A197137 - OEIS

A197137

Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x.

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

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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..99.

EXAMPLE

length of Philo line: 3.350162315943772...

endpoint on x axis: (4.55402, 0); see A197036)

endpoint on line y=x: (2.93048, 2.93048)

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 = 1; h = 4; k = 1; (* slope m; point (h, k) *)

t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]