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A246521 List of free polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000105. 3
0, 1, 3, 7, 11, 15, 23, 27, 30, 75, 31, 47, 62, 79, 91, 94, 143, 181, 182, 188, 406, 1099, 63, 95, 111, 126, 159, 175, 183, 189, 190, 207, 219, 221, 222, 252, 347, 350, 378, 407, 413, 476, 504, 1103, 1115, 1118, 1227, 1244, 2127, 2229, 2230, 2236, 2292, 2451, 2454, 2460, 33867, 127 (list; graph; refs; listen; history; text; internal format)
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 ...

5 8 ...

2 4 7 ...

0 1 3 6 10 ...

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.

CROSSREFS

See A246533 and A246559 for enumerations of fixed and one-sided polyominoes.

Sequence in context: A071849 A165197 A246559 * A160785 A095100 A036994

Adjacent sequences:  A246518 A246519 A246520 * A246522 A246523 A246524

KEYWORD

nonn

AUTHOR

M. F. Hasler, Aug 28 2014

EXTENSIONS

More terms from John Mason, Aug 29 2014

STATUS

approved

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Last modified October 18 07:19 EDT 2019. Contains 328146 sequences. (Running on oeis4.)