A254979 -id:A254979 - 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

A117653 Coefficients in an asymptotic expansion for mean distance from origin of a point uniformly distributed over the n-dimensional unit cube [0,1]^n (numerators).

Original entry on oeis.org

1, -1, -13, -101, -37533

Offset: 0

N. J. A. Sloane, Apr 12 2006

			(n/3)^(1/2)*(1-n/10 -13n^2/280 - ...)

R. S. Anderssen et al., Concerning Integral_{0..1} ... Integral_{0..1} (x_1^2+...+x_k^2)^(1/2) dx_1 ... dx_k and a Taylor series method, SIAM J. Appl. Math., 30 (1976), 22-30.
Eric Weisstein's MathWorld, Hypercube Point Picking.

Cf. A117654, A254979.

A117654 Coefficients in an asymptotic expansion for mean distance from origin of a point uniformly distributed over the n-dimensional unit cube [0,1]^n (denominators).

Original entry on oeis.org

1, 10, 280, 2800, 1232000

Offset: 0

N. J. A. Sloane, Apr 12 2006

			(n/3)^(1/2)*(1-n/10 -13n^2/280 - ...)

R. S. Anderssen et al., Concerning Integral_{0..1} ... Integral_{0..1} (x_1^2+...+x_k^2)^(1/2) dx_1 ... dx_k and a Taylor series method, SIAM J. Appl. Math., 30 (1976), 22-30.
Eric Weisstein's MathWorld, Hypercube Point Picking.

Cf. A117653, A254979.

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

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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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A117653 Coefficients in an asymptotic expansion for mean distance from origin of a point uniformly distributed over the n-dimensional unit cube [0,1]^n (numerators).

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Author

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A117654 Coefficients in an asymptotic expansion for mean distance from origin of a point uniformly distributed over the n-dimensional unit cube [0,1]^n (denominators).

Views

Author

Keywords

Examples

Links

Crossrefs

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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