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.

Previous Showing 21-25 of 25 results.

A197153 Decimal expansion of the shortest distance from the x axis through (3,1) to the line y=x/2.

Original entry on oeis.org

1, 4, 8, 1, 5, 0, 6, 5, 0, 5, 8, 4, 4, 3, 0, 9, 1, 9, 4, 0, 3, 5, 9, 2, 5, 3, 0, 6, 6, 1, 1, 1, 4, 1, 7, 3, 6, 8, 1, 0, 5, 2, 2, 1, 1, 7, 1, 5, 7, 4, 1, 6, 1, 8, 6, 8, 5, 0, 4, 8, 6, 5, 0, 0, 7, 1, 1, 5, 4, 4, 9, 9, 4, 7, 0, 5, 6, 5, 9, 1, 4, 7, 2, 5, 7, 2, 4, 3, 2, 3, 1, 9, 0, 7, 7, 4, 3, 3, 8, 7
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:   1.481506505...
endpoint on x axis:    (3.15091, 0); see A197152
endpoint on line y=3x: (2.92984, 1.46492)

Crossrefs

Cf. A197032, A197152, A197008, A195284.

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 = 1/2; 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]

A197154 Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x/2.

Original entry on oeis.org

4, 2, 2, 3, 6, 0, 3, 3, 0, 4, 1, 9, 3, 9, 8, 1, 1, 4, 7, 6, 0, 7, 9, 3, 9, 5, 2, 5, 9, 6, 1, 6, 9, 0, 1, 4, 2, 0, 9, 3, 1, 8, 6, 8, 4, 1, 9, 3, 2, 0, 8, 6, 3, 1, 1, 7, 6, 6, 3, 2, 5, 7, 8, 0, 7, 1, 2, 4, 8, 7, 6, 2, 3, 0, 6, 5, 6, 6, 0, 5, 7, 7, 8, 3, 5, 2, 3, 0, 8, 5, 0, 5, 5, 3, 0, 6, 3, 0, 8
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:    1.94634640...; see A197155
endpoint on x axis:    (4.2236, 0)
endpoint on line y=3x: (3.79888, 1.89944)

Crossrefs

Cf. A197032, A197155, A197008, A195284.

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 = 1/2; h = 4; k = 1;(* slop m, point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197154 *)
{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]  (* A197155 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4.5}],
ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4.5}, {y, 0, 3}],
PlotRange -> {0, 2}, AspectRatio -> Automatic]

Extensions

Last digit removed (repr. truncated, not rounded up) by Georg Fischer, Nov 11 2022

A197155 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2.

Original entry on oeis.org

1, 9, 4, 6, 3, 4, 6, 4, 0, 2, 3, 7, 8, 4, 8, 3, 8, 5, 6, 1, 6, 6, 4, 0, 9, 1, 1, 4, 2, 3, 0, 0, 8, 0, 6, 8, 1, 8, 5, 8, 2, 1, 0, 6, 7, 1, 1, 7, 6, 0, 3, 8, 5, 7, 0, 1, 8, 9, 2, 3, 8, 5, 0, 9, 1, 0, 4, 9, 9, 8, 9, 5, 6, 0, 1, 8, 8, 6, 8, 0, 1, 9, 1, 0, 7, 7, 4, 4, 3, 2, 0, 7, 0, 6, 5, 2, 2, 4, 1, 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:  1.94634640...
endpoint on x axis:    (4.2236, 0); see A197154
endpoint on line y=3x: (3.79888, 1.89944)

Crossrefs

Cf. A197032, A197155, A197008, A195284.

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 = 1/2; h = 4; k = 1;(* slop m, point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197154 *)
{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]  (* A197155 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4.5}],
 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4.5}, {y, 0, 3}],
 PlotRange -> {0, 2}, AspectRatio -> Automatic]

A197033 Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

For discussions and guides to related sequences, see A197032, A197008 and A195284.

Examples

			length of Philo line:  1.8442716817001718647799577442735702984134...
endpoint on x axis:  (2.35321..., 0); see A197032
endpoint on line y=x:  (1.73898, 1.73898)

Crossrefs

Cf. A197032, A197008, A195284.

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 (* root of p[t] minimizes f *)
m = 1; h = 2; k = 1; (* m=slope; (h,k)=point *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197032 *)
{N[t], 0} (* lower endpoint of minimal segment [Philo line] *)
{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] (* A197033 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}],
 ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]

A357469 Decimal expansion of the real root of x^3 - x^2 + x - 2.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Oct 17 2022

Keywords

Comments

This equals r0 + 1/3 where r0 is the real root of y^3 + (2/3)*y - 47/27, after 1/3.
The other (complex) roots of x^3 - x^2 + x - 2 are (w1*(4*(47 + 3*sqrt(249)))^(1/3) + (4*(47 - 3*sqrt(249)))^(1/3) + 2)/6 = -0.1766049820... + 1.2028208192...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (-sqrt(2)*(sinh((1/3)*arcsinh((47/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((47/8)*sqrt(2)))*i) + 1)/3, and its complex conjugate.

Examples

			1.3532099641993244294831013325773884572707056138568468268066930426515189723220920859165...

Links

Crossrefs

Cf. A197032, A137421, A357468.

Programs

  • Maple
    Digits := 140 ;
r := (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3 ;
evalf(%) ; # R. J. Mathar, Nov 08 2022
  • Mathematica
    RealDigits[x /. FindRoot[x^3 - x^2 + x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 18 2022 *)

Formula

r = ((4*(47 + 3*sqrt(249)))^(1/3) - 8*(4*(47 + 3*sqrt(249)))^(-1/3) + 2)/6.
r = ((4*(47 + 3*sqrt(249)))^(1/3) + w1*(4*(47 - 3*sqrt(249)))^(1/3) + 2)/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
r = (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3.
Equals A197032 minus one.
Previous Showing 21-25 of 25 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.