%I #25 Jun 28 2026 15:48:14

%S 1,1,3,3,17,17,163,163,2753,2753,84731,84731,4879497,4879497,

%T 535376723,535376723,112921823249,112921823249,45931435159067,

%U 45931435159067,36048888105745113,36048888105745113,54568015172025197171,54568015172025197171,159197415409641803530753,159197415409641803530753

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

%C 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.

%H Craig Knecht, <a href="/A268339/a268339.png">Change of state - math becomes art</a>

%H Craig Knecht, <a href="/A268311/a268311.pdf">Polyominoe enumeration</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/File:Connective_polyominoes_with_4_sym-axis.jpg">Connective polyominoes with 4 sym-axis</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Water retention on mathematical surfaces">Water Retention on Mathematical Surfaces</a>

%F a(2*n) = a(2*n-1) = A268758(n). - _Andrew Howroyd_, May 03 2020

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

%e 0 1 1 1 0

%e 1 0 1 0 1

%e 1 1 1 1 1

%e 1 0 1 0 1

%e 0 1 1 1 0

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

%K nonn

%O 1,3

%A _Craig Knecht_, Feb 02 2016

%E Terms a(17) and beyond from _Andrew Howroyd_, May 03 2020