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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.

%I A355415 #9 Jul 01 2022 10:11:03
%S A355415 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,
%T A355415 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,
%U A355415 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
%N 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.
%C A355415 If the point is uniformly chosen at random on the surface, then the average is A097047.
%H A355415 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4207524/average-width-of-the-cube-or-average-radius-of-a-regular-polyhedron">Average Width of the Cube, or Average Radius of a Regular Polyhedron</a>, 2021.
%H A355415 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3433906/what-is-the-average-radius-of-the-hypercube">What is the average radius of the hypercube?</a>, 2019.
%F A355415 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.
%F A355415 Equals (3/Pi) * Integral_{x=0..1} arccot(sqrt(1+x^2))/sqrt(1+x^2) dx.
%F A355415 Equals (6/Pi) * Integral_{x=0..Pi/4} log(sqrt(1+cos(x)^2)/cos(x)) dx.
%F A355415 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).
%e A355415 0.61068740195158385653466722967371662846911552581907...
%t A355415 RealDigits[N[(3/Pi)*Integrate[ArcCot[Sqrt[1 + x^2]]/Sqrt[1 + x^2], {x, 0, 1}], 101], 10, 100][[1]]
%t A355415 (* or *)
%t A355415 RealDigits[3 * ((Im[PolyLog[2, (3 - 2*Sqrt[2])*I]] - Catalan)/Pi - Log[17 - 12*Sqrt[2]]/8), 10, 100][[1]]
%Y A355415 Cf. A006752, A073012, A093066, A097047, A130590, A135691, A348680, A348681, A348682, A348683, A355186 (2D analog).
%K A355415 nonn,cons
%O A355415 0,1
%A A355415 _Amiram Eldar_, Jul 01 2022