A355415 Decimal expansion of the average distance between the center of a unit cube to a point on its surface uniformly chosen by a random direction from the center.

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

Offset

0,1

Comments

If the point is uniformly chosen at random on the surface, then the average is A097047.

Links

Table of n, a(n) for n=0..104.

Mathematics Stack Exchange, Average Width of the Cube, or Average Radius of a Regular Polyhedron, 2021.

Mathematics Stack Exchange, What is the average radius of the hypercube?, 2019.

Formula

Equals (1/2) * Integral_{x=-1..1, y=-1..1} (1 + x^2 + y^2)^(-1) dx dy / Integral_{x=-1..1, y=-1..1} (1 + x^2 + y^2)^(-3/2) dx dy.

Equals (3/Pi) * Integral_{x=0..1} arccot(sqrt(1+x^2))/sqrt(1+x^2) dx.

Equals (6/Pi) * Integral_{x=0..Pi/4} log(sqrt(1+cos(x)^2)/cos(x)) dx.

Equals 3 * ((Im(Li_2((3-2*sqrt(2))*i)) - G)/Pi - log(17-12*sqrt(2))/8), where Li_2 is the dilogarithm function, i is the imaginary unit, and G is Catalan's constant (A006752).

Example

0.61068740195158385653466722967371662846911552581907...

Mathematica

RealDigits[N[(3/Pi)*Integrate[ArcCot[Sqrt[1 + x^2]]/Sqrt[1 + x^2], {x, 0, 1}], 101], 10, 100][[1]]
(* or *)
RealDigits[3 * ((Im[PolyLog[2, (3 - 2*Sqrt[2])*I]] - Catalan)/Pi - Log[17 - 12*Sqrt[2]]/8), 10, 100][[1]]

Crossrefs

Cf. A006752, A073012, A093066, A097047, A130590, A135691, A348680, A348681, A348682, A348683, A355186 (2D analog).

Sequence in context: A370563 A120113 A074395 * A262704 A335245 A195402

Adjacent sequences: A355412 A355413 A355414 * A355416 A355417 A355418

Keyword

nonn,cons

Author

Amiram Eldar, Jul 01 2022

Status

approved