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

1,9

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

Links

Table of n, a(n) for n=1..36.

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

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)

Example

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
...

Crossrefs

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

Cf. A195739, A049430.

Cf. A000055, A045649, A036364, A036365.

Sequence in context: A351111 A154738 A355653 * A322632 A184082 A198868

Adjacent sequences: A195735 A195736 A195737 * A195739 A195740 A195741

Keyword

nonn,tabl,more

Author

N. J. A. Sloane, Sep 22 2011

Extensions

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

Status

approved