A394469 - OEIS

%I #6 Mar 25 2026 21:36:57

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

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

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

%N Decimal expansion of the average reciprocal of Euclidean distance between two randomly picked points, one of which is located in a unit cube and the other in an adjacent unit cube, both axis-parallel, with the two cubes sharing one common vertex.

%H Wolfgang Hackbusch, <a href="https://doi.org/10.1007/s00607-001-1443-8">Direct Integration of the Newton Potential over Cubes</a>, Computing, Vol. 68 (2002), pp. 193-216; <a href="https://www.researchgate.net/publication/227274152_Direct_Integration_of_the_Newton_Potential_over_Cubes">ResearchGate preprint</a>.

%F Equals 66/5 - 4*Pi + 24*arctan(sqrt(2/3)) - 16*arctan(4/3) + 8*arctan(3) - 9*sqrt(2)/5 - 12*sqrt(3)/5 - sqrt(5) - 9*sqrt(6)/5 - 24*log(2) - 7*log(5) - 9*log(1+sqrt(2)) + 12*log(1+sqrt(3)) + 8*log(1+sqrt(5)) + 7*log(1+sqrt(6)) + log(2+sqrt(5)) + 4*log(2+sqrt(6)).

%F Equals 3 * (A242588 - A394467 - A394468).

%e 0.578797001778540201893744559735783933977210440901809...

%t RealDigits[66/5 - 4*Pi + 24*ArcTan[Sqrt[2/3]] - 16*ArcTan[4/3] + 8*ArcTan[3] - (9*Sqrt[2])/5 - (12*Sqrt[3])/5 - Sqrt[5] - (9*Sqrt[6])/5 - 24*Log[2] - 7*Log[5] - 9*Log[1+Sqrt[2]] + 12*Log[1+Sqrt[3]] + 8*Log[1+Sqrt[5]] + 7*Log[1+Sqrt[6]] + Log[2+Sqrt[5]] + 4*Log[2+Sqrt[6]], 10, 120][[1]]

%o (PARI) 66/5 - 4*Pi + 24*atan(sqrt(2/3)) - 16*atan(4/3) + 8*atan(3) - 9/5*sqrt(2) - 12/5*sqrt(3) - sqrt(5) - 9/5*sqrt(6) - 24*log(2) - 7*log(5) - 9*log(1+sqrt(2)) + 12*log(1+sqrt(3)) + 8*log(1+sqrt(5)) + 7*log(1+sqrt(6)) + log(2+sqrt(5)) + 4*log(2+sqrt(6))

%Y Cf. A242588, A336274, A392514, A394467, A394468.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 21 2026