

A268404


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


5



1, 5, 111, 7943, 1890403, 1562052227, 4617328590967, 49605487608825311, 1951842619769780119767, 282220061839181920696642671, 150134849621798165832163223922131, 293909551918134914019004192289440616787, 2116817972794640259940977362779552773322908743
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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 unique water patterns equals 102 and includes lakes and ponds. 2) The number of free laketype polyominoes equals 24. 3) The number of fixed laketype polyominoes equals 111. See the explanatory graphics in the link section.
John Mason has looked at free polyominoes in rectangles 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 laketype polyominoes 3x3
Craig Knecht, Polyominoe enumeration
Craig Knecht, Walter Trump's 111 fixed laketype polyominoes 3x3
Wikipedia, Water Retention on Mathematical Surfaces


EXAMPLE

There are many interesting ways to connect all boundaries of the square with the fewest number of edgejoined 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] (* JeanFrançois 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: A219161 A284461 A002400 * A258795 A263531 A258177
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



