p. 129 - 140 Counting all equilateral triangles in {0, 1, ...,
n}^{3}E. J. Ionascu Received: January 3, 2007;
Accepted: September 21, 2007
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}^{3}. 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 [3]. Using this we calculated the values ET(n) for
n = 1 ... 55 extending previous calculations known for
n £ 34. Some
facts and conjectures about this sequence are stated. The main
conjecture raised here is thatlim _{n ® ¥}((ln ET(n)) / (ln n + 1))
exists.
Keywords:
diophantine equations; integers.
AMS Subject classification:
Primary: 11B99;
Secondary: 11D09, 11C08.
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