|
| |
|
|
A246559
|
|
List of one-sided polyominoes in binary coding, ordered by number of bits, then value of the binary code. Can be read as irregular table with row lengths A000988.
|
|
5
|
|
|
|
0, 1, 3, 7, 11, 15, 23, 27, 30, 39, 54, 75, 31, 47, 55, 62, 79, 91, 94, 143, 181, 182, 188, 203, 286, 314, 406, 551, 566, 1099, 63, 95, 111, 126, 159, 175, 183, 189, 190, 207, 219, 221, 222, 252, 287, 303, 315, 318, 347, 350, 378, 407, 413, 476, 504
(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 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 x direction), is 2^0+2^1+2^3+...+2^A000217(n-1) = A181388(n-1).
|
|
|
LINKS
|
|
|
|
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 dominos ".." and ":" would be represented by 2^0+2^1 = 3 and 2^0+2^2 = 5. Since they are equivalent up to rotation, only 3 = a(3) is listed.
The A000988(3) = 2 one-sided trominoes are represented by 2^0+2^1+2^3 = 11 (...) and 2^0+2^1+2^2 = 7 (:.). Again these values are listed in increasing size as a(4) and a(5).
|
|
|
PROG
|
(PARI) rot(P, T=[0, 1; -1, 0])=P=Set(apply(x->x*T, P)); apply(x->x-[P[1][1], 0], P)
onesided(L, N=apply(p2n, L))={ local(L=L, R=apply(P->setsearch(L, rot(P)), L), cleanup(i)=my(m=N[i]); while(m!=N[i=R[i]], if( m>N[i], m=N[i], L[i]=0))); for(i=1, #L, L[i] && cleanup(i)); if(#L>1, select(P->P, L), L)}
for(i=0, 5, print(Set(apply(p2n, onesided(L=if(i, grow(L), [[]])))))) \\ see A246533 for grow() and p2n()
|
|
|
CROSSREFS
|
See A246521 and A246533 for enumeration of free and fixed polyominoes.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
