cp's OEIS Frontend

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.

%I A381057 #18 Feb 16 2025 10:26:25
%S A381057 1,2,2,3,5,5,6,17,24,12,10,41,101,89,35,20,106,353,535,382,108,36,243,
%T A381057 1091,2355,2769,1566,369,72,567,3095,8937,14841,13739,6569,1285,136,
%U A381057 1259,8209,29744,65651,86322,66499,27205,4655,272,2806,20804,90914,252277,439879,479343,314445,112886,17073,528,6113,50801,259078,872526
%N 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.
%C A381057 (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.
%C A381057 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:
%C A381057    XO  XOO
%C A381057     OX   X
%H A381057 Dmitry Kamenetsky and Tristrom Cooke, <a href="https://arxiv.org/abs/1411.2699">Tiling rectangles with holey polyominoes</a>, arXiv:1411.2699 [cs.CG], 2015.
%e A381057 The table begins as follows:
%e A381057   n\k|     0      1       2       3       4       5      6      7     8    9  10
%e A381057    --+--------------------------------------------------------------------------
%e A381057     2|     1      2       3       6      10      20     36     72   136  272 528
%e A381057     3|     2      5      17      41     106     243    567   1259  2806 6113
%e A381057     4|     5     24     101     353    1091    3095   8209  20804 50801
%e A381057     5|    12     89     535    2355    8937   29744  90914 259078
%e A381057     6|    35    382    2769   14841   65651  252277 872526
%e A381057     7|   108   1566   13739   86322  439879 1917387
%e A381057     8|   369   6569   66499  479343 2759969
%e A381057     9|  1285  27205  314445 2555903
%e A381057    10|  4655 112886 1461335
%e A381057    11| 17073 466178
%e A381057    12| 63600
%Y A381057 Row 2 gives A005418.
%Y A381057 Column 0 gives A000105.
%Y A381057 Cf. A381030.
%K A381057 nonn,tabl
%O A381057 2,2
%A A381057 _John Mason_, Feb 12 2025