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

1,3

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.

Links

Table of n, a(n) for n=1..26.

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

Formula

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

Example

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

Crossrefs

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

Sequence in context: A226610 A332840 A279172 * A224750 A014783 A090524

Adjacent sequences: A268336 A268337 A268338 * A268340 A268341 A268342

Keyword

nonn

Author

Craig Knecht, Feb 02 2016

Extensions

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

Status

approved