A338323 a(n) is the number of regular k-gons in three dimensions with all k vertices (x,y,z) in the set {1,2,...,n}^3.

0, 0, 14, 138, 640, 2190, 6042, 13824, 28400, 53484, 94126, 156462, 248568, 380802, 564242, 813528, 1146472, 1581936, 2143878, 2857194, 3749240, 4854942, 6210442, 7856340, 9832056, 12194784, 15002678, 18312486, 22183672, 26693382, 31909362, 37916916, 44802728

Offset: 0

Author

Peter Kagey, Oct 22 2020

Keywords

• nonn

Comments

The only regular polygons that can appear are equilateral triangles, squares, and regular hexagons.

Examples

			For the 3 X 3 X 3 grid, the a(3) = 138 regular polygons are A102698(3-1) = 80 triangles, A334881(3) = 54 squares, and A338322(3) = 4 hexagons.
An example of each shape, listed by the coordinates of their vertices:
Triangle: (1,2,1), (2,1,3), (3,3,2)
Square:   (1,1,1), (2,1,1), (2,2,1), (1,2,1)
Hexagon:  (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2)
		

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), A338322 (regular hexagons).

The two-dimensional case is given by A002415.

Formula

a(n) = A102698(n-1) + A334881(n) + A338322(n) for n >= 2.