A268339 -id:A268339 - cp's OEIS frontend

A268404 Number of fixed polyominoes that have a width and height of n.

1, 5, 111, 7943, 1890403, 1562052227, 4617328590967, 49605487608825311, 1951842619769780119767, 282220061839181920696642671, 150134849621798165832163223922131, 293909551918134914019004192289440616787, 2116817972794640259940977362779552773322908743

Offset: 1

Craig Knecht, Feb 03 2016

nonn

Iwan Jensen originally provided this sequence.
The sequence also describes the water patterns of lakes in the water retention model.
A lake is defined as a body of water with dimensions of n X n when the size of the square is (n+2) X (n+2). All other bodies of water are ponds.
The 3 X 3 square serves as a tutorial for the following three nomenclatures: (1) The total number of distinct water patterns is 102 and includes lakes and ponds. (2) The number of free lake-type polyominoes is 24. (3) The number of fixed lake-type polyominoes is 111. See the explanatory graphics in the link section.
John Mason has looked at free polyominoes in rectangles; see A268371.
Anna Skelt initiated the discussion on the definition of a lake.

			There are many interesting ways to connect all boundaries of the square with the smallest number of edge-joined cells.
  0 0 0 0 1 0
  0 0 0 0 1 1
  0 0 1 1 1 0
  0 0 1 0 0 0
  1 1 1 0 0 0
  0 1 0 0 0 0

Andrew Howroyd, Table of n, a(n) for n = 1..15
Craig Knecht, 4x4 minimal lake area patterns
Craig Knecht, 5x5 minimal lake area patterns
Craig Knecht, 6x6 minimal lake area patterns
Craig Knecht, 7x7 minimal lake area patterns
Craig Knecht, 24 free lake-type polyominoes 3x3
Craig Knecht, Polyominoe enumeration
Craig Knecht, Walter Trump's 111 fixed lake-type polyominoes 3x3
Wikipedia, Water Retention on Mathematical Surfaces

Main diagonal of A292357.

Cf. A054247 (all unique water retention patterns for an n X n square), A268311 (free polyominoes that connect all boundaries on a square), A268339 (lake patterns that are invariant to all transformations).

A292357 = Cases[Import["https://oeis.org/A292357/b292357.txt", "Table"], {, }][[All, 2]];
a[n_] := A292357[[2n^2 - 2n + 1]];
Array[a, 15] (* Jean-François Alcover, Sep 10 2019 *)

a(12)-a(13) from Andrew Howroyd, Oct 02 2017

A268758 Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.

Original entry on oeis.org

1, 3, 17, 163, 2753, 84731, 4879497, 535376723, 112921823249, 45931435159067, 36048888105745113, 54568015172025197171, 159197415409641803530753, 894444473815989281612355579, 9671160618112663336510127727593, 201110001346886305066013828873025811

Offset: 1

Craig Knecht, Feb 14 2016

nonn

Also number of polyominoes with width and height equal to 2n - 1 that are invariant under all symmetries of the square.
Bisection of A268339.
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."
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?
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.

			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.
3 X 3:
   0 1 0     1 1 1     1 1 1
   1 1 1     1 0 1     1 1 1
   0 1 0     1 1 1     1 1 1
4 X 4:
  0 1 1 0   1 1 1 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  0 1 1 0   1 1 1 1   1 1 1 1

Andrew Howroyd, Table of n, a(n) for n = 1..20
Craig Knecht, Connections between boundaries of the square
Craig Knecht Retention capacity of a random surface, arXiv:1110.6166 [cond-mat.dis-nn], 2011-2012.
Wikipedia, Water Retention on Mathematical Surfaces

Cf. A268339, A268404, A331878.

a(n) = A331878(n) - 3*A331878(n-1) + 3*A331878(n-2) - A331878(n-3) for n >= 4. - Andrew Howroyd, May 03 2020

Terms a(9) and beyond from Andrew Howroyd, May 03 2020

cp's OEIS Frontend

A268404 Number of fixed polyominoes that have a width and height of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions