Counting 3-Term Arithmetic Progressions in Integer Hypercubes

Abstract

In this research note, we generalize the combinatorial problem of counting 3-term arithmetic progressions in the $n\times n$ grid to the integer hypercube of arbitrary dimension $d\ge1$ 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: $$T(n,d) = \frac{(a^{2}+b^{2})^{d}-n^{d}}{2}$$ where $a=\lceil n/2\rceil$ and $b=\lfloor n/2\rfloor$. 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.