Pensieve header: December 7, 2020: The cube of oranges riddle.

{Graphics[{Orange,Disk /@ Tuples[{1,-1},2],Blue, Disk[{0,0},(Sqrt[2])-1]}, Frame->True, ImageSize -> 360],
Graphics3D[{Orange,Ball /@ Tuples[{1,-1},3],Blue, Ball[{0,0,0},(Sqrt[3])-1]}, ImageSize -> 360]}

(Subscript[σ, 0])=2; (Subscript[σ, 1])=2π; (Subscript[σ, n_]):=(Subscript[σ, n])=((2π)/(n-1))(Subscript[σ, n-2]);
(Subscript[β, n_]):=((Subscript[σ, n-1])/n);

Table[(Subscript[σ, n]),{n,0,10}]

Table[(Subscript[β, n]),{n,1,10}]

r[n_]:=(((Subscript[β, n])((((Sqrt[n])-1))^n))/(4^n))

Table[N[r[n],5], {n,1,10}]

ListPlot[Table[N[r[n],5], {n,1,2000}]]

ListPlot[Table[N[Log[r[n]],5], {n,1,2000}]]