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A268311 Number of free polyominoes that form a continuous path of edge joined cells spanning an n X n square in both dimensions. 8
1, 2, 24, 1051, 238048, 195228256, 577169894573, 6200686124225191 (list; graph; refs; listen; history; text; internal format)



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 nxn.  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.


Table of n, a(n) for n=1..8.

Craig Knecht, Polyominoe enumeration

Craig Knecht, Connective polyominoes 3x3

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


The cells with value one show the smallest possible lake in this 4x4 square.

1 1 1 1

0 0 0 1

0 0 0 1

0 0 0 1


Cf. A054247 (all unique water retention patterns for an n X n square).

Sequence in context: A122551 A136524 A213984 * A137887 A232310 A028365

Adjacent sequences:  A268308 A268309 A268310 * A268312 A268313 A268314




Craig Knecht, Jan 31 2016



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Last modified January 24 00:58 EST 2019. Contains 319404 sequences. (Running on oeis4.)