Regular Octahedrons in {0, 1, K, n} 3

In this paper we describe a procedure for calculating the number of regular octahedra, RO(n), which have vertices with coordinates in the set {0, 1, ..., n}. As a result, we introduce a new sequence in The Online Encyclopedia of Integer Sequences (A178797) and list the first one hundred terms of it. We improve the method appeared in [12] which was used to find the number of regular tetrahedra with coordinates of their vertices in {0, 1, ..., n}. A new fact proved here helps increasing considerably the speed of all programs used before. The procedure is put together in a series of commands written for Maple and it is included in an earlier version of this paper in the matharxiv. Our technique allows us to find a series of cubic polynomials p1(n) = (n − 1)3 , p2(n) = 5(n − 3)3 , p3(n) = (n − 5)3 , p4(n) = 5(n − 7)3 , p5(n) = (n − 9)(7n 2 − 102n + 375),..., such that RO(n) = p1(n)χx≥1(n) + p2(n)χx≥3(n) + p3(n)χx≥5(n) + p4(n)χx≥7(n) + p5(n)χx≥9(n) + · · ·