A197155 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2.

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

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:  1.94634640...
endpoint on x axis:    (4.2236, 0); see A197154
endpoint on line y=3x: (3.79888, 1.89944)

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 = 1/2; h = 4; k = 1; (* slop m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197154 *)
{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=x/2 *)
d = N[Sqrt[f[t]], 100]
RealDigits[d]  (* A197155 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4.5}],
 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4.5}, {y, 0, 3}],
 PlotRange -> {0, 2}, AspectRatio -> Automatic]

Crossrefs

Cf. A197032, A197155, A197008, A195284.

Sequence in context: A086240 A288240 A289267 * A153756 A322088 A343199

Adjacent sequences: A197152 A197153 A197154 * A197156 A197157 A197158

Keyword

nonn,cons

Author

Clark Kimberling, Oct 11 2011

Status

approved