A392687 - OEIS

%I #14 Feb 18 2026 04:19:22

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

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

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

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

%F Equals 3*sqrt(2)/5 + sqrt(5)/3 - log(2)/12 - log(8) + 3*log(1+sqrt(2)) + (1/3)*log(7-3*sqrt(5)) + (61/24)*log(sqrt(5)-1) + (5/24)*log(sqrt(5)+1) - 1.

%F Equals 3*sqrt(2)/5 + sqrt(5)/3 - 1 + 3*log(1 + sqrt(2)) - 11*log((1 + sqrt(5))/2)/3. - _Vaclav Kotesovec_, Feb 18 2026

%e 1.47356153243053669629011641648627...

%t d[x1_, y1_, x2_, y2_] := Sqrt[(x1 - x2)^2 + (y1 - y2)^2]

%t N[Integrate[d[x1, y1, x2, y2], {x1, -1, 0}, {x2, 0, 1}, {y1, -1, 0}, {y2, 0, 1}], 100]

%o (PARI) 3*sqrt(2)/5 + sqrt(5)/3 - log(2)/12 - log(8) + 3*log(1+sqrt(2)) + (1/3)*log(7-3*sqrt(5)) + (61/24)*log(sqrt(5)-1) + (5/24)*log(sqrt(5)+1) - 1

%Y Cf. A135707.

%K nonn,cons

%O 1,2

%A _Hugo Pfoertner_, Jan 20 2026