A381057 - OEIS

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

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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, 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

LINKS

Table of n, a(n) for n=2..61.

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

EXAMPLE

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

CROSSREFS

Row 2 gives A005418.

Column 0 gives A000105.

Cf. A381030.

Sequence in context: A029039 A316185 A131429 * A105605 A237710 A090473

Adjacent sequences: A381054 A381055 A381056 * A381058 A381059 A381060

KEYWORD

nonn,tabl

AUTHOR

John Mason, Feb 12 2025

STATUS

approved