A286344 Number of (n,1)-polyominoes.

0, 2, 4, 20, 68, 289, 1151, 4792, 19603, 80820, 331373

Offset: 1

Author

Dmitry Kamenetsky, May 07 2017

Keywords

• nonn
• more

Comments

(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.

Number of distinct n-cell subsets of (n+1)-celled polyominoes that are not polyominoes. - Charlie Neder, Feb 12 2019

Examples

			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:
   1101 100 100 010
        101 011 101
		

Links

• Dmitry Kamenetsky and Tristrom Cooke, Tiling rectangles with holey polyominoes, arXiv:1411.2699 [cs.CG], 2015.

Crossrefs

Cf. A286194, A286345, A381030.

Extensions

a(6)-a(7) corrected and a(8)-a(11) added by John Mason, Feb 15 2025