Regular tetrahedra whose vertices have integer coordinates

In this paper we introduce theoretical arguments for constructing a procedure that allows one to find the number of all regular tetrahedra that have coordinates in the set {0, 1, ..., n}. The terms of this sequence are twice the values of the sequence A103158 in the Online Encyclopedia of Integer Sequences [16]. These results lead to the consideration of an infinite graph having fractal nature which is tightly connected to the set of orthogonal 3-by-3 matrices with rational coefficients. The vertices of this graph are the primitive integer solutions of the Diophantine equation a 2 + b 2 + c 2 = 3d 2 . Our aim here is to lay down the basis of finding good estimates, if not exact formulae, for the sequence A103158