OFFSET
1,3
COMMENTS
The binary coding (as suggested in a post to the SeqFan list by F. T. Adams-Watters) is obtained by summing the powers of 2 corresponding to the numbers covered by the polyomino, when the points of the quarter-plane are numbered by antidiagonals, and the animal is placed (and flipped/rotated) as to obtain the smallest possible value, which in particular implies pushing it to both borders. See example for further details.
The smallest value for an n-omino is the sum 2^0 + ... + 2^(n-1) = 2^n - 1 = A000225(n), and the largest value, obtained for the straight n-omino, is 2^0 + 2^1 + 2^3 + ... + 2^A000217(n-1) = A181388(n-1).
See A246533 for the variant that lists fixed polyominoes.
LINKS
John Mason, Table of n, a(n) for n = 1..87147
F. T. Adams-Watters, Re: Sequence proposal by John Mason, SeqFan list, Aug 24 2014
EXAMPLE
Number the points of the first quadrant as follows:
... ... ...
9 13 18 24 31 ...
5 8 12 17 23 ...
2 4 7 11 16 ...
0 1 3 6 10 ...
An animal occupying squares numbered k1, ..., kN will be represented by a term a(n) = 2^k1 + ... + 2^kN, the position and orientation being chosen as to minimize this value:
The "empty" 0-omino is represented by the empty sum equal to 0 = a(1).
The monomino is represented by a square on 0, and the binary code 2^0 = 1 = a(2).
The free domino is rotated to the ".." configuration represented by 2^0 + 2^1 (since this is smaller than the ":" configuration with value 2^0 + 2^2).
The A000105(3) = 2 free triominoes are represented by 2^0 + 2^1 + 2^3 = [...] and 2^0 + 2^1 + 2^2 = [:.]. The latter value is smaller, therefore the L-shaped triomino is listed before the straight one.
From M. F. Hasler, Jan 25 2021: (Start)
Writing all N-ominoes on row N, the table begins:
----+--------------------------------------------------------------
0 | a(1) = 0 = []
1 | a(2) = 1 = 2^0 = [.]
2 | a(3) = 3 = 2^0 + 2^1 = [..]
3 | a(4) = 7 = [:.], a(5) = 11 = [...]
4 | 15 = [:..], 23 = [::], 27 = [.:.], 30 = [':.], 75 = [....]
... | ...
(End)
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Aug 28 2014
EXTENSIONS
More terms from John Mason, Aug 29 2014
STATUS
approved