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).

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

0,1

Links

Table of n, a(n) for n=0..103.

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.

Formula

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).

Example

0.96741202124116589866183643817815839013593700929996...

Mathematica

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

Prog

(Python)
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

Crossrefs

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

Sequence in context: A260315 A242713 A155129 * A318336 A340577 A229923

Adjacent sequences: A254977 A254978 A254979 * A254981 A254982 A254983

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Feb 11 2015

Extensions

Name corrected by Amiram Eldar, Jun 04 2023

Status

approved