|
%I #47 May 03 2020 22:19:14
%S 1,3,17,163,2753,84731,4879497,535376723,112921823249,45931435159067,
%T 36048888105745113,54568015172025197171,159197415409641803530753,
%U 894444473815989281612355579,9671160618112663336510127727593,201110001346886305066013828873025811
%N Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.
%C Also number of polyominoes with width and height equal to 2n - 1 that are invariant under all symmetries of the square.
%C Bisection of A268339.
%C 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."
%C 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?
%C 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.
%H Andrew Howroyd, <a href="/A268758/b268758.txt">Table of n, a(n) for n = 1..20</a>
%H Craig Knecht, <a href="/A054247/a054247.png">Connections between boundaries of the square</a>
%H Craig Knecht <a href="http://arxiv.org/abs/1110.6166">Retention capacity of a random surface</a>, arXiv:1110.6166 [cond-mat.dis-nn], 2011-2012.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Water retention on mathematical surfaces">Water Retention on Mathematical Surfaces</a>
%F a(n) = A331878(n) - 3*A331878(n-1) + 3*A331878(n-2) - A331878(n-3) for n >= 4. - _Andrew Howroyd_, May 03 2020
%e 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.
%e 3 X 3:
%e 0 1 0 1 1 1 1 1 1
%e 1 1 1 1 0 1 1 1 1
%e 0 1 0 1 1 1 1 1 1
%e 4 X 4:
%e 0 1 1 0 1 1 1 1 1 1 1 1
%e 1 1 1 1 1 0 0 1 1 1 1 1
%e 1 1 1 1 1 0 0 1 1 1 1 1
%e 0 1 1 0 1 1 1 1 1 1 1 1
%Y Cf. A268339, A268404, A331878.
%K nonn
%O 1,2
%A _Craig Knecht_, Feb 14 2016
%E Terms a(9) and beyond from _Andrew Howroyd_, May 03 2020
| 