Counting all equilateral triangles in 0,1,2,...,n³

2007/01/03 by Eugen J. Ionaşcu, Eugen J. Ionascu, Ionascu, Eugen J.
Mathematics · #11D09 #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Mathematical Dynamics and Fractals #Mathematics and Applications #Number Theory (math.NT) #math.GM #math.NT #msc:11D09

paper · pdf · doi:10.48550/arxiv.math/0701111

12 pages, 1 figure, Maple code

arxiv created 2007/01/03 · openalex publication_date 2007/01/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a procedure of counting all equilateral triangles in the three dimensional space whose coordinates are allowed only in the set \0,1,...,n\. This sequence is denoted here by ET(n) and it has the entry A102698 in "The On-Line Encyclopedia of Integer Sequences". The procedure is implemented in Maple and its main idea is based on the results in \citeeji. Using this we calculated the values ET(n) for n=1..55 which are included here. Some facts and conjectures about this sequence are stated. The main of them is that \ds lim_{n→ ∞} (ln ET(n))/(ln n+1) exists.

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