

A197144


Decimal expansion of the xintercept of the shortest segment from the x axis through (3,1) to the line y=2x.


2



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

1,1


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: 3.7423891424451...; see A197145
endpoint on x axis: (3.82891, 0)
endpoint on line y=2x: (1.44062, 2.88124)


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 = 3; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision > 100]
RealDigits[t] (* A197144 *)
{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] (* A197145 *)
Show[Plot[{k*(x  t)/(h  t), m*x}, {x, 0, 4}],
ContourPlot[(x  h)^2 + (y  k)^2 == .002, {x, 0, 4}, {y, 0, 3}], PlotRange > {0, 3}, AspectRatio > Automatic]


CROSSREFS

Cf. A197032, A197145, A197008, A195284.
Sequence in context: A086178 A016669 A094964 * A138714 A336054 A341814
Adjacent sequences: A197141 A197142 A197143 * A197145 A197146 A197147


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 11 2011


STATUS

approved



