A338323 -id:A338323 - cp's OEIS frontend

A339483 Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.

0, 9, 75, 294, 810, 1815, 3549, 6300, 10404, 16245, 24255, 34914, 48750, 66339, 88305, 115320, 148104, 187425, 234099, 288990, 353010, 427119, 512325, 609684, 720300, 845325, 985959, 1143450, 1319094, 1514235, 1730265, 1968624, 2230800, 2518329, 2832795

Offset: 0

Peter Kagey, Dec 06 2020

The only regular polygons that can be drawn with vertices on the centered hexagonal grid are equilateral triangles and regular hexagons.

			There are a(2) = 75 regular polygons that can be drawn on the centered hexagonal grid with side length 2: A000537(2) = 9 regular hexagons and A008893(n) = 66 equilateral triangles.
The nine hexagons are:
    * . *       . * .       * * .
   . . . .     * . . *     * . * .
  * . . . *   . . . . .   . * * . .
   . . . .     * . . *     . . . .
    * . *       . * .       . . .
      1           1           7
which are marked with the number of ways to draw the hexagons up to translation.
The 66 equilateral triangles are:
    * . .       * . .       * . .       * . *       * . .       . . .
   * * . .     . . * .     . . . .     . . . .     . . . .     * . . *
  . . . . .   . * . . .   . . . * .   . . * . .   . . . . *   . . . . .
   . . . .     . . . .     * . . .     . . . .     . . . .     . . . .
    . . .       . . .       . . .       . . .       * . .       . * .
     24          14          12          12           2           2
which are marked with the number of ways to draw the triangles up to translation and dihedral action of the hexagon.

Peter Kagey, Table of n, a(n) for n = 0..10000
Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537 (regular hexagons), A008893 (equilateral triangles).
Cf. A338323 (cubic grid).
Cf. A003215, A180324, A185096, A322677, A348670.

a[n_] := n*(n+1)*(2*n+1)^2/2; Array[a, 35, 0] (* Amiram Eldar, Jun 20 2025 *)

a(n) = A000537(n) + A008893(n).
a(n) = (1/2)*(n+1)*n*(2*n+1)^2.
a(n) = 3*A180324(n).
Sum_{n>=1} 1/a(n) = 10 - Pi^2 (A348670). - Amiram Eldar, Jun 20 2025
From Elmo R. Oliveira, Aug 20 2025: (Start)
G.f.: -3*x*(x + 3)*(3*x + 1)/(x - 1)^5.
E.g.f.: exp(x)*x*(2 + x)*(9 + 24*x + 4*x^2)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A185096(n)/4 = A322677(n)/32. (End)

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.

A338791 a(n) is the number of Platonic solids in three dimensions with all vertices (x,y,z) in the set {1,2,...,n}^3.

Original entry on oeis.org

0, 0, 3, 28, 116, 340, 847, 1832, 3570, 6440, 10889, 17518, 26966, 40002, 57601, 80868, 111186, 150032, 199147, 260456, 336080, 428290, 539709, 673130, 831436, 1018154, 1237155, 1492352, 1787780, 2129250, 2521323, 2969584, 3479302, 4056636, 4707661, 5438808

Offset: 0

Peter Kagey, Dec 05 2020

nonn

Dodecahedra and icosahedra with integer coordinates cannot be formed in Euclidean space (of any dimension) because pentagons with integer coordinates cannot be formed in Euclidean space, and both polyhedra contain a subset of vertices that form a pentagon. Therefore, this sequence counts the regular tetrahedra, cubes, and octahedra in the bounded cubic lattice.

Peter Kagey, Table of n, a(n) for n = 0..100, based on the b-files for A098928, A103158, and A178797.
Eugen J. Ionascu and Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, Pages 138-145.

Cf. A098928 (cubes), A103158 (tetrahedra), A178797 (octahedra), A338323 (regular polygons).

a(n) = A098928(n) + 2*A103158(n-1) + A178797(n-1) for n >= 2.

cp's OEIS Frontend

A339483 Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.

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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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Formula

A338791 a(n) is the number of Platonic solids in three dimensions with all vertices (x,y,z) in the set {1,2,...,n}^3.

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Author

Keywords

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Crossrefs

Formula