A103983 Decimal expansion of average length of a line segment picked at random in a unit 4-cube.
7, 7, 7, 6, 6, 5, 6, 5, 3, 5, 8, 6, 2, 6, 7, 1, 1, 5, 3, 3, 7, 9, 3, 4, 0, 9, 4, 6, 1, 7, 8, 1, 9, 5, 0, 9, 9, 6, 2, 8, 8, 2, 7, 2, 4, 4, 1, 7, 1, 3, 3, 0, 5, 8, 0, 2, 3, 4, 4, 5, 9, 6, 4, 8, 6, 5, 0, 5, 7, 3, 5, 3, 1, 5, 9, 2, 6, 5, 4, 0, 1, 1, 4, 6, 1, 5, 1, 6, 5, 6, 8, 9, 3, 1, 6, 8, 1, 8, 8, 4, 6, 5
Offset: 0
Examples
0.777665653...
References
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.1, p. 480.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
- Eric Weisstein's World of Mathematics, Hypercube Line Picking.
Programs
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Mathematica
A:= ( 0 - 3024*Im[PolyLog[2, I*(3 - 2*Sqrt[2])]] + 6048*I*(PolyLog[2, (1 - I)/2*(-2 + Sqrt[2])] - PolyLog[2, (1 + I)/2*(-2 + Sqrt[2])]) + 6048*I*(PolyLog[2, I*(1 - Sqrt[2])] - PolyLog[2, I*(-1 + Sqrt[2])]) )/3780; B := (-644 + 438*Sqrt[2] + 288*Sqrt[3] + 1344*Catalan + 12*Pi*(-16 + 7*(-8 + 9*Sqrt[2])*Pi) - 2448*Sqrt[2]*ArcCot[2*Sqrt[2]] + 3744*ArcCsch[Sqrt[2]] + 270*ArcSinh[1] + 1773*Log[3] - 189*Sqrt[2]*(PolyGamma[1, 1/8] + PolyGamma[1, 3/8]) + 84*(PolyGamma[1, 1/12] + PolyGamma[1, 5/12]))/3780; RealDigits[Re[A] + B, 10, 50][[1]] (* G. C. Greubel, Jan 11 2017 *)
Formula
(-644 + 438*sqrt(2) + 288*sqrt(3) + 1344*Catalan + 12*Pi(-16 + 7(-8 + 9*sqrt(2))*Pi) - 2448*sqrt(2)*arccot(2*sqrt(2)) + 3744*arccsch(sqrt(2)) + 270*arcsinh(1) - 3024*Im(PolyLog(2, i*(3 - 2*sqrt(2)))) + 1773*log(3) - 189*sqrt(2)*(PolyGamma(1, 1/8) + PolyGamma(1, 3/8)) + 84*(PolyGamma(1, 1/12) + PolyGamma(1, 5/12)) + 6048*i(PolyLog(2, (1/2 - i/2)*(-2 + sqrt(2))) - PolyLog)2, (1/2 + i/2)*(-2 + sqrt(2)))) + 6048*i(PolyLog(2, i*(1 - sqrt(2))) - PolyLog(2, i*(-1 + sqrt(2)))))/3780, where i=sqrt(-1). - Eric W. Weisstein, Mar 02 2005
Extensions
Offset corrected by Vaclav Kotesovec, Aug 23 2025 following a suggestion of Artur Jasinski