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A197155 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2. 2
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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..100.

EXAMPLE

length of Philo line:  1.94634640...

endpoint on x axis:    (4.2236, 0); see A197154

endpoint on line y=3x: (3.79888, 1.89944)

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]

CROSSREFS

Cf. A197032, A197155, A197008, A195284.

Sequence in context: A086240 A288240 A289267 * A153756 A322088 A139720

Adjacent sequences:  A197152 A197153 A197154 * A197156 A197157 A197158

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 11 2011

STATUS

approved

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Last modified October 16 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)