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A103983 Decimal expansion of average length of a line segment picked at random in a unit 4-cube.

%I A103983 #27 Aug 23 2025 09:50:13
%S A103983 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,
%T A103983 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,
%U A103983 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
%N A103983 Decimal expansion of average length of a line segment picked at random in a unit 4-cube.
%D A103983 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.1, p. 480.
%H A103983 G. C. Greubel, <a href="/A103983/b103983.txt">Table of n, a(n) for n = 1..5000</a>
%H A103983 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HypercubeLinePicking.html">Hypercube Line Picking</a>.
%F A103983 (-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
%e A103983 0.777665653...
%t A103983 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;
%t A103983 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;
%t A103983 RealDigits[Re[A] + B, 10, 50][[1]] (* _G. C. Greubel_, Jan 11 2017 *)
%Y A103983 Cf. A103984, A103985, A103986, A103987.
%K A103983 nonn,cons
%O A103983 0,1
%A A103983 _Eric W. Weisstein_, Feb 24 2005
%E A103983 Offset corrected by _Vaclav Kotesovec_, Aug 23 2025 following a suggestion of _Artur Jasinski_