A309339 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.

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, 7, 3, 3, 0, 3, 0, 7, 0, 3, 9, 5, 0, 5, 3, 3, 0

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

Table of n, a(n) for n=1..50.

Gaston A. Brouwer, Jonathan Joe, and Matt Noble, Odd Vector Cycles in Z^m, arXiv:2305.07770 [math.NT], 2023.

T. Chow, Distances forbidden by two-colorings of Q^3 and A_n, Discrete Math. Vol. 115 (1993), 95-102.

E. J. Ionascu, A parameterization of equilateral triangles having integer coordinates, J. Integer Sequences Vol. 10 (2007), #07.6.7.

E. 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

Sequence in context: A261922 A078907 A282135 * A333453 A278923 A210485

Adjacent sequences: A309336 A309337 A309338 * A309340 A309341 A309342

Keyword

nonn,more

Author

Matt Noble, Jul 24 2019

Status

approved