A381057 Array read by diagonals downwards: A(n,k) for n>=2 and k>=0 is the number of (n,k)-polyominoes counting as distinct different formations of transparent squares.

1, 2, 2, 3, 5, 5, 6, 17, 24, 12, 10, 41, 101, 89, 35, 20, 106, 353, 535, 382, 108, 36, 243, 1091, 2355, 2769, 1566, 369, 72, 567, 3095, 8937, 14841, 13739, 6569, 1285, 136, 1259, 8209, 29744, 65651, 86322, 66499, 27205, 4655, 272, 2806, 20804, 90914, 252277, 439879, 479343, 314445, 112886, 17073, 528, 6113, 50801, 259078, 872526

Offset: 2

Author

John Mason, Feb 12 2025

Keywords

• nonn
• tabl

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, different sets of the same number of transparent squares that connect in distinct ways the same set of visible squares, are counted separately. E.g. these 2 different formations count as 2:

XO XOO

OX X

Examples

			The table begins as follows:
  n\k|     0      1       2       3       4       5      6      7     8    9  10
   --+--------------------------------------------------------------------------
    2|     1      2       3       6      10      20     36     72   136  272 528
    3|     2      5      17      41     106     243    567   1259  2806 6113
    4|     5     24     101     353    1091    3095   8209  20804 50801
    5|    12     89     535    2355    8937   29744  90914 259078
    6|    35    382    2769   14841   65651  252277 872526
    7|   108   1566   13739   86322  439879 1917387
    8|   369   6569   66499  479343 2759969
    9|  1285  27205  314445 2555903
   10|  4655 112886 1461335
   11| 17073 466178
   12| 63600
		

Links

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

Crossrefs

Row 2 gives A005418.

Column 0 gives A000105.

Cf. A381030.