A197145 - OEIS

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A197145

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

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

(list; constant; graph; refs; listen; history; text; internal format)

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: 3.7423891424451...

endpoint on x axis: (3.82891, 0); see A197144

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]