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

1,2

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.

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

Example

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

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 *)

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).

Sequence in context: A367248 A284461 A002400 * A258795 A263531 A351148

Adjacent sequences: A268401 A268402 A268403 * A268405 A268406 A268407

Keyword

nonn

Author

Craig Knecht, Feb 03 2016

Extensions

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

Status

approved