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

2,2

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.

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

Links

Table of n, a(n) for n=2..64.

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

Formula

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

Example

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

Crossrefs

Cf. A381057.

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

Sequence in context: A299204 A230141 A151680 * A231351 A024681 A240327

Adjacent sequences: A381025 A381026 A381028 * A381031 A381032 A381033

Keyword

nonn,tabl,new

Author

John Mason, Feb 12 2025

Status

approved