A246533 List of fixed polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A001168.

0, 1, 3, 5, 7, 11, 19, 21, 22, 37, 15, 23, 27, 30, 39, 53, 54, 75, 139, 147, 149, 150, 156, 275, 277, 278, 293, 306, 549, 31, 47, 55, 62, 79, 91, 94, 143, 151, 155, 157, 158, 181, 182, 188, 203, 220, 279, 283, 286, 295, 307, 309, 310, 314, 403, 405, 406, 412, 434, 440

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 pushed to both borders as to obtain the smallest possible value. 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 in y direction, is 2^0+2^2+2^5+...+2^(A000217(n)-1) = A246534(n).

Links

John Mason, Table of n, a(n) for n = 1..50149

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 two fixed dominos are ".." and ":", represented by 2^0+2^1 = 3 = a(3) and 2^0+2^2 = 5 = a(4).
The A001168(3) = 6 fixed trominoes are represented by 2^0+2^1+2^3 = 11 (...), 2^0+2^1+2^2 = 7 (:.), 2^0+2^1+2^4 =19 (.:), ..., 2^0+2^2+2^5 = 37; again these 6 values are listed in increasing size as a(5), ..., a(10).

Prog

(PARI) grow(L, N=[], D=[[1, 0], [0, 1], [-1, 0], [0, -1]])={ for(i=1, #L, for(j=1, #P=L[i], for(k=1, #P, for(d=1, #D, vecmin(P[k]+D[d])<0 && P-=vector(#P, i, D[d])/*shift if needed*/; !setsearch(P, P[k]+D[d]) && N=setunion([setunion([P[k]+D[d]], P)], N); P!=L[i] && P+=vector(#P, i, D[d])/*undo...*/)))); if(N, N, [[[0, 0]]])}
p2n(P)=sum(i=1, #P, 2^(P[i][2]+A000217(P[i][1]+P[i][2])))
for(i=0, 5, print(vecsort(apply(p2n, L=if(i, grow(L), [[]])))))

Crossrefs

See A246521 and A246559 for enumeration of free and one-sided polyominoes.

Sequence in context: A119573 A001366 A093329 * A111052 A138536 A093929

Adjacent sequences: A246530 A246531 A246532 * A246534 A246535 A246536

Keyword

nonn

Author

M. F. Hasler, Aug 28 2014

Status

approved