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A332526 Decimal expansion of the minimal distance between distinct branches of the tangent function; see Comments. 1
2, 3, 7, 5, 0, 6, 9, 1, 4, 6, 0, 4, 0, 1, 7, 6, 3, 4, 9, 4, 3, 9, 8, 5, 1, 5, 5, 8, 7, 7, 8, 9, 8, 2, 4, 8, 7, 8, 6, 6, 2, 6, 7, 8, 0, 6, 5, 0, 8, 8, 4, 1, 7, 9, 2, 9, 2, 6, 9, 8, 5, 6, 4, 5, 9, 7, 5, 4, 8, 6, 6, 7, 0, 2, 9, 6, 9, 1, 3, 1, 6, 3, 3, 4, 1, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let T0 and T1 be the branches of the graph of y = tan x that passes through (0,0,) and (Pi,0), respectively. There exist points P = (p,q) on T0 and U = (u,v) on T1 such that the distance between P and U is the minimal distance, d, between points on T0 and T1.

u = 2.549082584017596768984130292562154758705824602711...

v = -0.67319711901285205370684801604861382107848678888...

p = Pi - u

q = - v

d = 2.375069146040176349439851558778982487866267806508...

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

minimal distance = 2.375069146040176349439851558778982487866267806508...

MATHEMATICA

min = Quiet[FindMinimum[Sqrt[(#[[1]][[1]] - #[[2]][[1]])^2 + (#[[1]][[2]] - \

#[[2]][[2]])^2] &[{{#, Tan[#]} &[x /. FindRoot[# Cos[#]^2 - x Cos[#]^2 + Tan[#] == Tan[x], {x, 0}, WorkingPrecision -> 500]], {#, Tan[#]} &[#]} &[y]], {y, 2}, WorkingPrecision -> 100]]

Show[Plot[{Tan[x], (-# Sec[#]^2) + x Sec[#]^2 + Tan[#], {(# Cos[#]^2) - x Cos[#]^2 + Tan[#]}}, {x, 0, Pi}, AspectRatio -> Automatic, ImageSize -> 300, PlotRange -> {-2, 2}], Graphics[{PointSize[Large], Point[{Pi/2, 0}], Point[{#, Tan[#]}], Point[{Pi - #, -Tan[#]}]}]] &[y /. min[[2]][[1]]]

(* Peter J. C. Moses, May 06 2020 *)

CROSSREFS

Cf. A332500, A332525, A332527.

Sequence in context: A229794 A331318 A082734 * A021425 A329412 A335292

Adjacent sequences:  A332523 A332524 A332525 * A332527 A332528 A332529

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jun 15 2020

STATUS

approved

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Last modified April 20 03:02 EDT 2021. Contains 343121 sequences. (Running on oeis4.)