A366767 Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

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

Offset: 1

Author

Pontus von Brömssen, Oct 22 2023

Keywords

• nonn
• tabl

Comments

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

Examples

			Array begins:
  n\k| 1  2  3   4   5    6     7      8      9      10       11        12
  ---+--------------------------------------------------------------------
   1 | 1  0  0   0   0    0     0      0      0       0        0         0
   2 | 1  1  1   1   1    1     1      1      1       1        1         1
   3 | 2  0  0   0   0    0     0      0      0       0        0         0
   4 | 2  2  2   2   2    2     2      2      2       2        2         2
   5 | 2  4 12  38 126  432  1520   5450  19820   72892   270536   1011722
   6 | 2  6 22  88 372 1628  7312  33466 155446  730534  3466170  16576874
   7 | 1  0  0   0   0    0     0      0      0       0        0         0
   8 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
   9 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
  10 | 1  4 20 110 638 3832 23592 147941 940982 6053180 39299408 257105146
  11 | 3  0  0   0   0    0     0      0      0       0        0         0
  12 | 3  3  3   3   3    3     3      3      3       3        3         3
		

Links

• Pontus von Brömssen, Python programs that relate row numbers and parameter sets.
• Wikipedia, Polyominoid.
• Index entries for sequences related to polyominoes.

Crossrefs

Cf. A366766 (free), A366768.

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:

X: Allowed connection.

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

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

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

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

| | | connections |

| | | g:112223 |

n | D | d | h:010120 | sequence

----+---+---+-------------+----------

1 | 1 | 1 | * -..... | A063524

2 | 1 | 1 | * X..... | A000012

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

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

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

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

7 | 2 | 2 | * -.-... | A063524

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

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

10 | 2 | 2 | * X.X... | A006770

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

31 | 3 | 3 | * -.-..- | A063524

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

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

34 | 3 | 3 | * X.X..- |

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

36 | 3 | 3 | * X.-..X |

37 | 3 | 3 | * -.X..X |

38 | 3 | 3 | * X.X..X |

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

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

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

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

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

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

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

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

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

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