A197142 -id:A197142

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A197032

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

+10
26

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

(list; constant; graph; refs; listen; history; text; internal format)

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.

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

LINKS

Table of n, a(n) for n=1..89.

R. J. Mathar, OEIS A197032, Nov. 8, 2022

M. F. Hasler, Philo line - oeis.org/A197032 (google drawing), Nov. 8, 2022

Wikipedia, Philo line

Index entries for algebraic numbers, degree 3

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

EXAMPLE

length of Philo line: 1.8442716817001... (see A197033)

endpoint on x axis: (2.35321..., 0)

endpoint on y=x: (1.73898, 1.73898)

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]

PROG

(PARI) solve(x=2, 3, x^3 - 4*x^2 + 6*x - 5)

CROSSREFS

Cf. A197033, A197008, A195284.

Cf. A357469 (= this constant - 1).

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 10 2011

EXTENSIONS

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

STATUS

approved

A197143

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

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 A197032, A197008 and A195284.

LINKS

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

EXAMPLE

length of Philo line: 2.7463941076100...

endpoint on x axis: (2.69141, 0); see A197142

endpoint on line y=2x: (1.1295, 2.25901)

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]