

A197140


Decimal expansion of the xintercept of the shortest segment from the x axis through (1,1) to the line y=2x.


2



1, 4, 4, 0, 6, 1, 9, 7, 0, 0, 5, 3, 8, 1, 9, 9, 1, 1, 7, 6, 3, 3, 2, 5, 2, 3, 0, 2, 5, 8, 9, 2, 7, 7, 4, 3, 5, 3, 7, 9, 9, 0, 9, 4, 7, 2, 6, 0, 8, 9, 0, 3, 3, 7, 7, 3, 9, 8, 4, 6, 7, 3, 6, 4, 2, 5, 6, 5, 6, 3, 7, 3, 8, 9, 3, 2, 7, 7, 8, 9, 2, 9, 4, 2, 8, 1, 7, 1, 4, 8, 8, 0, 4, 1, 0, 3, 9, 7, 9, 2
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OFFSET

1,2


COMMENTS

The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197032, A197008 and A195284.


LINKS

Table of n, a(n) for n=1..100.


EXAMPLE

length of Philo line: 1.6736473041529...; see A197139
endpoint on x axis: (1.44062, 0)
endpoint on line y=2x: (0.765782, 1.53156)


MATHEMATICA

f[t_] := (t  k*t/(k + m*t  m*h))^2 + (m*k*t/(k + m*t  m*h))^2;
g[t_] := D[f[t], t]; Factor[g[t]]
p[t_] := h^2 k + k^3  h^3 m  h k^2 m  3 h k t + 3 h^2 m t + 2 k t^2  3 h m t^2 + m t^3 m = 2; h = 1; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision > 100]
RealDigits[t] (* A197140 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t  m*h)],
N[m*k*t/(k + m*t  m*h)]} (* endpt. on line y=2x *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197141 *)
Show[Plot[{k*(x  t)/(h  t), m*x}, {x, 0, 2}],
ContourPlot[(x  h)^2 + (y  k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange > {0, 1.7}, AspectRatio > Automatic]


CROSSREFS

Cf. A197032, A197141, A197008, A195284.
Sequence in context: A069523 A021231 A246811 * A155502 A228048 A016705
Adjacent sequences: A197137 A197138 A197139 * A197141 A197142 A197143


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 11 2011


STATUS

approved



