

Comments

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

x=1.24529475848935327220740793075336892358...

y=1.17612050304358700012836336674764176692...

slope=-1.148377151582141158728147357192644...

(The Mathematica program includes a graph.)

Links

Table of n, a(n) for n=1..98.

Example

radius=0.70991492686043992140959388916255887...

Mathematica

r = .71; c = 3;
Show[Plot[c/(1 + x^2), {x, 0, 2}],
 ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, 0, 2}, {y, 0, 2}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
u[x_] := (x*(1 + x^2)^3 - 2*x*c^2)/((1 + x^2)^3 - 2*c*x*(1 + x^2))
v = x /. FindRoot[c/(1 + x^2) == u[x] + Sqrt[2*u[x]*x - x^2], {x, .9, 1.45}, WorkingPrecision -> 100]
t = Re[v]; RealDigits[t] (* x coord. of tangency pt. *)
y = c/(1 + t^2)          (* y coord. of tangency pt. *)
radius = u[t]
RealDigits[radius] (* A197025 *)
slope = -2*c*t/(1 + t^2)^2   (* slope at tangency pt *)

Crossrefs

Cf. A197023, A196024.

Sequence in context: A014565 A073115 A176444 * A335763 A096408 A005481

Adjacent sequences: A197022 A197023 A197024 * A197026 A197027 A197028

Keyword

nonn,cons

Author

Clark Kimberling, Oct 08 2011

Status

approved