A338322 a(n) is the number of regular hexagons with all six vertices (x,y,z) in the set {1,2,...,n}^3.

0, 0, 0, 4, 32, 116, 320, 728, 1472, 2796, 5056, 8584, 13792, 21136, 31168, 45464, 64704, 90036, 122784, 164472, 216864, 281584, 360416, 457400, 574304, 714644, 881312, 1077612, 1306720, 1575088, 1884928, 2245336, 2658592, 3130028, 3665376, 4277376, 4967424

Offset

0,4

Links

Peter Kagey, Table of n, a(n) for n = 0..100

Code Golf Stack Exchange, Polygons in a cube

Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).

Formula

a(n) >= 4*(n-2)^3 for n >= 2.

Example

The a(3) = 4 hexagons with integer coordinates in {1,2,3} have vertices:
  (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2);
  (1,1,2), (1,2,1), (2,1,3), (2,3,1), (3,2,3), (3,3,2);
  (1,2,1), (1,3,2), (2,1,1), (2,3,3), (3,1,2), (3,2,3); and
  (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
One of the a(5) = 116 hexagons has vertices:
  (2,2,1), (1,4,2), (2,5,4), (4,4,5), (5,2,4), (4,1,2).

Crossrefs

Cf. A102698 (equilateral triangles), A334881 (squares), A338323 (regular polygons).

Sequence in context: A211627 A033430 A267668 * A239056 A088658 A088802

Adjacent sequences: A338319 A338320 A338321 * A338323 A338324 A338325

Keyword

nonn

Author

Peter Kagey, Oct 22 2020

Status

approved