A197033 Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.

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

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: A356810 A011360 A244091 * A245720 A229495 A370198

Adjacent sequences: A197030 A197031 A197032 * A197034 A197035 A197036

Keyword

nonn,cons

Author

Clark Kimberling, Oct 10 2011

Status

approved