[seqfan] Re: A polyomino sequence
David Newman
davidsnewman at gmail.com
Wed Jun 12 08:41:34 CEST 2013
You might want to mention that the polyominos are free and whether holes
are allowed.
On Wed, Jun 12, 2013 at 12:32 AM, Allan Wechsler <acwacw at gmail.com> wrote:
> 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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