A286344 -id:A286344 - cp's OEIS frontend

A381030 Array read by diagonals downwards: A(n,k) for n>=2 and k>=0 is the number of (n,k)-polyominoes.

1, 2, 2, 2, 4, 5, 3, 11, 20, 12, 3, 17, 60, 68, 35, 4, 32, 151, 302, 289, 108, 4, 45, 322, 955, 1523, 1151, 369, 5, 71, 633, 2617, 5942, 7384, 4792, 1285, 5, 94, 1132, 6179, 19061, 33819, 35188, 19603, 4655, 6, 134, 1930, 13374, 52966, 125940, 184938, 164036, 80820, 17073, 6, 170, 3095, 26567, 131717, 400119, 778318, 969972

Offset: 2

John Mason, Feb 12 2025

(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.
Note that, in this sequence, 2 different sets of the same number of transparent squares that connect in distinct ways the same set of visible squares, count as 1. E.g. these 2 different formations count as 1:
   XO  XOO
    OX   X

			The table begins as follows:
  n\k|     0      1      2      3      4      5      6     7    8   9 10
  ---+------------------------------------------------------------------
    2|     1      2      2      3      3      4      4     5    5   6  6
    3|     2      4     11     17     32     45     71    94  134 170
    4|     5     20     60    151    322    633   1132  1930 3095
    5|    12     68    302    955   2617   6179  13374 26567
    6|    35    289   1523   5942  19061  52966 131717
    7|   108   1151   7384  33819 125940 400119
    8|   369   4792  35188 184938 778318
    9|  1285  19603 164036 969972
   10|  4655  80820 753310
   11| 17073 331373
   12| 63600

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

Cf. A381057.

Columns 0..4: A000105, A286344, A286194, A286345.

First row, a(2,k) = floor((k+3)/2).

A286194 Number of (n,2)-polyominoes.

Original entry on oeis.org

0, 2, 11, 60, 302, 1523, 7384, 35188, 164036, 753310

Offset: 1

Dmitry Kamenetsky, May 05 2017

(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.

			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) two 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 3. Here are all such polyominoes for n=2:
   1001 100
        001

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

Cf. A286344, A286345, A381030.

a(6) corrected and a(7)-a(10) from John Mason, Feb 15 2025

A286345 Number of (n,3)-polyominoes.

Original entry on oeis.org

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

Offset: 1

Dmitry Kamenetsky, May 07 2017

			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

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

Cf. A286194, A286344, A381030.

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

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A381030 Array read by diagonals downwards: A(n,k) for n>=2 and k>=0 is the number of (n,k)-polyominoes.

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A286194 Number of (n,2)-polyominoes.

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A286345 Number of (n,3)-polyominoes.

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