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

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

1,1

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).

Sequence in context: A352377 A202694 A123221 * A321781 A254862 A340641

Adjacent sequences: A197029 A197030 A197031 * A197033 A197034 A197035

Keyword

nonn,cons

Author

Clark Kimberling, Oct 10 2011

Extensions

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

Status

approved