A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

1, 0, 1, 0, 1, 1, 0, 1, 6, 3, 0, 1, 17, 17, 4, 0, 1, 59, 131, 52, 7, 0, 1, 195, 915, 709, 153, 13, 0, 1, 703, 6553, 8946, 3350, 454, 28

Offset: 1

Author

N. J. A. Sloane, Sep 22 2011

Keywords

• nonn
• tabl
• more

Comments

From Petros Hadjicostas, Jan 11 2019: (Start)

Table 1 (p. 366) in Lunnon (1975) contains more terms. Because the table there (in the reference) has incomplete columns, the extra terms do not appear in this triangular sequence (array).

Entry DR(n=11, d=2) in Table 1 (p. 366) must be a typo. It should not be 33890, but 33895. This was corrected by N. J. A. Sloane in 2011 in the documentation of sequence A006758. (See also sequence A000988.)

(End)

The number of oriented polyominoes (chiral pairs counted as two) here is the sum of the number of unoriented polyominoes (chiral pairs counted as one) in A049430 and the number of chiral pairs. - Robert A. Russell, May 03 2020

Examples

			Triangle begins:
n\d| 0    1    2    3    4    5    6    7
---+---------------------------------=---
1  | 1
2  | 0    1
3  | 0    1    1
4  | 0    1    6    3
5  | 0    1   17   17    4
6  | 0    1   59  131   52    7
7  | 0    1  195  915  709  153   13
8  | 0    1  703 6553 8946 3350  454   28
...
		

Links

• W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (4) (1975) 366-367.

Crossrefs

Columns give A006758 (and A000988), A006759, A006760, A006761.

Cf. A195739, A049430.

Cf. A000055, A045649, A036364, A036365.

Formula

From Robert A. Russell, May 03 2020: (Start)

For n > 1, DR(n,n-1) = A000055(n) + A045649(n).

DR(n,n-2) = A036364(n) + A036365(n).

We can add unoriented and chiral pairs for the top two diagonals. The summands have quick algorithms. (End)

Extensions

Sequence corrected by Petros Hadjicostas, Jan 11 2019 after observation by Jon E. Schoenfield