A385715 -id:A385715 - cp's OEIS frontend

A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

1, 2, 12, 6, 140, 320, 19, 1554, 10368, 13520, 63, 17622, 265344, 892864, 786432, 216, 206747, 6390484, 41998840, 89389920, 58383808, 760, 2503578, 152166240, 1749529040, 6773387520

Offset: 1

Pontus von Brömssen, Aug 14 2025

A (d,2)-polyominoid consists of unit square cells with integer coordinates in the d-dimensional grid, where two cells are connected if they share an edge. The polyominoid is properly d-dimensional if it is not contained in a (d-1)-dimensional affine subspace.

			Triangle begins:
  n\d |     2          3           4           5          6        7  8  9 10 11
  ----+-------------------------------------------------------------------------
   1  |     1
   2  |     2         12
   3  |     6        140         320
   4  |    19       1554       10368       13520
   5  |    63      17622      265344      892864     786432
   6  |   216     206747     6390484    41998840   89389920 58383808
   7  |   760    2503578   152166240  1749529040 6773387520        ?  ?
   8  |  2725   31117536  3644734836 69246650605          ?        ?  ?  ?
   9  |  9910  394953243 88344741448           ?          ?        ?  ?  ?  ?
  10  | 36446 5098388985           ?           ?          ?        ?  ?  ?  ?  ?

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A001168 (column d=2), A195739 (polyominoes), A385582 (polysticks), A385715, A387004 (free).

T(n,d) = Sum_{k=2..d} (-1)^(d-k)*binomial(d,k)*A385715(k,n), i.e., the n-th row is the inverse binomial transform of the n-th column of A385715 (with the convention that T(n,d) = A385715(d,n) = 0 when d <= 1).

A387003 Triangle read by rows: T(n,d) is the number of free (d,2)-polyominoids of size n, 2 <= d <= n+1.

Original entry on oeis.org

1, 1, 2, 2, 9, 12, 5, 54, 95, 103

Offset: 1

Pontus von Brömssen, Aug 14 2025

If d > n+1, there are T(n,n+1) such polyominoids. The triangle only includes the values for d <= n+1.

			Triangle begins:
  n\d| 2  3  4   5
  ---+------------
  1  | 1
  2  | 1  2
  3  | 2  9 12
  4  | 5 54 95 103

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Columns: A000105 (d=2), A075679 (d=3), A366334 (d=4).

Cf. A330891 (polyominoes), A385583 (polysticks), A385715 (fixed), A387002, A387004, A387005 (main diagonal).

T(n,d) = Sum_{k=1..d} A387004(n,k).

cp's OEIS Frontend

A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

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