For all Platonic solids (excluding the tetrahedron), the expected number of steps of a random walk from one vertex to its opposite vertex is indeed an integer.
Table of n, a(n) for n=1..4.
Cf. comment to A063723
Sequence in context: A049004 A104645 A181107 * A248593 A038308 A185221
Adjacent sequences: A234971 A234972 A234973 * A234975 A234976 A234977
Jens Voß, Jan 02 2014