cp's OEIS Frontend

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.

%I A246533 #18 Dec 23 2024 14:53:43
%S A246533 0,1,3,5,7,11,19,21,22,37,15,23,27,30,39,53,54,75,139,147,149,150,156,
%T A246533 275,277,278,293,306,549,31,47,55,62,79,91,94,143,151,155,157,158,181,
%U A246533 182,188,203,220,279,283,286,295,307,309,310,314,403,405,406,412,434,440
%N 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.
%C A246533 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.
%C A246533 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).
%H A246533 John Mason, <a href="/A246533/b246533.txt">Table of n, a(n) for n = 1..50149</a>
%H A246533 F. T. Adams-Watters, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-August/013511.html">Re: Sequence proposal by John Mason</a>, SeqFan list, Aug 24 2014
%e A246533 Number the points of the first quadrant as follows:
%e A246533 ...
%e A246533 9 ...
%e A246533 5 8 ...
%e A246533 2 4 7 ...
%e A246533 0 1 3 6 10 ...
%e A246533 The "empty" 0-omino is represented by the empty sum equal to 0 = a(1).
%e A246533 The monomino is represented by a square on 0, and the binary code 2^0 = 1 = a(2).
%e A246533 The two fixed dominos are ".." and ":", represented by 2^0+2^1 = 3 = a(3) and 2^0+2^2 = 5 = a(4).
%e A246533 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).
%o A246533 (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]]])}
%o A246533 p2n(P)=sum(i=1,#P,2^(P[i][2]+A000217(P[i][1]+P[i][2])))
%o A246533 for(i=0,5,print(vecsort(apply(p2n,L=if(i,grow(L),[[]])))))
%Y A246533 See A246521 and A246559 for enumeration of free and one-sided polyominoes.
%K A246533 nonn
%O A246533 1,3
%A A246533 _M. F. Hasler_, Aug 28 2014