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

Author

Peter Kagey, Oct 22 2020

Keywords

• nonn

Examples

			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).
		

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).

Crossrefs

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

Formula

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