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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
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Let (x,y) denote the point of tangency. Then
x=0.611116305530271280094961817959748153764566...
y=0.728086522083031555694579423798015759485165...
slope=-0.64791770945231344102369199673001177755...
(The Mathematica program includes a graph.)
LINKS
EXAMPLE
radius=0.39586243784748079824070772256631552733434...
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A332564 A199053 A200595 * A096418 A100811 A223652
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 08 2011
STATUS
approved

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Last modified May 28 12:36 EDT 2024. Contains 372913 sequences. (Running on oeis4.)