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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. 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 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: A090203 A120113 A074395 * A262704 A335245 A195402 Adjacent sequences: A355412 A355413 A355414 * A355416 A355417 A355418 KEYWORD nonn,cons AUTHOR Amiram Eldar, Jul 01 2022 STATUS approved

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Last modified December 4 08:59 EST 2023. Contains 367558 sequences. (Running on oeis4.)