A395334 Number of free polyominoes with n cells (up to the symmetries of the square) that contain a saturated cell (a cell all four of whose edge-neighbors are present).

0, 0, 0, 0, 1, 2, 9, 36, 157, 667, 2852, 12081, 50786, 211971

Offset

1,6

Comments

The smallest polyomino with a saturated cell is the X-pentomino, so a(n) = 0 for n < 5. a(n) < A000105(n), since a straight strip of n cells has no saturated cell; A000105(n) - a(n) is the number of free polyominoes in which every cell has at least one exposed edge.

By Madras's pattern theorem for lattice clusters, a saturated cell occurs in all but an exponentially small fraction of large polyominoes, so a(n)/A000105(n) tends to 1; the ratio is 0.06 at n = 6 and 0.24 at n = 14.

Links

Table of n, a(n) for n=1..14.

Neal Madras, A pattern theorem for lattice clusters, Annals of Combinatorics 3 (1999), 357-384; arXiv:math/9902161 [math.PR], 1999.

Index entries for sequences related to polyominoes.

Crossrefs

Cf. A000105, A390515, A395331, A395552.

Sequence in context: A150968 A073156 A150969 * A150970 A150971 A150972

Adjacent sequences: A395331 A395332 A395333 * A395335 A395336 A395337

Keyword

nonn,more,new

Author

Peter Exley, Jun 15 2026

Status

approved