A268758 Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.

1, 3, 17, 163, 2753, 84731, 4879497, 535376723, 112921823249, 45931435159067, 36048888105745113, 54568015172025197171, 159197415409641803530753, 894444473815989281612355579, 9671160618112663336510127727593, 201110001346886305066013828873025811

Offset: 1

Author

Craig Knecht, Feb 14 2016

Keywords

• nonn

Comments

Also number of polyominoes with width and height equal to 2n - 1 that are invariant under all symmetries of the square.

Bisection of A268339.

The water retention model for mathematical surfaces is described in the link below. The definition of a "lake" in this model is related to a class of polyominoes in A268339. Percolation theory may refer to these structures as "clusters that touch all boundaries."

Transportation across the square lattice requires a path of continuous edge connected cells. Is a pattern that only connects two opposite boundaries of the square ranked differently from one that connects all four boundaries?

This sequence is part of a effort to classify water retention patterns in a square by their symmetry, their capacity to connect boundaries of the square and the number of edge cells that are connected across opposite boundaries.

Examples

			For a(2) = 3: the three polyominoes of width and height 2*2 - 1 = 3 and the corresponding three polynomial of width and height 2*2 = 4 are shown below. Note that each even-dimension polyomino is produced by duplicating the center row/column of an odd-dimension polyomino.
3 X 3:
   0 1 0     1 1 1     1 1 1
   1 1 1     1 0 1     1 1 1
   0 1 0     1 1 1     1 1 1
4 X 4:
  0 1 1 0   1 1 1 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  0 1 1 0   1 1 1 1   1 1 1 1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..20
• Craig Knecht, Connections between boundaries of the square
• Craig Knecht Retention capacity of a random surface, arXiv:1110.6166 [cond-mat.dis-nn], 2011-2012.
• Wikipedia, Water Retention on Mathematical Surfaces

Crossrefs

Cf. A268339, A268404, A331878.

Formula

a(n) = A331878(n) - 3*A331878(n-1) + 3*A331878(n-2) - A331878(n-3) for n >= 4. - Andrew Howroyd, May 03 2020

Extensions

Terms a(9) and beyond from Andrew Howroyd, May 03 2020