

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
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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=xintercept of L
V=yintercept of L
d=UV
...
Although Philo lines are not generally Euclideanconstructible, 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...
xintercept: U=(2.5874...,0)
yintercept: 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] (* xintercept; 1+4^(1/3); cf. A005480 *)
y = N[k*t/(t  h)] (* yintercept *)
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



