%I #19 Apr 26 2021 06:33:38

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

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

%U 2,3,9,9,1,0,3,5,0,7,1,2,8,8,9,1,6,3,6,9,5,7,7,9,8,6,9,0,5,5,2,9,1,8,5

%N Decimal expansion of the average reciprocal length of a line segment picked at random in a unit 4-cube.

%H D. H. Bailey, J. M. Borwein, and R. E. Crandall, <a href="https://doi.org/10.1090/S0025-5718-10-02338-0">Advances in the theory of box integrals</a>, Math. Comp. 79 (2010), 1839-1866. See p. 25.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HypercubeLinePicking.html">Hypercube Line Picking</a>

%H Eric Weisstein World of Mathematics, <a href="http://mathworld.wolfram.com/InverseTangentIntegral.html">Inverse Tangent Integral</a>

%F Delta_4(-1) = Integral over a unit 4-cube of 1/sqrt((r1-q1)^2+(r2-q2)^2+(r3-q3)^2+(r4-q4)^2) dr dq.

%F Delta_4(-1) = -152/315 - 8*Pi/15 - 16/5*log(2) + 2/5*log(3) + 68/105*sqrt(2) - 16/35*sqrt(3) + 4/5*log(1 + sqrt(2)) + 32/5*log(1 + sqrt(3)) - 8/3*Catalan + 8*Ti2(3 - 2*sqrt(2)) - 8/5*sqrt(2)*arctan(sqrt(2)/4), where Ti2 is Lewin's arctan integral.

%e 1.481432636521064749748769140727658302570952634154861...

%t Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); Delta4[-1]=-152/315 - 8*Pi/15 - 16/5*Log[2] + 2/5*Log[3] + 68/105*Sqrt[2] - 16/35*Sqrt[3] + 4/5*Log[1 + Sqrt[2]] + 32/5*Log[1 + Sqrt[3]] - 8/3*Catalan + 8*Ti2[3 - 2 Sqrt[2]] - 8/5*Sqrt[2]*ArcTan[Sqrt[2]/4] // Re; RealDigits[Delta4[-1], 10, 103] // First

%o (Python)

%o from mpmath import *

%o mp.dps=104

%o x=3 - 2*sqrt(2)

%o Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))

%o C=-152/315 - 8*pi/15 - 16/5*log(2) + 2/5*log(3) + 68/105*sqrt(2) - 16/35*sqrt(3) + 4/5*log(1 + sqrt(2)) + 32/5*log(1 + sqrt(3)) - 8/3*catalan + 8*Ti2x - 8/5*sqrt(2)*atan(sqrt(2)/4)

%o print([int(n) for n in list(str(C.real).replace('.','')[:-1])]) # _Indranil Ghosh_, Jul 03 2017

%Y Cf. A073012, A091505, A103983, A242588, A254140.

%K nonn,cons,easy

%O 1,2

%A _Jean-François Alcover_, Jan 26 2015