

A197033


Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.


2



1, 8, 4, 4, 2, 7, 1, 6, 8, 1, 7, 0, 0, 1, 7, 1, 8, 6, 4, 7, 7, 9, 9, 5, 7, 7, 4, 4, 2, 7, 3, 5, 7, 0, 2, 9, 8, 4, 1, 3, 4, 8, 7, 6, 3, 3, 8, 7, 7, 0, 9, 5, 0, 9, 1, 5, 7, 4, 7, 9, 4, 0, 1, 7, 8, 6, 4, 8, 7, 6, 8, 3, 4, 3, 8, 5, 3, 8, 8, 6, 1, 2, 4, 8, 5, 0, 6, 4, 4, 7, 0, 9, 9, 7, 5, 8, 1, 8, 5, 0
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OFFSET

1,2


COMMENTS

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: 1.8442716817001718647799577442735702984134...
endpoint on x axis: (2.35321..., 0); see A197032
endpoint on line y=x: (1.73898, 1.73898)


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]


CROSSREFS

Cf. A197032, A197008, A195284.
Sequence in context: A093822 A011360 A244091 * A245720 A229495 A131921
Adjacent sequences: A197030 A197031 A197032 * A197034 A197035 A197036


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 10 2011


STATUS

approved



