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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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified January 19 03:54 EST 2020. Contains 331031 sequences. (Running on oeis4.)