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Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.
3

%I #38 Nov 05 2023 07:49:31

%S 1,1,3,7,20,60,204,702,2526,9180,33989,126713,476597,1802109,6850969,

%T 26151529,100207548,385217382,1485216987,5741240989,22246000726,

%U 86383317470,336093551268,1309997856337,5114452295933,19998171631076,78306014924606,307022177714062

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

%C 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

%H Andrey Zabolotskiy, <a href="/A361625/b361625.txt">Table of n, a(n) for n = 1..50</a>

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.

%F 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.

%e 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.

%e For n = 3, the a(3) = 3 allowed polysticks are:

%e _ _

%e _| _| _|_

%Y A122675 is the 3-dimensional analog based on polycubes.

%Y Cf. A001933, A019988, A359689.

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

%Y 5th row of A366766.

%K nonn

%O 1,3

%A _Andrey Zabolotskiy_, Mar 19 2023; thanks to _John Mason_ for his help.