Congruence Classes of Triangles in a Cubical Grid

Abstract

For n >= 0, let G_n = {0,1,...,n}^3. This note studies the number a(n) of Euclidean congruence classes of nondegenerate triangles whose vertices lie in G_n, the three-dimensional analogue of the square-grid sequence A028419. An exact finite encoding is obtained from the 3*n*(n+1)/2 + 1 possible one-coordinate contributions to the three squared side lengths. It yields an O(n^6) upper bound, a direct finite enumeration algorithm, and strict monotonicity of the sequence. For the lower bound, primitive ordered edge pairs are counted using the density theorem for rectangular unimodular integer matrices, while a lattice-theoretic multiplicity estimate bounds the number of primitive realizations of a fixed Gram matrix. This gives, for some constant C > 0 and all sufficiently large n, a(n) >= n^6 * exp(-C*log(n)/log(log(n))), and in particular lim_{n -> infinity} log(a(n))/log(n) = 6. The record contains the research note, an 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. The mathematical proofs are computer-free.