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A197008 Decimal expansion of the shortest distance from x axis through (1,2) to y axis. 32
4, 1, 6, 1, 9, 3, 8, 1, 8, 4, 9, 4, 1, 4, 6, 2, 7, 5, 2, 3, 9, 0, 0, 8, 0, 7, 2, 2, 9, 4, 6, 6, 9, 9, 6, 3, 7, 7, 8, 9, 3, 2, 5, 5, 8, 7, 5, 5, 0, 9, 3, 0, 3, 0, 2, 4, 2, 9, 6, 2, 3, 8, 5, 2, 7, 0, 6, 8, 8, 5, 0, 3, 6, 5, 0, 2, 9, 1, 5, 9, 3, 8, 2, 4, 6, 1, 3, 8, 8, 2, 2, 0, 6, 7, 8, 3, 6, 1, 2, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P.  Suppose that T is the angle formed by the positive x and y axes and that h>0 and k>0.  Notation:

...

P=(h,k)

L=the Philo line of P across T

U=x-intercept of L

V=y-intercept of L

d=|UV|

...

Although Philo lines are not generally Euclidean-constructible, exact expressions for U, V, and d can be found for the angle T under consideration.  Write u(t)=(t,0), let v(t) the corresponding point on the y axis, and let d(t) be the distance between u(t) and v(t). Then d is found by minimizing d(t)^2:

  d=w*sqrt(1+(k/h)^(2/3)), where w=(h+(h*k^2))^(1/3).

...

Guide:

h....k...........d

1....2........A197008

1....3........A197012

1....4........A197013

2....3........A196014

1....e........A196015

1..sqrt(2)....A196031

...

For a guides other Philo lines, see A195284 and A197132.

LINKS

Table of n, a(n) for n=1..100.

EXAMPLE

d=4.161938184941462752390080...

x-intercept: U=(2.5874...,0)

y-intercept: V=(0,3.2599...)

MATHEMATICA

f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);

h = 1; k = 2; d = N[f[t]^(1/2), 100]

RealDigits[d] (* A197008 *)

x = N[t] (* x-intercept; -1+4^(1/3); cf. A005480 *)

y = N[k*t/(t - h)] (* y-intercept *)

Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],

ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 4}], PlotRange -> All, AspectRatio -> Automatic]

CROSSREFS

Cf. A197012, A005480, A195284.

Sequence in context: A135683 A113520 A232597 * A316223 A087652 A072195

Adjacent sequences:  A197005 A197006 A197007 * A197009 A197010 A197011

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 10 2011

STATUS

approved

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Last modified November 15 08:55 EST 2019. Contains 329144 sequences. (Running on oeis4.)