cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A197150 Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) to the line y=3x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 11 2011

Keywords

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 3*x^3-16*x^2+30*x-25. - R. J. Mathar, Nov 08 2022

Examples

			length of Philo line:    3.160946973065...; see A197151
endpoint on x axis:    (2.85106, 0)
endpoint on line y=3x: (0.802397, 2.40719)
		

Crossrefs

Programs

  • 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 = 3; h = 2; k = 1;(* slope m, point (h,k) *)
    t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
    RealDigits[t]  (* A197150 *)
    {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=3x *)
    d = N[Sqrt[f[t]], 100]
    RealDigits[d]  (* A197151 *)
    Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3}],
    ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],
    PlotRange -> {0, 2.5}, AspectRatio -> Automatic]

Extensions

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