A197029 - OEIS

%I #5 Mar 30 2012 18:57:52

%S 5,0,6,0,6,4,3,3,3,2,1,6,5,2,4,5,1,0,0,5,4,6,3,7,6,2,1,7,7,3,4,7,1,4,

%T 4,1,1,6,9,4,8,7,3,8,8,6,1,8,3,2,2,7,7,3,2,8,6,6,4,0,3,6,7,1,7,8,8,6,

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

%N Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(4x) at points (x,y), (-x,y).

%C Let (x,y) denote the point of tangency, where x>0:

%C x=0.488618197079923270050681129865078039260837...

%C y=0.374332154777652501331094642913853652491893...

%C slope=3.709178750935618333987343550424591912283...

%C (The Mathematica program includes a graph.)

%e radius=0.5060643332165245100546376217734714411...

%t r = .5; c = 4;

%t Show[Plot[-Cos[c*x], {x, -1, 1}],

%t  ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1.5, 2}], PlotRange -> All, AspectRatio -> Automatic]

%t u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);

%t t1 = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, .4, .5}, WorkingPrecision -> 100]

%t t = Re[t1]    (* x coordinate of tangency point *)

%t y = -Cos[c*t] (* y coordinate of tangency point *)

%t radius = u[t]

%t RealDigits[radius] (* A197029 *)

%t slope = c*Sin[c*t] (* slope at tangency point *)

%Y Cf. A197026, A196027, A196028, A196022.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 09 2011