A197138 - OEIS

A197138

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

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

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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..81.

EXAMPLE

length of Philo line: 2.886117...; see A197139

endpoint on x axis: (3.4883, 0)

endpoint on line y=x: (2.80376, 2.80376)

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 = 3; k = 2; (* slope m; point (h, k) *)

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