A268339 Number of polyominoes with width and height equal to n that are invariant under all symmetries of the square.

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

Offset: 1

Author

Craig Knecht, Feb 02 2016

Keywords

• nonn

Comments

Percolation theory focuses on patterns that provide connectivity. Polyominoes that connect all boundaries of a square are in the percolation neighborhood. This subclass of symmetric polyominoes distinguishes itself for its beauty and its unusual enumeration pattern.

Examples

			The ones in this example provide the connective pattern that joins all boundaries of the square.
0 1 1 1 0
1 0 1 0 1
1 1 1 1 1
1 0 1 0 1
0 1 1 1 0
		

Links

• Craig Knecht, Change of state - math becomes art
• Craig Knecht, Polyominoe enumeration
• Wikipedia, Connective polyominoes with 4 sym-axis
• Wikipedia, Water Retention on Mathematical Surfaces

Crossrefs

Cf. A054247 (all unique water retention patterns for an n X n square), A268311 (polyominoes that connect all boundaries on a square), A268758.

Formula

a(2*n) = a(2*n-1) = A268758(n). - Andrew Howroyd, May 03 2020

Extensions

Terms a(17) and beyond from Andrew Howroyd, May 03 2020