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a(n) is the minimum odd number of Z^3 vectors, each of magnitude square root of 2n, that together sum to the zero vector. When no such minimum exists for a particular n, we set a(n) = 0.
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%I #30 Oct 23 2024 16:54:28

%S 3,0,3,3,5,0,3,0,3,0,9,3,3,0,5,3,5,0,3,5,3,0,5,0,3,0,3,3,11,0,3,0,5,0,

%T 7,3,3,0,3,0,7,0,3,9,5,0,5,3,3,0

%N a(n) is the minimum odd number of Z^3 vectors, each of magnitude square root of 2n, that together sum to the zero vector. When no such minimum exists for a particular n, we set a(n) = 0.

%C The following observations are shown by Chow and Ionascu, respectively, in the first two referenced articles.

%C 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.

%C 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).

%H Gaston A. Brouwer, Jonathan Joe, and Matt Noble, <a href="https://arxiv.org/abs/2305.07770">Odd Vector Cycles in Z^m</a>, arXiv:2305.07770 [math.NT], 2023. See also <a href="https://math.colgate.edu/~integers/a5Proc23/a5Proc23.pdf">Integers</a> (2024) Vol. 24A, Art. No. A5. See p. 3, also <a href="https://doi.org/10.1515/9783111395593-005">Combinatorial Number Theory: Proc. Integers Conf.</a> (2023), 65-82. See p. 66.

%H Timothy Chow, <a href="https://doi.org/10.1016/0012-365X(93)90481-8">Distances forbidden by two-colorings of Q^3 and A_n</a>, Discrete Math. Vol. 115 (1993), 95-102.

%H Eugen J. Ionascu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Ionascu/ionascu2.html">A parameterization of equilateral triangles having integer coordinates</a>, J. Integer Sequences Vol. 10 (2007), #07.6.7.

%H Eugen J. Ionascu, <a href="http://www.iam.fmph.uniba.sk/amuc/_vol-77/_no_1/_ionascu/ionascu.html">Counting all equilateral triangles in {0, 1, ..., n}</a>, Acta Math. Univ. Comenianae Vol. 77 (1) (2008), 129-140.

%e As an example, a(1) = 3 as evidenced by the vectors (1,0,1), (0,1,-1), and (-1,-1,0).

%e 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).

%K nonn,more

%O 1,1

%A _Matt Noble_, Jul 24 2019