A130590 -id:A130590 - cp's OEIS frontend

A135691 Decimal expansion of (6sqrt(3) + 12log(2+sqrt(3)) - Pi)/48.

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

Offset: 0

N. J. A. Sloane, Mar 03 2008

Decimal expansion of the expected distance from a randomly selected point in the unit cube to its center.

			0.4802959782275264797125539756969...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 8.1, p. 479.

G. C. Greubel, Table of n, a(n) for n = 0..5000
D. M. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208, multiply B_3(1) by 24.

Cf. A130590.

RealDigits[(6Sqrt[3]+12Log[2+Sqrt[3]]-Pi)/48,10,120][[1]] (* Harvey P. Dale, Jan 30 2013 *)

(6*sqrt(3)+12*log(2+sqrt(3))-Pi)/48 \\ Stefano Spezia, Dec 21 2024

Equals A130590 / 2. - Amiram Eldar, Jun 04 2023

A254979 Decimal expansion of the mean Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(1) in Bailey's paper).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 11 2015

Also, decimal expansion of twice the expected distance from a randomly selected point in the unit 4D cube to the center. - Amiram Eldar, Jun 04 2023

			1.12189961871586097735161517556754270920080795643954583...

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.
Eric Weisstein's World of Mathematics, Inverse Tangent Integral.
Eric Weisstein's World of Mathematics, Polylogarithm.
Eric Weisstein's World of Mathematics, Box Integral.

Cf. A117653, A117654, A244920, A254968.

Analogous constants: A244921 (square), A130590 (cube).

Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[1] = 2/5 - Catalan/10 + (3/10)*Ti2[3 - 2*Sqrt[2]] + Log[3] - (7*Sqrt[2]/10)*ArcTan[1/Sqrt[8]] // Re; RealDigits[B4[1], 10, 105] // First
N[Integrate[1/u^2 - Pi^2*Erf[u]^4/(16*u^6), {u, 0, Infinity}]/Sqrt[Pi], 50] (* Vaclav Kotesovec, Aug 13 2019 *)

from mpmath import *
mp.dps=106
x=3 - 2*sqrt(2)
Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
C = 2/5 - catalan/10 + (3/10)*Ti2x + log(3) - (7*sqrt(2)/10)*atan(1/sqrt(8))
print([int(n) for n in str(C.real).replace('.', '')]) # Indranil Ghosh, Jul 04 2017

Equals B_4(1) = 2/5 - Catalan/10 + (3/10)*Ti_2(3-2*sqrt(2)) + log(3) - (7*sqrt(2)/10) * arctan(1/sqrt(8)), where Ti_2(x) = (i/2)*(polylog(2, -i*x) - polylog(2, i*x)) (Ti_2 is the inverse tangent integral function).

Name corrected by Amiram Eldar, Jun 04 2023

A254968 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit cube to a given vertex of the cube (named B_3(-1) in Bailey's paper).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 11 2015

			1.1900386819897767533219086751420769449911806073574982644...

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.
Eric Weisstein's MathWorld, Box Integral.

Cf. A130590, A244920, A244921.

RealDigits[(3/2)*Log[2 + Sqrt[3]] - Pi/4, 10, 105] // First

Equals B_3(-1) = (3/2)*log(2 + sqrt(3)) - Pi/4.

Equals log(7 + 4*sqrt(3)) - Pi/4 - arcsinh(1/sqrt(2)).

Name corrected by Amiram Eldar, Jun 04 2023

A254980 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(-1) in Bailey's paper).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Feb 11 2015

			0.96741202124116589866183643817815839013593700929996...

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.
Eric Weisstein's MathWorld, Inverse Tangent Integral.
Eric Weisstein's MathWorld, Polylogarithm.
Eric Weisstein's MathWorld, Box Integral.

Cf. A130590, A244920, A244921, A254968, A254979.

Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[-1] = 2*Log[3] - (2/3) * Catalan + 2*Ti2[3 - 2*Sqrt[2]] - Sqrt[8]*ArcTan[1/Sqrt[8]] // Re; RealDigits[ B4[-1], 10, 104] // First

from mpmath import *
mp.dps=105
x=3 - 2*sqrt(2)
Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
C = 2*log(3) - (2/3)*catalan + 2*Ti2x - sqrt(8) * atan(1/sqrt(8))
print([int(n) for n in list(str(C.real)[2:-1])]) # Indranil Ghosh, Jul 03 2017

B_4(-1) = 2*log(3) - (2/3)*Catalan + 2*Ti_2(3-2*sqrt(2)) - sqrt(8) * arctan( 1/sqrt(8) ), where Ti_2(x) = (i/2)*(polylog(2, -i*x) - polylog(2, i*x)) (Ti_2 is the inverse tangent integral function).

Name corrected by Amiram Eldar, Jun 04 2023

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).

cp's OEIS Frontend

A135691 Decimal expansion of (6*sqrt(3) + 12*log(2+sqrt(3)) - Pi)/48.

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A254979 Decimal expansion of the mean Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(1) in Bailey's paper).

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A254968 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit cube to a given vertex of the cube (named B_3(-1) in Bailey's paper).

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A254980 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(-1) in Bailey's paper).

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Examples

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Crossrefs

Programs

Mathematica

Python

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Extensions

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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Examples

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Programs

Mathematica

Formula

A135691 Decimal expansion of (6sqrt(3) + 12log(2+sqrt(3)) - Pi)/48.