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%I #16 Feb 18 2019 16:25:23
%S 0,2,4,20,68,288,1138
%N Number of (n,1)-polyominoes.
%C (n,k)-polyominoes are disconnected polyominoes with n visible squares and k transparent squares. Importantly, k must be the least number of transparent squares that need to be converted to visible squares to make all the visible squares connected. Note that a regular polyomino of order n is a (n,0)-polyomino, since all its visible squares are already connected. For more details see the paper by Kamenetsky and Cooke.
%C Number of distinct n-cell subsets of (n+1)-celled polyominoes that are not polyominoes. - _Charlie Neder_, Feb 12 2019
%H Dmitry Kamenetsky and Tristrom Cooke, <a href="https://arxiv.org/abs/1411.2699">Tiling rectangles with holey polyominoes</a>, arXiv:1411.2699 [cs.CG], 2015.
%e We can represent these polyominoes as binary matrices, where 1 means visible square and 0 means transparent square. Note that we need to flip (change to 1) one 0 to make all the 1s connected. This also means that the Manhattan distance between any pair of 1s is at most 2. Here are all such polyominoes for n=3:
%e 1101 100 100 010
%e 101 011 101
%Y Cf. A286194, A286345.
%K nonn,more
%O 1,2
%A _Dmitry Kamenetsky_, May 07 2017
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