A135691 -id:A135691 - cp's OEIS frontend

A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

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

Offset: 0

R. J. Mathar, Aug 10 2007

			0.960591956455052959425107951...

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2. [From _R. J. Mathar_, Oct 13 2010]
Eric Weisstein's World of Mathematics, Box Integral.

Cf. A010527, A019691, A065914, A135691.

Analogous constants: A244921 (square), A254979 (4-cube).

evalf( sqrt(3)/4+log(2+sqrt(3))/2-Pi/24);

RealDigits[Sqrt[3]/4 + Log[2+Sqrt[3]]/2 - Pi/24, 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

Equals sqrt(3)/4 + log(2+sqrt(3))/2 - Pi/24 = A010527/2 + A065914/2 - A019691.

Equals 2 * A135691. - Amiram Eldar, Jun 04 2023

Name corrected by Amiram Eldar, Jun 04 2023

A137209 Decimal expansion of (1/2)*sqrt(3/Pi).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Mar 05 2008

Decimal expansion of the radius x (in units of cube edge length) of sphere with volume x (in units of cube volume).

Appears in the asymptotic expansions of A228484 and A006588. - Johannes W. Meijer, Aug 22 2013

			0.488602511902919921586384622

Amir Behrouzi-Far, and Doron Zeilberger, On the Average Maximal Number of Balls in a Bin Resulting from Throwing r Balls into n Bins T times, arXiv:1905.07827 [math.CO], 2019.
Marcus Michelen, A Short Note on the Average Maximal Number of Balls in a Bin, Journal of Integer Sequences, Vol. 23 (2020), Article 20.1.7. See C 3,1 Table 2 p. 6. And also on arXiv, arXiv:1905.08933 [math.CO], 2019.
Index entries for transcendental numbers

Cf. A135691.

RealDigits[1/2 Sqrt[3/Pi],10,120][[1]] (* Harvey P. Dale, Jul 11 2017 *)

sqrt(3)/(2*sqrt(Pi)) \\ Michel Marcus, Jun 05 2020

A242588 Decimal expansion of the expected reciprocal Euclidean distance between two random points in the unit cube.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 20 2014

			1.88231264438966016010560083886836758785246288...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.1, p. 480.

D. H. Bailey, J. M. Borwein, R. E. Crandall, Advances in the theory of box integrals Math. Comp. 79 (2010), 1839-1866, p. 24.
Eric Weisstein's MathWorld, Cube Point Picking

Cf. A073012, A097047, A135691.

2*(1/5*(Sqrt[2] + 1 - 2*Sqrt[3]) - Log[(Sqrt[2] - 1)*(2 - Sqrt[3])] - Pi/3) // RealDigits[#, 10, 100]& // First

Integral over a unit cube of 1/sqrt((r1-q1)^2 + (r2-q2)^2 + (r3-q3)^2) dr1 dr2 dr3 dq1 dq2 dq3 = 2*(1/5*(sqrt(2) + 1 - 2*sqrt(3)) - log((sqrt(2) - 1)*(2 - sqrt(3))) - Pi/3).

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.

Original entry on oeis.org

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

Amiram Eldar, Jul 01 2022

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

			0.61068740195158385653466722967371662846911552581907...

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.

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

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]]

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).

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A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

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A137209 Decimal expansion of (1/2)*sqrt(3/Pi).

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A242588 Decimal expansion of the expected reciprocal Euclidean distance between two random points in the unit cube.

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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.

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