A246521 - OEIS

%I #50 Aug 27 2023 01:25:07

%S 0,1,3,7,11,15,23,27,30,75,31,47,62,79,91,94,143,181,182,188,406,1099,

%T 63,95,111,126,159,175,183,189,190,207,219,221,222,252,347,350,378,

%U 407,413,476,504,1103,1115,1118,1227,1244,2127,2229,2230,2236,2292,2451,2454,2460,33867,127

%N 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 (in which case the offset is 0).

%C 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.

%C 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).

%C See A246533 for the variant that lists fixed polyominoes.

%H John Mason, <a href="/A246521/b246521.txt">Table of n, a(n) for n = 1..87147</a>

%H F. T. Adams-Watters, <a href="http://list.seqfan.eu/oldermail/seqfan/2014-August/013511.html">Re: Sequence proposal by John Mason</a>, SeqFan list, Aug 24 2014

%e Number the points of the first quadrant as follows:

%e    ...   ...   ...

%e     9 13 18 24 31 ...

%e     5  8 12 17 23 ...

%e     2  4  7 11 16 ...

%e     0  1  3  6 10 ...

%e 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:

%e The "empty" 0-omino is represented by the empty sum equal to 0 = a(1).

%e The monomino is represented by a square on 0, and the binary code 2^0 = 1 = a(2).

%e 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).

%e 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.

%e From _M. F. Hasler_, Jan 25 2021: (Start)

%e Writing all N-ominoes on row N, the table begins:

%e   N | a(m .. m+k), m = 1 + Sum_{j<N} A000105(j), k = A000105(N) - 1

%e ----+--------------------------------------------------------------

%e   0 | a(1) = 0 = []

%e   1 | a(2) = 1 = 2^0 = [.]

%e   2 | a(3) = 3 = 2^0 + 2^1 = [..]

%e   3 | a(4) = 7 = [:.], a(5) = 11 = [...]

%e   4 | 15 = [:..], 23 = [::], 27 = [.:.], 30 = [':.], 75 = [....]

%e ... | ...

%e (End)

%Y See A246533 and A246559 for lists of fixed and one-sided polyominoes.

%Y Cf. A000105, A000217, A000225, A181388.

%K nonn

%O 1,3

%A _M. F. Hasler_, Aug 28 2014

%E More terms from _John Mason_, Aug 29 2014