Counting 3-Term Arithmetic Progressions in Integer Hypercubes

In this research note, we generalize the combinatorial problem of counting 3-term arithmetic progressions in the n× n grid to the integer hypercube of arbitrary dimension d≥1 denoted [1,n]ᵈ. Counting exclusively strictly increasing progressions using a bijection based on the uniqueness of the midpoint and the partition of the hypercube into parity classes, we derive the closed formula: T(n,d) = (a²+b²)ᵈ-nᵈ2 where a= n/2 and b= n/2. This approach offers a combinatorial perspective distinct from classical analytic methods, providing explicit formulas, tables of values, and establishing results not yet recorded in the OEIS.