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

0,3

Comments

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

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

Example

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)

Crossrefs

Cf. A102698 (equilateral triangles), A334881 (squares), A338322 (regular hexagons).

The two-dimensional case is given by A002415.

Sequence in context: A155625 A016296 A021044 * A121034 A374513 A377200

Adjacent sequences: A338320 A338321 A338322 * A338324 A338325 A338326

Keyword

nonn

Author

Peter Kagey, Oct 22 2020

Status

approved