%I #8 Mar 30 2012 18:57:52
%S 2,3,4,0,0,5,1,4,0,5,9,5,1,3,7,9,0,1,7,3,4,7,2,7,6,2,3,7,6,7,2,2,9,9,
%T 6,0,6,2,0,4,5,8,8,8,6,4,7,4,9,5,1,1,9,4,1,4,4,3,8,1,0,3,3,4,0,3,0,6,
%U 3,4,2,1,9,4,1,8,8,9,9,4,7,3,6,2,2,0,5,9,8,6,0,2,2,8,6,1,5,2,2,0
%N Decimal expansion of the radius of the circle tangent to the curve y=(1/2)/(1+x^2) and to the positive x and y axes.
%C Let (x,y) denote the point of tangency. Then
%C x=0.290074091667981539080192147132694221247...
%C y=0.461193781487549868098884143492334039544...
%C slope=-0.24679469383945033223474847695422791...
%C (The Mathematica program includes a graph.)
%e radius=0.23400514059513790173472762376722996062...
%t r = .234; c = 1/2;
%t Show[Plot[c/(1 + x^2), {x, 0, 0.8}],
%t ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
%t u[x_] := (x*(1 + x^2)^3 - 2*x*c^2)/((1 + x^2)^3 - 2*c*x*(1 + x^2))
%t v = x /. FindRoot[c/(1 + x^2) == u[x] + Sqrt[2*u[x]*x - x^2], {x, .4, 1}, WorkingPrecision -> 100]
%t t = Re[v]; RealDigits[t] (* x coord. of tangency pt. *)
%t y = c/(1 + t^2) (* y coord. of tangency pt. *)
%t radius = u[t]
%t RealDigits[radius] (* A197024 *)
%t slope = -2*c*t/(1 + t^2)^2 (* slope at tangency pt. *)
%Y Cf. A197023, A197025.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 08 2011