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%I #33 Apr 13 2016 17:26:45
%S 1,1,1,1,2,2,2,1,2,2,3,3,4,4,4,4,4,4,3,3,4,5
%N Number of disk polyominoes of order n (see Comments for definition).
%C Any closed disk in the real plane includes a finite set (possibly empty) of points from the square lattice Z^2.
%C 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 "disc polyominoes".
%C 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.
%C That is, given a polyomino, determine whether it is a disk polyomino.
%C I have been enumerating small disk polyominoes, to see how many configurations are possible for various numbers of lattice points.
%C 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.
%e The following is a list of the polyominoes that have been shown to be disks.
%e I use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum description.
%e For order 0, we of course have only (0), and for order 1 only (1).
%e Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
%e Order 6: (273), (333).
%e Order 7: (373).
%e Order 8: (377), (2772).
%e Order 9: (777), (2773).
%e Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
%e Order 11: (3777), (27f7), (67f6).
%e Order 12: (7777), (2ff7), (27f72), (6ff6).
%e Order 13: (77f7), (6ff7), (27ff2),(4eve4). (The "v" represents a decimal 31, binary 11111, a row of five lattice-points.)
%e Order 14: (7ff7), (2fff2), (27ff6), (4eve6).
%e Order 15: (7fff), (2fff6), (4evee), (4evf6).
%K nonn,more
%O 0,5
%A _Allan C. Wechsler_, Apr 30 2009
%E a(12) added by _Allan C. Wechsler_, May 12 2011, and a(13)-a(14) on Apr 09 2012
%E a(15) added by _Allan C. Wechsler_, Apr 10 2012
%E a(16)-a(21) added by _Allan C. Wechsler_, Apr 12 2012
%E a(20) corrected from 3 to 4 by _Allan C. Wechsler_, Nov 07 2013
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