A254149 Decimal expansion of the average reciprocal length of a line segment picked at random in a unit 4-cube.

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Jan 26 2015

Keywords

• nonn
• cons
• easy

Examples

			1.481432636521064749748769140727658302570952634154861...
		

Links

• D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (2010), 1839-1866. See p. 25.
• Eric Weisstein's World of Mathematics, Hypercube Line Picking
• Eric Weisstein World of Mathematics, Inverse Tangent Integral

Crossrefs

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

Programs

Mathematica

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

Python

from mpmath import *
mp.dps=104
x=3 - 2*sqrt(2)
Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
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)
print([int(n) for n in list(str(C.real).replace('.','')[:-1])]) # Indranil Ghosh, Jul 03 2017

Formula

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.

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.