A385581 -id:A385581 - cp's OEIS frontend

A385582 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional polysticks of size n.

1, 1, 4, 1, 20, 32, 1, 86, 420, 400, 1, 370, 4164, 10368, 6912, 1, 1626, 38205, 186552, 301840, 153664, 1, 7310, 343380, 2934560, 8637760, 10223616, 4194304, 1, 33464, 3086049, 43517697, 207353960, 427708848, 396809280, 136048896

Offset: 1

Pontus von Brömssen, Jul 04 2025

			Triangle begins:
  n\d| 1     2       3        4         5         6         7         8
  ---+-----------------------------------------------------------------
  1  | 1
  2  | 1     4
  3  | 1    20      32
  4  | 1    86     420      400
  5  | 1   370    4164    10368      6912
  6  | 1  1626   38205   186552    301840    153664
  7  | 1  7310  343380  2934560   8637760  10223616   4194304
  8  | 1 33464 3086049 43517697 207353960 427708848 396809280 136048896

Pontus von Brömssen, Table of n, a(n) for n = 1..78 (first 12 rows)
Stephan Mertens and Cristopher Moore, Series expansion of the percolation threshold on hypercubic lattices, J. Phys. A: Math. Theor., 51 (2018), 475001; arXiv:1805.02701 [cond-mat.stat-mech], 2018.
Index entries for sequences related to polyominoes.

Cf. A127670 (main diagonal), A195739 (polyominoes), A365566 (free), A385581.

T(n,d) = Sum_{k=1..d} (-1)^(d-k)*binomial(d,k)*A385581(n,k).

A385583 Triangle read by rows: T(n,d) is the number of free d-dimensional polysticks of size n.

Original entry on oeis.org

1, 1, 2, 1, 5, 7, 1, 16, 28, 31, 1, 55, 160, 199, 205, 1, 222, 1085, 1651, 1768, 1779

Offset: 1

Pontus von Brömssen, Jul 04 2025

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

			Triangle begins:
  n\d| 1   2    3    4    5    6
  ---+--------------------------
  1  | 1
  2  | 1   2
  3  | 1   5    7
  4  | 1  16   28   31
  5  | 1  55  160  199  205
  6  | 1 222 1085 1651 1768 1779

Index entries for sequences related to polyominoes.

Cf. A330891 (polyominoes), A365565 (main diagonal), A365566, A385581 (fixed).

Columns: A019988 (d=2), A365559 (d=3), A365561 (d=4), A365563 (d=5).

T(n,d) = Sum_{k=1..d} A365566(n,k).

A385715 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional (n,2)-polyominoids, n >= 2, of size k >= 1.

Original entry on oeis.org

1, 2, 3, 6, 18, 6, 19, 158, 60, 10, 63, 1611, 916, 140, 15, 216, 17811, 16698, 3060, 270, 21, 760, 207395, 336210, 81090, 7690, 462, 28, 2725, 2505858, 7218768, 2396434, 268005, 16226, 728, 36, 9910, 31125711, 162185112, 76020890, 10477161, 701589, 30408, 1080, 45

Offset: 2

John Mason, Jul 07 2025

			The top corner of the array (size on horizontal axis, dimensions on vertical):
              1    2     3       4         5          6           7           8         9         10
(A001168) 2:  1    2     6      19        63        216         760        2725      9910      36446
(A075678) 3:  3   18   158    1611     17811     207395     2505858    31125711 394982973 5098498323
(A366335) 4:  6   60   916   16698    336210    7218768   162185112  3769221330
          5: 10  140  3060   81090   2396434   76020890  2535403620 87781527395
          6: 15  270  7690  268005  10477161  441378400 19603138320
          7: 21  462 16226  701589  34160301 1796996509
          8: 28  728 30408 1570436  91583156
          9: 36 1080 52296 3141108 213477012

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

Rows: A001168 (n=2), A075678 (n=3), A366335 (n=4).
Columns: A000217 (k=1), A213820 (k=2).
Cf. A385291 (polyominoes), A385581 (polysticks).

cp's OEIS Frontend

A385582 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional polysticks of size n.

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A385583 Triangle read by rows: T(n,d) is the number of free d-dimensional polysticks of size n.

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Examples

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Formula

A385715 Square array read by descending antidiagonals: A(n,k) is the number of fixed n-dimensional (n,2)-polyominoids, n >= 2, of size k >= 1.

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Author

Keywords

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