[seqfan] A polyomino sequence

Allan Wechsler acwacw at gmail.com
Tue Jun 11 23:32:17 CEST 2013


For P a polyomino, let V(P) be the number of cells in P, and let W(P) be
the number of cells not in P but orthogonally adjacent to a cell in P.

Let A(n) be the number of (congruency classes of) polyominoes P for which V
+ W = n.
A(2) = A(3) = A(4) = 0.
A(5) = 1; the monomino has four neighbors.
A(6) = A(7) = 0.
A(8) = 1; the domino has six neighbors.
A(9) = 0.
A(10) = 1; this is the L tromino with its 7 neighbors.
A(11) = 1; this is the straight tromino, with 8 neighbors.
A(12) = 3; the square, skew, and T tetrominoes have 8 neighbors.
A(13) = 2; the L tetromino has 9 neighbors, and the X pentomino has 8.
I am pretty sure A(14) = 4, with the straight tetromino and the P, S, R,
and W pentominos qualifying.

I can't find anything like this data in OEIS. Does anyone know any higher
values, or have I made an error? The sequence came up while I was thinking
about a percolation problem on the square lattice.


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