OFFSET
1,1
COMMENTS
The following observations are shown by Chow and Ionascu, respectively, in the first two referenced articles.
There exists an odd number of Z^3 vectors, each of magnitude square root of z, that together sum to the zero vector if and only if the squarefree part of z is even.
There exist three Z^3 vectors, each of magnitude square root of z, that together sum to the zero vector if and only if the squarefree part of z is even, but contains no odd prime factor congruent to 2 (mod 3).
LINKS
Gaston A. Brouwer, Jonathan Joe, and Matt Noble, Odd Vector Cycles in Z^m, arXiv:2305.07770 [math.NT], 2023. See also Integers (2024) Vol. 24A, Art. No. A5. See p. 3, also Combinatorial Number Theory: Proc. Integers Conf. (2023), 65-82. See p. 66.
Timothy Chow, Distances forbidden by two-colorings of Q^3 and A_n, Discrete Math. Vol. 115 (1993), 95-102.
Eugen J. Ionascu, A parameterization of equilateral triangles having integer coordinates, J. Integer Sequences Vol. 10 (2007), #07.6.7.
Eugen J. Ionascu, Counting all equilateral triangles in {0, 1, ..., n}, Acta Math. Univ. Comenianae Vol. 77 (1) (2008), 129-140.
EXAMPLE
As an example, a(1) = 3 as evidenced by the vectors (1,0,1), (0,1,-1), and (-1,-1,0).
We have a(5) > 3 by Ionascu's result, and to see that a(5) = 5, consider the vectors (-3,1,0), (1,0,3), (1,0,3), (1,0,-3), and (0,-1,-3).
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Matt Noble, Jul 24 2019
STATUS
approved