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.

Showing 1-10 of 25 results. Next

A197008 Decimal expansion of the shortest distance from x axis through (1,2) to y axis.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P. Suppose that T is the angle formed by the positive x and y axes and that h>0 and k>0. Notation:
...
P=(h,k)
L=the Philo line of P across T
U=x-intercept of L
V=y-intercept of L
d=|UV|
...
Although Philo lines are not generally Euclidean-constructible, exact expressions for U, V, and d can be found for the angle T under consideration. Write u(t)=(t,0), let v(t) the corresponding point on the y axis, and let d(t) be the distance between u(t) and v(t). Then d is found by minimizing d(t)^2:
d=w*sqrt(1+(k/h)^(2/3)), where w=(h+(h*k^2))^(1/3).
...
Guide:
h....k...........d
1....2........A197008
1....3........A197012
1....4........A197013
2....3........A197014
3....4........A197015
1..sqrt(2)....A197031
...
For guides to other Philo lines, see A195284 and A197032.
The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - Raul Prisacariu, Apr 06 2024

Examples

			d=4.161938184941462752390080...
x-intercept: U=(2.5874..., 0)
y-intercept: V=(0, 3.2599...)
		

Crossrefs

Programs

  • Maple
    (1+2^(2/3))^(3/2); evalf(%) ; # R. J. Mathar, Nov 08 2022
  • Mathematica
    f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);
    h = 1; k = 2; d = N[f[t]^(1/2), 100]
    RealDigits[d] (* this sequence *)
    x = N[t] (* x-intercept; -1+4^(1/3); cf. A005480 *)
    y = N[k*t/(t - h)] (* y-intercept *)
    Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],
    ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 4}], PlotRange -> All, AspectRatio -> Automatic]
  • PARI
    polrootsreal(x^6 - 15*x^4 - 33*x^2 - 125)[2] \\ Charles R Greathouse IV, Feb 03 2025

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 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 A197008 and A195284.
A root of the polynomial x^3-7*x^2+18*x-20. - R. J. Mathar, Nov 08 2022

Examples

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

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 = 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]

Extensions

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

A197035 Decimal expansion of the shortest distance from the x axis through (3,1) to the line y=x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 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 A197008 and A195284.

Examples

			length of Philo line:  2.60819402496101...
endpoint on x axis:   (3.47797, 0); see A197034
endpoint on line y=x: (2.35321, 2.35321)
		

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 = 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]

A197136 Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 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 A197008 and A195284.

Examples

			length of Philo line:   3.350162315943772... (see A197137)
endpoint on x axis:   (4.55402, 0)
endpoint on line y=x: (2.93048, 2.93048)
		

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 = 1; h = 4; k = 1;(* slope m; point (h,k) *)
    t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
    RealDigits[t]  (* A197136 *)
    {N[t], 0} (* endpoint on x axis *)
    {N[k*t/(k + m*t - m*h)],
    N[m*k*t/(k + m*t - m*h)]} (* endpoint on line y=mx *)
    d = N[Sqrt[f[t]], 100]
    RealDigits[d]  (* A197137 *)
    Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 5}],
    ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 5}, {y, 0, 3}],
    PlotRange -> {0, 3}, AspectRatio -> Automatic]

Extensions

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

A197137 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 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.

Examples

			length of Philo line:   3.350162315943772...
endpoint on x axis:   (4.55402, 0);  (see A197136)
endpoint on line y=x: (2.93048, 2.93048)
		

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 = 1; h = 4; k = 1;(* slope m; point (h,k) *)
    t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
    RealDigits[t]  (* A197136 *)
    {N[t], 0} (* endpoint on x axis *)
    {N[k*t/(k + m*t - m*h)],
    N[m*k*t/(k + m*t - m*h)]} (* endpoint on line y=mx *)
    d = N[Sqrt[f[t]], 100]
    RealDigits[d]  (* A197137 *)
    Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 5}],
    ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 5}, {y, 0, 3}],
    PlotRange -> {0, 3}, AspectRatio -> Automatic]

A197138 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,2) to the line y=x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 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.

