A268311 Number of free polyominoes that form a continuous path of edge joined cells spanning an n X n square in both dimensions.

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

Offset: 1

Author

Craig Knecht, Jan 31 2016

Keywords

• nonn
• more

Comments

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".

Iwan Jensen with his transfer matrix algorithm provided the number of symmetrically redundant solutions. Walter Trump enumerated the symmetrically unique solutions.

Examples

			The cells with value 1 show the smallest possible lake in this 4 X 4 square:
1 1 1 1
0 0 0 1
0 0 0 1
0 0 0 1
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
		

Links

• Craig Knecht, Polyominoes enumeration
• Craig Knecht, Connective polyominoes 3x3
• R. J. Mathar, Corrigendum to "Polyomino Enumeration Results (Parkin et al, SIAM Fall Meeting 1967)" viXra:1905.0474 (2019)
• R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy) page 9 (incorrect at n=15).
• Wikipedia, Connective Polyominoes 4x4
• Wikipedia, Connective Polyominoes 5x5
• Wikipedia, Connective polyominoes with 4 sym-axis
• Wikipedia, Pond larger than a lake
• Wikipedia, Water Retention on Mathematical Surfaces
• Index entries for sequences related to polyominoes

Crossrefs

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

Cf. A259088.

Extensions

a(6) corrected. Craig Knecht, May 25 2020