A254979 - OEIS

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).

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

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OFFSET

1,3

COMMENTS

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

LINKS

Table of n, a(n) for n=1..105.

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.

FORMULA

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).

EXAMPLE

1.12189961871586097735161517556754270920080795643954583...

MATHEMATICA

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 *)

PROG

(Python)

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

CROSSREFS

Cf. A117653, A117654, A244920, A254968.

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