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

1,1

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.

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

Example

0.777665653...

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 *)

Crossrefs

Cf. A103984, A103985, A103986, A103987.

Sequence in context: A341908 A216159 A276615 * A232127 A195413 A083947

Adjacent sequences: A103980 A103981 A103982 * A103984 A103985 A103986

Keyword

nonn,cons,changed

Author

Eric W. Weisstein, Feb 24 2005

Status

approved