A361625 Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.

1, 1, 3, 7, 20, 60, 204, 702, 2526, 9180, 33989, 126713, 476597, 1802109, 6850969, 26151529, 100207548, 385217382, 1485216987, 5741240989, 22246000726, 86383317470, 336093551268, 1309997856337, 5114452295933, 19998171631076, 78306014924606, 307022177714062

Offset: 1

Author

Andrey Zabolotskiy, Mar 19 2023; thanks to John Mason for his help

Keywords

• nonn

Comments

Also, number of polysticks of size n (see A019988), with the requirement that any two sticks are connected by a sequence of adjacent, alternately horizontal and vertical sticks. - Pontus von Brömssen, Sep 01 2023

Examples

			There are 2 ways to color the 4 corners of a monomino with black and white colors alternatingly, but they are related by a rotation or a reflection, so a(1) = 1. a(2) is also 1 because the two ways to color the 6 vertices of a domino with black and white colors in the checkerboard pattern are related to each other by a reflection or a rotation. The same is true for the stick tromino, but the two ways to color the 8 vertices of the L-tromino are inequivalent, so a(3) = 3.
For n = 3, the a(3) = 3 allowed polysticks are:
  _     _
  _|  _|   _|_
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..50
• Index entries for sequences related to polyominoes.

Crossrefs

A122675 is the 3-dimensional analog based on polycubes.

Cf. A001933, A019988, A359689.

Cf. A000105, A351190, A351142, A351127, A349328, A346799, A234008.

5th row of A366766.

Formula

a(n) = 2 * A000105(n) - (A351190(n) + A351142(n) + A351127(n) + A349328(n) + A346799(n/2) + A234008(n/2)), where the last two terms are only included if 2|n. I.e., every free polyomino is counted twice here unless it is symmetric with respect to a Pi/2 rotation centered at a cell, or a Pi rotation centered at an edge, or a reflection with respect to an axis parallel to the grid and passing through cells.