A197034 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x.

3, 4, 7, 7, 9, 6, 7, 2, 4, 3, 0, 0, 9, 0, 1, 2, 4, 7, 4, 6, 4, 6, 9, 2, 5, 0, 8, 1, 3, 4, 2, 1, 7, 5, 1, 0, 1, 4, 4, 7, 5, 4, 9, 5, 5, 2, 7, 5, 8, 1, 9, 3, 4, 4, 4, 2, 3, 5, 9, 0, 9, 9, 3, 8, 6, 0, 4, 6, 0, 4, 0, 6, 3, 1, 9, 6, 0, 1, 1, 8, 7, 6, 9, 8, 4, 9, 7, 7, 5, 3, 6, 2, 6, 5, 5, 3, 0, 8, 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 A197008 and A195284.

A root of the polynomial x^3-7*x^2+18*x-20. - R. J. Mathar, Nov 08 2022

Links

Table of n, a(n) for n=1..99.

Example

length of Philo line:   2.60819402496101...; see A197035
endpoint on x axis:   (3.47797, 0)
endpoint on line y=x: (2.35321, 2.35321)

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; h = 3; k = 1;  (* slope m; point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197034 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)], N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197035 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 3.5}, {y, 0, 3}], PlotRange -> {0, 3}, AspectRatio -> Automatic]

Crossrefs

Cf. A197032, A197035, A197008, A195284.

Sequence in context: A021746 A081826 A023849 * A180020 A162431 A024605

Adjacent sequences: A197031 A197032 A197033 * A197035 A197036 A197037

Keyword

nonn,cons

Author

Clark Kimberling, Oct 10 2011

Extensions

Last digit removed (representation truncated, not rounded up). - R. J. Mathar, Nov 08 2022

Status

approved