A332526 Decimal expansion of the minimal distance between distinct branches of the tangent function; see Comments.

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

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 A355278 A329412

Adjacent sequences: A332523 A332524 A332525 * A332527 A332528 A332529

Keyword

nonn,cons

Author

Clark Kimberling, Jun 15 2020

Status

approved