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Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.
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%I #9 Oct 30 2023 11:20:05

%S 1,0,1,0,1,2,0,1,0,2,0,1,0,2,2,0,1,0,2,4,2,0,1,0,2,12,6,1,0,1,0,2,38,

%T 22,0,1,0,1,0,2,126,88,0,2,1,0,1,0,2,432,372,0,6,2,1,0,1,0,2,1520,

%U 1628,0,19,6,4,3,0,1,0,2,5450,7312,0,63,19,20,0,3

%N Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

%C See A366766 (corresponding array for free polyominoids) for details.

%H Pontus von Brömssen, <a href="https://oeis.org/A366766/a366766.py.txt">Python programs that relate row numbers and parameter sets</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyominoid">Polyominoid</a>.

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.

%e Array begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12

%e ---+--------------------------------------------------------------------

%e 1 | 1 0 0 0 0 0 0 0 0 0 0 0

%e 2 | 1 1 1 1 1 1 1 1 1 1 1 1

%e 3 | 2 0 0 0 0 0 0 0 0 0 0 0

%e 4 | 2 2 2 2 2 2 2 2 2 2 2 2

%e 5 | 2 4 12 38 126 432 1520 5450 19820 72892 270536 1011722

%e 6 | 2 6 22 88 372 1628 7312 33466 155446 730534 3466170 16576874

%e 7 | 1 0 0 0 0 0 0 0 0 0 0 0

%e 8 | 1 2 6 19 63 216 760 2725 9910 36446 135268 505861

%e 9 | 1 2 6 19 63 216 760 2725 9910 36446 135268 505861

%e 10 | 1 4 20 110 638 3832 23592 147941 940982 6053180 39299408 257105146

%e 11 | 3 0 0 0 0 0 0 0 0 0 0 0

%e 12 | 3 3 3 3 3 3 3 3 3 3 3 3

%Y Cf. A366766 (free), A366768.

%Y The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C (see A366766). Some sequences occur in more than one row. Notation used in the table:

%Y X: Allowed connection.

%Y -: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).

%Y .: Not applicable for (D,d) in this row.

%Y !: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.

%Y *: Whether a connection of type (g,h) is allowed or not is independent of h.

%Y | | | connections |

%Y | | | g:112223 |

%Y n | D | d | h:010120 | sequence

%Y ----+---+---+-------------+----------

%Y 1 | 1 | 1 | * -..... | A063524

%Y 2 | 1 | 1 | * X..... | A000012

%Y 3 |!2 | 1 | * --.... | 2*A063524

%Y 4 |!2 | 1 | X-.... | 2*A000012

%Y 5 | 2 | 1 | -X.... | 2*A001168

%Y 6 | 2 | 1 | * XX.... | A096267

%Y 7 | 2 | 2 | * -.-... | A063524

%Y 8 | 2 | 2 | * X.-... | A001168

%Y 9 | 2 | 2 | * -.X... | A001168

%Y 10 | 2 | 2 | * X.X... | A006770

%Y 11 |!3 | 1 | * --.... | 3*A063524

%Y 12 |!3 | 1 | X-.... | 3*A000012

%Y 13 | 3 | 1 | -X.... | A365655

%Y 14 | 3 | 1 | * XX.... | A365560

%Y 15 |!3 | 2 | * ----.. | 3*A063524

%Y 16 |!3 | 2 | X---.. | 3*A001168

%Y 17 | 3 | 2 | -X--.. | A365655

%Y 18 | 3 | 2 | * XX--.. | A075678

%Y 19 |!3 | 2 | --X-.. | 3*A001168

%Y 20 |!3 | 2 | X-X-.. | 3*A006770

%Y 21 | 3 | 2 | -XX-.. | A365996

%Y 22 | 3 | 2 | XXX-.. | A365998

%Y 23 | 3 | 2 | ---X.. | A366000

%Y 24 | 3 | 2 | X--X.. | A366002

%Y 25 | 3 | 2 | -X-X.. | A366004

%Y 26 | 3 | 2 | XX-X.. | A366006

%Y 27 | 3 | 2 | * --XX.. | A365653

%Y 28 | 3 | 2 | X-XX.. | A366008

%Y 29 | 3 | 2 | -XXX.. | A366010

%Y 30 | 3 | 2 | * XXXX.. | A365651

%Y 31 | 3 | 3 | * -.-..- | A063524

%Y 32 | 3 | 3 | * X.-..- | A001931

%Y 33 | 3 | 3 | * -.X..- | A039742

%Y 34 | 3 | 3 | * X.X..- |

%Y 35 | 3 | 3 | * -.-..X | A039741

%Y 36 | 3 | 3 | * X.-..X |

%Y 37 | 3 | 3 | * -.X..X |

%Y 38 | 3 | 3 | * X.X..X |

%Y 39 |!4 | 1 | * --.... | 4*A063524

%Y 40 |!4 | 1 | X-.... | 4*A000012

%Y 41 | 4 | 1 | -X.... | A366341

%Y 42 | 4 | 1 | * XX.... | A365562

%Y 43 |!4 | 2 | * -----. | 6*A063524

%Y 44 |!4 | 2 | X----. | 6*A001168

%Y 45 | 4 | 2 | -X---. | A366339

%Y 46 | 4 | 2 | * XX---. | A366335

%Y 47 |!4 | 2 | --X--. | 6*A001168

%Y 48 |!4 | 2 | X-X--. | 6*A006770

%K nonn,tabl

%O 1,6

%A _Pontus von Brömssen_, Oct 22 2023