A049429 Triangle T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (0 < d < n).

1, 1, 1, 1, 4, 2, 1, 11, 11, 3, 1, 34, 77, 35, 6, 1, 107, 499, 412, 104, 11, 1, 368, 3442, 4888, 2009, 319, 23, 1, 1284, 24128, 57122, 36585, 8869, 951, 47, 1, 4654, 173428, 667959, 647680, 231574, 36988, 2862, 106, 1, 17072, 1262464, 7799183, 11173880, 5712765, 1297366, 146578, 8516, 235

Offset: 2

Author

Richard C. Schroeppel

Keywords

• nonn
• nice
• tabl
• hard

Comments

These are unoriented polyominoes of the regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). For unoriented polyominoes, chiral pairs are counted as one. The dimension of the convex hull of the cell centers determines the dimension d. - Robert A. Russell, Aug 09 2022

Examples

			From _Robert A. Russell_, Aug 09 2022: (Start)
Triangle begins with T(2,1):
n\d 1     2       3       4        5       6       7      8    9  10
2   1
3   1     1
4   1     4       2
5   1    11      11       3
6   1    34      77      35        6
7   1   107     499     412      104      11
8   1   368    3442    4888     2009     319      23
9   1  1284   24128   57122    36585    8869     951     47
10  1  4654  173428  667959   647680  231574   36988   2862  106
11  1 17072 1262464 7799183 11173880 5712765 1297366 146578 8516 235
(End)
		

References

• Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Links

• R. C. Schroeppel, A few mathematical experiments

Crossrefs

Cf. A049430 (col. d=0 added), A195738 (oriented), A195739 (fixed).

Diagonals (with algorithms) are A000055, A036364, A355053.

Row sums give A005519. Columns are A006765-A006768.

Extensions

Two more rows added by Robert A. Russell, Aug 09 2022.