A394469 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.

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, 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, 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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Wolfgang Hackbusch, Direct Integration of the Newton Potential over Cubes, Computing, Vol. 68 (2002), pp. 193-216; ResearchGate preprint.

Formula

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)).

Equals 3 * (A242588 - A394467 - A394468).

Example

0.578797001778540201893744559735783933977210440901809...

Mathematica

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]]

Prog

(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))

Crossrefs

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

Sequence in context: A197491 A385857 A338595 * A397003 A254274 A068001

Adjacent sequences: A394466 A394467 A394468 * A394470 A394471 A394472

Keyword

nonn,cons

Author

Amiram Eldar, Mar 21 2026

Status

approved