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

0,2

Comments

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

Links

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).

Formula

a(n) = A000537(n) + A008893(n).

a(n) = (1/2)*(n+1)*n*(2*n+1)^2.

a(n) = 3*A180324(n).

Example

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.

Crossrefs

Cf. A000537 (regular hexagons), A008893 (equilateral triangles).

Cf. A338323 (cubic grid).

Cf. A003215.

Sequence in context: A249396 A102094 A321234 * A274311 A281804 A210045

Adjacent sequences: A339480 A339481 A339482 * A339484 A339485 A339486

Keyword

nonn

Author

Peter Kagey, Dec 06 2020

Status

approved