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 31 results. Next

A197032 Decimal expansion of the x-intercept of the shortest segment from the positive x axis through (2,1) to the line y=x.

Original entry on oeis.org

2, 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
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.
Philo lines from positive x axis through (h,k) to line y=mx:
m......h......k....x-intercept.....distance
1......2......1.......A197032......A197033
1......3......1.......A197034......A197035
1......4......1.......A197136......A197137
1......3......2.......A197138......A197139
2......1......1.......A197140......A197141
2......2......1.......A197142......A197143
2......3......1.......A197144......A197145
2......4......1.......A197146......A197147
3......1......1.......A197148......A197149
3......2......1.......A197150......A197151
1/2....3......1.......A197152......A197153
1/2....4......1.......A197154......A197155

Examples

			length of Philo line:  1.8442716817001... (see A197033)
endpoint on x axis: (2.35321..., 0)
endpoint on y=x:    (1.73898, 1.73898)

Links

Crossrefs

Cf. A197033, A197008, A195284.
Cf. A357469 (= this constant - 1).

Programs

  • Maple
    Digits := 140 ;
x^3-4*x^2+6*x-5 ;
fsolve(%=0) ; # R. J. Mathar, Nov 08 2022
  • 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]
  • PARI
    solve(x=2,3, x^3 - 4*x^2 + 6*x - 5)

Formula

x = 2 + tan phi where 1 + 2 tan phi = 1/(sin phi + cos phi), whence x = 1 + A357469 = the only real root of x^3 - 4*x^2 + 6*x - 5. - M. F. Hasler, Nov 08 2022

Extensions

Invalid trailing digits corrected by R. J. Mathar, Nov 08 2022

A005480 Decimal expansion of cube root of 4.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane; entry revised Apr 23 2006

Keywords

Comments

Let h = 4^(1/3). Then (h+1,0) is the x-intercept of the shortest segment from the x-axis through (1,2) to the y-axis; see A197008. - Clark Kimberling, Oct 10 2011
Let h = 4^(1/3). The relative maximum of xy(x+y)=1 is (-1/sqrt(h), h). - Clark Kimberling, Oct 05 2020

Examples

			1.587401051968199474751705639272308260391493327899853...

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Horace S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data, Scripta Math. 18, (1952), p. 173-176.

Links

Crossrefs

Cf. A002947 (continued fraction). - Harry J. Smith, May 07 2009
Cf. A002580 (cube root of 2).

Programs

  • Mathematica
    RealDigits[N[4^(1/3), 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
  • PARI
    default(realprecision, 20080); x=4^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b005480.txt", n, " ", d));  \\ Harry J. Smith, May 07 2009, with a correction made May 19 2009

Formula

Equals Product_{k>=0} (1 + (-1)^k/(3*k + 1)). - Amiram Eldar, Jul 25 2020
Equals A002580^2. - Michel Marcus, Jan 08 2022
Equals hypergeom([1/3, 1/6], [2/3], 1). - Peter Bala, Mar 02 2022

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

Cf. A197032, A197035, 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; 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

Cf. A197032, A197034, 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; 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

Cf. A197032, A197137, 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; 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

Cf. A197032, A197136, 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; 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

Cf. A197032, A197139, 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; 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

Cf. A197032, A197138, 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; 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

Cf. A197032, A197141, 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 = 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

Cf. A197032, A197140, 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 = 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]
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