Examples

			length of Philo line:   2.886117...; see A197139
endpoint on x axis:   (3.4883, 0)
endpoint on line y=x: (2.80376, 2.80376)
		

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 = 1; h = 3; k = 2;(* slope m; point (h,k) *)
    t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
    RealDigits[t]  (* A197138 *)
    {N[t], 0} (* endpoint on x axis *)
    {N[k*t/(k + m*t - m*h)],
    N[m*k*t/(k + m*t - m*h)]} (* endpoint on line y=x *)
    d = N[Sqrt[f[t]], 100]
    RealDigits[d]  (* A197139 *)
    Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
    ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 4}, {y, 0, 3}],
    PlotRange -> {0, 3}, AspectRatio -> Automatic]

Extensions

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

A197139 Decimal expansion of the shortest distance from the x axis through (3,2) to the line y = x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 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.
Any Philo line can be constructed using the intersections of circles and hyperbolas. In this case, the Philo line passes though the two points at which the circle (x - 3/2)^2 + (y - 1)^2 = 13/4 and the hyperbola x*y - y^2 = 2 intersect. The circle has segment OP as diameter, where O(0,0) is the origin and P is the point (3,2). The asymptotes of the hyperbola are the x axis and the line y = x. Point P is one of the two points at which the circle and the hyperbola intersect. - Raul Prisacariu, Apr 06 2024

Examples

			Length of Philo line:   2.8861171058980012915367...
Endpoint on x axis:     (3.4883, 0); see A197138
Endpoint on line y = x: (2.80376, 2.80376)
		

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 = 1; h = 3; k = 2;(* slope m; point (h,k) *)
    t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
    RealDigits[t]  (* A197138 *)
    {N[t], 0} (* endpoint on x axis *)
    {N[k*t/(k + m*t - m*h)],
    N[m*k*t/(k + m*t - m*h)]} (* endpoint on line y=x *)
    d = N[Sqrt[f[t]], 100]
    RealDigits[d]  (* this sequence *)
    Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
    ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 4}, {y, 0, 3}],
    PlotRange -> {0, 3}, AspectRatio -> Automatic]

Extensions

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

A197140 Decimal expansion of the x-intercept of the shortest segment from the x axis through (1,1) to the line y=2x.

Original entry on oeis.org

1, 4, 4, 0, 6, 1, 9, 7, 0, 0, 5, 3, 8, 1, 9, 9, 1, 1, 7, 6, 3, 3, 2, 5, 2, 3, 0, 2, 5, 8, 9, 2, 7, 7, 4, 3, 5, 3, 7, 9, 9, 0, 9, 4, 7, 2, 6, 0, 8, 9, 0, 3, 3, 7, 7, 3, 9, 8, 4, 6, 7, 3, 6, 4, 2, 5, 6, 5, 6, 3, 7, 3, 8, 9, 3, 2, 7, 7, 8, 9, 2, 9, 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 2*x^3-4*x^2+3*x-2. - R. J. Mathar, Nov 08 2022

Examples

			length of Philo line:    1.6736473041529...; see A197139
endpoint on x axis:    (1.44062, 0)
endpoint on line y=2x: (0.765782, 1.53156)
		

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 = 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=2x *)
    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]

Extensions

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

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

Original entry on oeis.org

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

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.

Examples

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

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 = 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]

A197142 Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) to the line y=2x.

Original entry on oeis.org

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

Examples

			length of Philo line:    2.7463941076100...; see A197143
endpoint on x axis:    (2.69141, 0)
endpoint on line y=2x: (1.1295, 2.25901)
		

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 = 2; h = 2; k = 1;(* slope m, point (h,k) *)
    t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
    RealDigits[t] (* A197142 *)
    {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=2x *)
    d = N[Sqrt[f[t]], 100]
    RealDigits[d] (* A197143 *)
    Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
    ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 2.5}, AspectRatio -> Automatic]
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