

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.


0



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



OFFSET

0,1


COMMENTS

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


LINKS



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((32*sqrt(2))*i))  G)/Pi  log(1712*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).


KEYWORD



AUTHOR



STATUS

approved



