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A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.

%I A365139 #7 Aug 27 2023 10:10:25
%S A365139 1,3,7,19,15,23,39,43,51,54,1043,31,47,55,59,87,118,173,179,182,199,
%T A365139 230,1047,1075,1078,2071,2075,2149,2150,2164,2214,2218,6182,1049619,
%U A365139 63,95,119,175,183,190,207,215,231,237,238,246,423,430,438,1055,1079,1083
%N A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.
%C A365139 The binary code used here is a straight-forward generalization of the binary code in A246521 to d > 2 dimensions. Order the d-tuples of nonnegative integers, first according to their sum, then colexicographically. (For the purposes of this definition, the result will be the same if we use lexicographic order instead.) Label the d-tuples 0, 1, 2, ... in this order. (For d = 3, this is the ordering of triples given by A144625.) Given a d-dimensional polyomino (represented as a finite set of integer d-tuples), consider all the d!*2^d ways of rotating/reflecting it. Translate each such rotation/reflection so that the minimum coordinate is 0 in each dimension, and add the powers of 2 with exponents equal to the labels of the d-tuples of the translation. The binary code of the polyomino (or any finite set of d-tuples) is the minimum of those sums.
%C A365139 Can be read as an irregular triangle, whose n-th row contains A038119(n) terms.
%H A365139 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%e A365139 Consider the pentacube consisting of a straight tricube with two monocubes attached to two adjacent faces of its middle cube. The following table shows the first few triples (with their ordinal number in front), with those triples appearing in the orientation of the pentacube that minimizes the binary code marked with an "X":
%e A365139   0. 000 X
%e A365139   1. 100 X
%e A365139   2. 010
%e A365139   3. 001
%e A365139   4. 200 X
%e A365139   5. 110 X
%e A365139   6. 020
%e A365139   7. 101 X
%e A365139   8. 011
%e A365139   9. 002
%e A365139 Consequently, the binary code of this pentacube is 2^0+2^1+2^4+2^5+2^7 = 179 = a(19).
%e A365139 As an irregular triangle:
%e A365139   1;
%e A365139   3;
%e A365139   7, 19;
%e A365139   15, 23, 39, 43, 51, 54, 1043;
%e A365139   ...
%Y A365139 Cf. A038119, A144625, A246521 (2 dimensions), A365140 (4 dimensions), A365141 (5 dimensions).
%K A365139 nonn,tabf
%O A365139 1,2
%A A365139 _Pontus von Brömssen_, Aug 23 2023