A254979 - OEIS

%I #33 Jun 04 2023 01:41:18

%S 1,1,2,1,8,9,9,6,1,8,7,1,5,8,6,0,9,7,7,3,5,1,6,1,5,1,7,5,5,6,7,5,4,2,

%T 7,0,9,2,0,0,8,0,7,9,5,6,4,3,9,5,4,5,8,3,0,8,3,6,7,9,2,4,6,6,9,1,6,4,

%U 0,3,5,4,8,6,0,6,9,1,5,3,4,9,0,2,4,6,7,3,1,4,5,5,7,8,6,3,7,6,4,4,9,7,6,3,4

%N Decimal expansion of the mean 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).

%C Also, decimal expansion of twice the expected distance from a randomly selected point in the unit 4D cube to the center. - _Amiram Eldar_, Jun 04 2023

%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 World of Mathematics, <a href="http://mathworld.wolfram.com/InverseTangentIntegral.html">Inverse Tangent Integral</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BoxIntegral.html">Box Integral</a>.

%F Equals B_4(1) = 2/5 - Catalan/10 + (3/10)*Ti_2(3-2*sqrt(2)) + log(3) - (7*sqrt(2)/10) * 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 1.12189961871586097735161517556754270920080795643954583...

%t Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[1] = 2/5 - Catalan/10 + (3/10)*Ti2[3 - 2*Sqrt[2]] + Log[3] - (7*Sqrt[2]/10)*ArcTan[1/Sqrt[8]] // Re; RealDigits[B4[1], 10, 105] // First

%t N[Integrate[1/u^2 - Pi^2*Erf[u]^4/(16*u^6), {u, 0, Infinity}]/Sqrt[Pi], 50] (* _Vaclav Kotesovec_, Aug 13 2019 *)

%o (Python)

%o from mpmath import *

%o mp.dps=106

%o x=3 - 2*sqrt(2)

%o Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))

%o C = 2/5 - catalan/10 + (3/10)*Ti2x + log(3) - (7*sqrt(2)/10)*atan(1/sqrt(8))

%o print([int(n) for n in str(C.real).replace('.', '')]) # _Indranil Ghosh_, Jul 04 2017

%Y Cf. A117653, A117654, A244920, A254968.

%Y Analogous constants: A244921 (square), A130590 (cube).

%K nonn,cons,easy

%O 1,3

%A _Jean-François Alcover_, Feb 11 2015

%E Name corrected by _Amiram Eldar_, Jun 04 2023