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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.

%I A381030 #21 Feb 16 2025 10:26:35
%S A381030 1,2,2,2,4,5,3,11,20,12,3,17,60,68,35,4,32,151,302,289,108,4,45,322,
%T A381030 955,1523,1151,369,5,71,633,2617,5942,7384,4792,1285,5,94,1132,6179,
%U A381030 19061,33819,35188,19603,4655,6,134,1930,13374,52966,125940,184938,164036,80820,17073,6,170,3095,26567,131717,400119,778318,969972
%N A381030 Array read by diagonals downwards: A(n,k) for n>=2 and k>=0 is the number of (n,k)-polyominoes.
%C A381030 (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 A381030 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:
%C A381030    XO  XOO
%C A381030     OX   X
%H A381030 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.
%F A381030 First row, a(2,k) = floor((k+3)/2).
%e A381030 The table begins as follows:
%e A381030   n\k|     0      1      2      3      4      5      6     7    8   9 10
%e A381030   ---+------------------------------------------------------------------
%e A381030     2|     1      2      2      3      3      4      4     5    5   6  6
%e A381030     3|     2      4     11     17     32     45     71    94  134 170
%e A381030     4|     5     20     60    151    322    633   1132  1930 3095
%e A381030     5|    12     68    302    955   2617   6179  13374 26567
%e A381030     6|    35    289   1523   5942  19061  52966 131717
%e A381030     7|   108   1151   7384  33819 125940 400119
%e A381030     8|   369   4792  35188 184938 778318
%e A381030     9|  1285  19603 164036 969972
%e A381030    10|  4655  80820 753310
%e A381030    11| 17073 331373
%e A381030    12| 63600
%Y A381030 Cf. A381057.
%Y A381030 Columns 0..4: A000105, A286344, A286194, A286345.
%K A381030 nonn,tabl
%O A381030 2,2
%A A381030 _John Mason_, Feb 12 2025