Counting all regular octahedrons in {0,1,...,n}^3

TL;DR

This paper presents a method to count all regular octahedrons with vertices in a 3D grid, introduces a new integer sequence, and improves computational efficiency using theoretical and algorithmic advancements.

Contribution

It adapts and extends previous methods to count regular octahedrons, introduces a new sequence in OEIS, and enhances computational speed with new theoretical insights.

Findings

Introduced a new sequence in OEIS (A178797).

Developed an efficient procedure for counting octahedrons in {0,...,n}^3.

Provided the first 100 terms of the sequence.

Abstract

In this paper we describe a procedure for calculating the number of regular octahedrons that have vertices with coordinates in the set {0,1,...,n}. As a result, we introduce a new sequence in ``The Online Encyclopedia of Integer Sequences" (A178797) and list the first one hundred terms of it. We adapt the method appeared in [11] which was used to find the number of regular tetrahedra with coordinates of their vertices in {0,1,...,n}. The idea of this calculation is based on the theoretical results obtained in [14]. A new fact proved here helps increasing the speed of all the programs used before. The procedure is put together in a series of commands written for Maple.