%I #52 Mar 21 2024 21:03:57

%S 1,2,24,1051,238048,195284973,577169894573,6200686124225191

%N Number of free polyominoes that form a continuous path of edge joined cells spanning an n X n square in both dimensions.

%C This idea originated from the water retention model for mathematical surfaces and is identical to the concept of a "lake". A lake is body of water that has dimensions of (n-2) X (n-2) when the square size is n X n. All other bodies of water are "ponds".

%C Iwan Jensen with his transfer matrix algorithm provided the number of symmetrically redundant solutions. _Walter Trump_ enumerated the symmetrically unique solutions.

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

%H Craig Knecht, <a href="/A268311/a268311.png">Connective polyominoes 3x3</a>

%H R. J. Mathar, <a href="http://www.vixra.org/abs/1905.0474">Corrigendum to "Polyomino Enumeration Results (Parkin et al, SIAM Fall Meeting 1967)"</a> viXra:1905.0474 (2019)

%H R. Parkin, L. J. Lander, and D. R. Parkin, <a href="/A000104/a000104.pdf">Polyomino Enumeration Results</a>, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy) page 9 (incorrect at n=15).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/File:Connective_polyominoes.jpeg">Connective Polyominoes 4x4</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/File:Connective_polyominoes_5x5.jpeg">Connective Polyominoes 5x5</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/File:Pond_larger_than_a_lake.png">Pond larger than a lake</a>

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

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>

%e The cells with value 1 show the smallest possible lake in this 4 X 4 square:

%e 1 1 1 1

%e 0 0 0 1

%e 0 0 0 1

%e 0 0 0 1

%e a(3)=24 = 6+7+7+3+1: There fit 6 5-ominoes in a 3x3 square, 7 6-ominoes in a 3x3 square, 7 7-ominoes in a 3x3 square, 3 8-ominoes in a 3x3 square, a 1 9-omino in a 3x3 square. - _R. J. Mathar_, Jun 07 2020

%Y Cf. A054247 (all unique water retention patterns). Diagonal of A268371.

%Y Cf. A259088.

%K nonn,more

%O 1,2

%A _Craig Knecht_, Jan 31 2016

%E a(6) corrected. _Craig Knecht_, May 25 2020