A286345 Number of (n,3)-polyominoes.

0, 3, 17, 151, 955, 5942, 33819, 184938, 969972

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.

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) three 0's to make all the 1's connected. This also means that the Manhattan distance between any pair of 1's is at most 4. Here are all such polyominoes for n=2:
   10001 1000 100
         0001 000
              001
		

Links

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

Crossrefs

Cf. A286194, A286344, A381030.

Extensions

a(6)-a(9) from John Mason, Feb 15 2025