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%I #22 Jun 04 2023 01:40:49
%S 9,6,7,4,1,2,0,2,1,2,4,1,1,6,5,8,9,8,6,6,1,8,3,6,4,3,8,1,7,8,1,5,8,3,
%T 9,0,1,3,5,9,3,7,0,0,9,2,9,9,9,6,0,7,0,7,2,7,4,8,2,5,7,9,2,6,6,9,5,2,
%U 4,8,4,1,9,6,7,2,3,8,4,0,5,6,6,7,2,3,1,0,2,5,3,2,3,4,2,7,7,0,0,6,6,6,6,9
%N Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(-1) in Bailey's paper).
%H D. H. Bailey, J. M. Borwein and R. E. Crandall, <a href="https://doi.org/10.1016/j.cam.2006.06.010">Box Integrals</a>, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
%H D. H. Bailey, J. M. Borwein, and R. E. Crandall, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02338-0">Advances in the theory of box integrals</a>, Math. Comp. 79 (271) (2010) 1839-1866, Table 2.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/InverseTangentIntegral.html">Inverse Tangent Integral</a>.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BoxIntegral.html">Box Integral</a>.
%F B_4(-1) = 2*log(3) - (2/3)*Catalan + 2*Ti_2(3-2*sqrt(2)) - sqrt(8) * arctan( 1/sqrt(8) ), where Ti_2(x) = (i/2)*(polylog(2, -i*x) - polylog(2, i*x)) (Ti_2 is the inverse tangent integral function).
%e 0.96741202124116589866183643817815839013593700929996...
%t Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[-1] = 2*Log[3] - (2/3) * Catalan + 2*Ti2[3 - 2*Sqrt[2]] - Sqrt[8]*ArcTan[1/Sqrt[8]] // Re; RealDigits[ B4[-1], 10, 104] // First
%o (Python)
%o from mpmath import *
%o mp.dps=105
%o x=3 - 2*sqrt(2)
%o Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
%o C = 2*log(3) - (2/3)*catalan + 2*Ti2x - sqrt(8) * atan(1/sqrt(8))
%o print([int(n) for n in list(str(C.real)[2:-1])]) # _Indranil Ghosh_, Jul 03 2017
%Y Cf. A130590, A244920, A244921, A254968, A254979.
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Feb 11 2015
%E Name corrected by _Amiram Eldar_, Jun 04 2023
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