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

Author

Craig Knecht, Feb 03 2016

Keywords

• nonn

Comments

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.

Examples

			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
		

Links

• 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

Crossrefs

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).

Programs

Mathematica

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 *)

Extensions

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