A147680 Number of disk polyominoes of order n (see Comments for definition).

1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 4, 5, 6, 6, 7, 7, 7, 6, 7, 7, 8, 8, 8, 8, 7, 7, 6, 7, 7, 7, 7, 8, 9, 11, 12, 12, 12, 11, 12, 12, 12, 11, 11, 11, 12, 13, 13, 12, 14, 15, 16, 16, 15, 13, 13, 13, 14, 15, 16, 18, 18, 16, 14, 15, 15, 14

Offset

0,5

Comments

Any closed disk in the real plane includes a finite set (possibly empty) of points from the square lattice Z^2.

These roughly-circular patches of lattice points are connected by chains of adjacent lattice points (this is an easy theorem) and hence they form a special class of polyominoes, which I call "disk polyominoes".

It's quite easy to calculate which lattice points are within a given radius of a given center, but the inverse problem can be a little challenging.

That is, given a polyomino, determine whether it is a disk polyomino.

I have been enumerating small disk polyominoes, to see how many configurations are possible for various numbers of lattice points.

There is one disk polyomino for each of the orders 0, 1, 2 and 3; two for each of the orders 4, 5 and 6; only one for order 7; two each for orders 9 and 10; and three each for orders 11 and 12.

Links

Martin Fuller, Table of n, a(n) for n = 0..1000

Martin Fuller, Table of disk polyominoes for n = 0..1000

Martin Fuller, Python program and explanation

Example

The following is a list of the polyominoes up through order 15 that have been shown to be disks.
We canonicalize a disk polyomino by translating, rotating, and reflecting it until the center of the disk is in the triangle with vertices (0, 0), (1/2, 0) and (1/2, 1/2); this is the fundamental domain of the symmetry group of the lattice Z^2.
Once the polyomino has been canonicalized, it can be represented uniquely as a list showing the number of cells on each horizontal row, starting with the topmost row. The uniqueness is due to the fact that each row must be "centered" at either x=0 (for odd rows) or x=1/2 (for even rows).
For each order, we sort the disk polyominoes into "lexical" order.
For order 0, we of course have only (0), and for order 1 only (1).
Order 2 gives (2), and order 3 gives the L-tromino (12).
Order 4 has two examples, the T-tetromino (121), and the block (22).
Order 5 gives the X-pentomino (131) and the P-pentomino (221).
Order 6: (222), (231).
Order 7: (232).
Order 8: (242), (332).
Order 9: (333), (342).
Order 10: (1342), (343), (442).
Order 11: (1343), (1442), (443).
Order 12: (13431), (1443), (2442), (444).
Order 13: (13531), (14431), (1444), (2443).
Order 14: (14441), (14531), (24431), (2444).
Order 15: (14541), (24531), (244421), (3444).

Prog

(Python) # See Fuller link.

Crossrefs

Cf. A365964.

Sequence in context: A109037 A366136 A366693 * A373688 A192895 A210685

Adjacent sequences: A147677 A147678 A147679 * A147681 A147682 A147683

Keyword

nonn,changed

Author

Allan C. Wechsler, Apr 30 2009

Extensions

a(12) added by Allan C. Wechsler, May 12 2011, and a(13)-a(14) on Apr 09 2012

a(15) added by Allan C. Wechsler, Apr 10 2012

a(16)-a(21) added by Allan C. Wechsler, Apr 12 2012

a(20) corrected from 3 to 4 by Allan C. Wechsler, Nov 07 2013

a(22)-a(50) added by Allan C. Wechsler, Jun 20 2026

a(45) and a(49) corrected, a(51) onward from Martin Fuller, Jul 10 2026

Status

approved