A197141 Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=2x.

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

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..99.

Example

length of Philo line:    1.6736473041529...
endpoint on x axis:    (1.44062, 0); see A197140
endpoint on line y=2x: (0.765782, 1.53156)

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 = 1; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197140 *)
{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=mx *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197141 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 1.7}, AspectRatio -> Automatic]

Crossrefs

Cf. A197032, A197140, A197008, A195284.

Sequence in context: A105831 A248650 A154339 * A139350 A092560 A272875

Adjacent sequences: A197138 A197139 A197140 * A197142 A197143 A197144

Keyword

nonn,cons

Author

Clark Kimberling, Oct 11 2011

Status

approved