Congruence Classes of Triangles in a Cubical Grid

For n >= 0, let G_n = {0,1,...,n}^3, and let a(n) be the number of Euclidean congruence classes of nondegenerate triangles with vertices in G_n. This is the three-dimensional analogue of the square-grid sequence A028419. Version 2.0 strengthens the asymptotic theory. In addition to the exact finite encoding by 3n(n+1)/2 + 1 one-coordinate contribution types and strict monotonicity, it proves the sharp order of growth a(n) = Theta(n^6/sqrt(log n)). Equivalently, log a(n) = 6*log n - (1/2)*log log n + O(1). The proof also gives an exact primitive representation count for integral Gram matrices: every primitively representable Gram matrix of determinant D has exactly 24*rho(D) ordered primitive representations in Z^3, where rho(D) is the number of square roots of 1 modulo D. The local representability conditions lead to a half-dimensional sieve, accounting for the factor 1/sqrt(log n). The record contains the research note and source, a companion illustration of the 40 congruence classes counted by a(2) = 40, a b-file for n = 0..42, and Maple code for computing the terms. No computation is used in the proofs.