cp's OEIS Frontend

A197023 Decimal expansion of the radius of the circle tangent to the curve y=1/(1+x^2) and to the positive x and y axes.

%I A197023 #9 Mar 30 2012 18:57:52
%S A197023 3,9,5,8,6,2,4,3,7,8,4,7,4,8,0,7,9,8,2,4,0,7,0,7,7,2,2,5,6,6,3,1,5,5,
%T A197023 2,7,3,3,4,3,4,3,9,2,9,2,4,9,1,0,0,0,8,2,5,0,4,4,2,7,9,6,3,2,5,1,2,6,
%U A197023 4,4,3,7,9,1,1,1,8,8,1,3,0,8,6,6,5,3,8,4,8,9,0,0,5,9,1,5,8,3,6
%N A197023 Decimal expansion of the radius of the circle tangent to the curve y=1/(1+x^2) and to the positive x and y axes.
%C A197023 Let (x,y) denote the point of tangency.  Then
%C A197023 x=0.611116305530271280094961817959748153764566...
%C A197023 y=0.728086522083031555694579423798015759485165...
%C A197023 slope=-0.64791770945231344102369199673001177755...
%C A197023 (The Mathematica program includes a graph.)
%e A197023 radius=0.39586243784748079824070772256631552733434...
%t A197023 r = .396; c = 1;
%t A197023 Show[Plot[c/(1 + x^2), {x, 0, 1.5}],
%t A197023 ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}],
%t A197023 PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
%t A197023 u[x_] := (x*(1 + x^2)^3 - 2*x*c^2)/((1 + x^2)^3 - 2*c*x*(1 + x^2))
%t A197023 v = x /. FindRoot[c/(1 + x^2) == u[x] + Sqrt[2*u[x]*x - x^2], {x, .4, 1},
%t A197023    WorkingPrecision -> 100]
%t A197023 t = Re[v] ; RealDigits[t] (* x coord. of tangency pt. *)
%t A197023 y = c/(1 + t^2)           (* y coord. of tangency pt. *)
%t A197023 radius = u[t]
%t A197023 RealDigits[radius] (* A197023 *)
%t A197023 slope = -2*c*t/(1 + t^2)^2 (* slope at tangency point *)
%Y A197023 Cf. A197020, A197024, A197025, A197016.
%K A197023 nonn,cons
%O A197023 0,1
%A A197023 _Clark Kimberling_, Oct 08 2011