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A197142 Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) to the line y=2x. 2
2, 6, 9, 1, 4, 1, 3, 9, 7, 9, 7, 8, 1, 7, 3, 6, 7, 9, 1, 9, 2, 8, 8, 6, 5, 6, 0, 5, 7, 5, 3, 2, 2, 5, 2, 1, 8, 3, 8, 5, 7, 6, 7, 6, 4, 6, 9, 2, 4, 6, 8, 9, 7, 0, 9, 7, 1, 2, 4, 7, 6, 5, 3, 6, 6, 0, 0, 4, 2, 2, 1, 8, 8, 2, 5, 9, 8, 6, 2, 1, 0, 6, 1, 9, 1, 0, 1, 9, 6, 9, 9, 3, 8, 3, 3, 7, 6, 0, 0, 4 (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..100.

EXAMPLE

length of Philo line:    2.7463941076100...; see A197143

endpoint on x axis:    (2.69141, 0)

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]

CROSSREFS

Cf. A197032, A197143, A197008, A195284.

Sequence in context: A021792 A200705 A154468 * A129635 A231985 A205527

Adjacent sequences:  A197139 A197140 A197141 * A197143 A197144 A197145

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 11 2011

STATUS

approved

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Last modified May 21 05:37 EDT 2022. Contains 353889 sequences. (Running on oeis4.)