Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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