A197152 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x/2.

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

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.

A root of the polynomial x^3/2 -5*x^2/2 +9*x/2 -5. - R. J. Mathar, Nov 08 2022

Links

Table of n, a(n) for n=1..95.

Example

length of Philo line:    1.481506505...; see A197153
endpoint on x axis:    (3.15091, 0)
endpoint on line y=3x: (2.92984, 1.46492)

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 = 3; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t] (* A197152 *)
{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]  (* A197153 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3.5}],
ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],
PlotRange -> {0, 1.5}, AspectRatio -> Automatic]

Crossrefs

Cf. A197032, A197153, A197008, A195284.

Sequence in context: A094919 A328373 A316631 * A337668 A176907 A143069

Adjacent sequences: A197149 A197150 A197151 * A197153 A197154 A197155

Keyword

nonn,cons

Author

Clark Kimberling, Oct 11 2011

Extensions

Incorrect trailing digits deleted. - R. J. Mathar, Nov 08 2022

Status

approved