%I #17 Feb 20 2024 08:35:32

%S 3,6,9,5,4,2,5,6,6,6,0,7,5,8,0,3,2,0,8,2,7,6,5,6,0,4,3,8,3,6,9,3,6,7,

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

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

%N Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/4) orthogonally.

%C See the Mathematica program for a graph.

%C xo=0.36954256660758032082765604383693...

%C yo=0.40397275329951720931896174006631...

%C m=1.093169744985016922088153214160579...

%C |OP|=0.547499492185436214325204150357...

%F Equals d/2 = A003957/2, where d is the Dottie number. - _Gleb Koloskov_, Jun 16 2021

%p evalf(solve(cos(x)=x,x)/2, 140); # _Alois P. Heinz_, Feb 20 2024

%t c = Pi/4;

%t xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]

%t RealDigits[xo] (* A197002 *)

%t m = 1/Sin[xo + c]

%t RealDigits[m] (* A197003 *)

%t yo = m*xo

%t d = Sqrt[xo^2 + yo^2]

%t Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, 1}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

%o (PARI) solve(x=0,1,cos(x)-x)/2 \\ _Gleb Koloskov_, Jun 16 2021

%Y Cf. A197003, A196996, A197000, A003957.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 09 2011