A334881 -id:A334881 - cp's OEIS frontend

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

Peter Kagey, Oct 22 2020

nonn

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

			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)

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).

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

The two-dimensional case is given by A002415.

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

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

Original entry on oeis.org

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

Peter Kagey, Oct 22 2020

nonn

			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).

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).

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

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

A334891 Number of ways to choose 4 points that form a square from the A000292(n) points in a regular tetrahedral grid where each side has n vertices.

Original entry on oeis.org

0, 0, 3, 12, 36, 84, 174, 336, 612, 1044, 1701

Offset: 0

Peter Kagey, May 14 2020

a(n) >= 3*A001752(n-2).

			For n = 4, three of the a(4) = 36 squares are (in barycentric coordinates)
  {(0,2,1,1),(1,1,0,2),(1,1,2,0),(2,0,1,1)},
  {(0,0,2,2),(0,2,0,2),(2,0,2,0),(2,2,0,0)}, and
  {(0,0,1,3),(0,1,0,3),(1,0,1,2),(1,1,0,2)}.
The other squares can be derived from these by translations or symmetries of the tetrahedron.

Cf. A000292, A001752.

Cf. A334581 (equilateral triangle), A334881 (cubic grid).

cp's OEIS Frontend

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.

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

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A334891 Number of ways to choose 4 points that form a square from the A000292(n) points in a regular tetrahedral grid where each side has n vertices.

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