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.

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

Original entry on oeis.org

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

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 x^3/2 -5*x^2/2 +9*x/2 -5. - R. J. Mathar, Nov 08 2022

Examples

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

Crossrefs

Cf. A197032, A197153, 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]

Extensions

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

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]

A197012 Decimal expansion of the shortest distance from x axis through (1,3) to y axis.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See A197008 for a discussion and guide to related sequences.

Examples

			d=5.40559872742348382543060865269562398196...
x-intercept=(3.0800...,0)
y-intercept=(0,4.4422...)

Crossrefs

Cf. A197008.

Programs

  • Maple
    (3^(1/2)+1/3^(1/6))*sqrt(3^(1/3)+3) ; 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 = 3; d = N[f[t]^(1/2), 100]
RealDigits[d] (* A197012 *)
x = N[t] (* x-intercept *)
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 == .002, {x, 0, 4}, {y, 0, 4}],PlotRange -> All, AspectRatio -> Automatic]

Extensions

Typo in definition corrected by Georg Fischer, Nov 11 2022

A197013 Decimal expansion of the shortest distance from x axis through (1,4) to y axis.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See A197008 for a discussion and guide to related sequences.

Examples

			d=6.60366102423402958588694523729286548481762327...
x-intercept=(3.5198...,0)
y-intercept=(0,5.5874...)

Crossrefs

Cf. A197008.

Programs

  • Mathematica
    f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);
h = 1; k = 4; d = N[f[t]^(1/2), 100]
RealDigits[d] (* A197013 *)
x = N[t] (* x-intercept *)
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 == .003, {x, 0, 4}, {y, 0, 5}], PlotRange -> All, AspectRatio -> Automatic]

A197014 Decimal expansion of the shortest distance from x axis through (2,3) to y axis.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See A197008 for a discussion and guide to related sequences.

Examples

			d=7.02348237921996592684456014412915048027327616603
x-intercept=(4.6207...,0)
y-intercept=(0,5.2894...)

Crossrefs

Cf. A197008.

Programs

  • Maple
    (18^(2/3)/9+1)*sqrt(18^(2/3)+9) ; 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 = 2; k = 3; d = N[f[t]^(1/2), 100]
RealDigits[d] (* A197014 *)
x = N[t] (* x-intercept *)
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 == .004, {x, 0, 4}, {y, 0, 5}], PlotRange -> All, AspectRatio -> Automatic]

A197015 Decimal expansion of the shortest distance from x axis through (3,4) to y axis.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See A197008 for a discussion and guide to related sequences.

Examples

			d=9.865662555435090192548544326689054243084...
x-intercept=(6.6342...,0)
y-intercept=(0,7.3019...)

Crossrefs

Cf. A197008.

Programs

  • Maple
     3*(1+(4/3)^(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 = 3; k = 4; d = N[f[t]^(1/2), 100]
RealDigits[d] (* A197015 *)
x = N[t] (* x-intercept *)
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 == .01, {x, 0, 4}, {y, 0, 5}], PlotRange -> All, AspectRatio -> Automatic]

A197031 Decimal expansion of the shortest distance from x axis through (1,sqrt(2)) to y axis.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See A197008 for a discussion and guide to related sequences.

Examples

			d=3.397346951017693441277913755501410790489483...
x-intercept=(2.2599...,0)
y-intercept=(0,2.5366...)

Crossrefs

Cf. A197008.

Programs

  • Maple
    (1+2^(1/3))^(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 = Sqrt[2]; d = N[f[t]^(1/2), 100]
RealDigits[d] (* A197031 *)
x = N[t] (* x-intercept *)
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, 5}], PlotRange -> All, 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]
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