A197025 -id:A197025 - 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.

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, 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, 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

Offset: 0

Clark Kimberling, Oct 08 2011

Let (x,y) denote the point of tangency.  Then
x=0.611116305530271280094961817959748153764566...
y=0.728086522083031555694579423798015759485165...
slope=-0.64791770945231344102369199673001177755...
(The Mathematica program includes a graph.)

			radius=0.39586243784748079824070772256631552733434...

Cf. A197020, A197024, A197025, A197016.

r = .396; c = 1;
Show[Plot[c/(1 + x^2), {x, 0, 1.5}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}],
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, .4, 1},
   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] (* A197023 *)
slope = -2*c*t/(1 + t^2)^2 (* slope at tangency point *)

A197024 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.

Original entry on oeis.org

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, 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, 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

Offset: 0

Clark Kimberling, Oct 08 2011

Let (x,y) denote the point of tangency.  Then
x=0.290074091667981539080192147132694221247...
y=0.461193781487549868098884143492334039544...
slope=-0.24679469383945033223474847695422791...
(The Mathematica program includes a graph.)

			radius=0.23400514059513790173472762376722996062...

Cf. A197023, A197025.

r = .234; c = 1/2;
Show[Plot[c/(1 + x^2), {x, 0, 0.8}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], 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, .4, 1}, 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] (* A197024 *)
slope = -2*c*t/(1 + t^2)^2  (* slope at tangency pt. *)

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.

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