A385291 -id:A385291 - cp's OEIS frontend

A385581 Square array read by antidiagonals: T(n,d) is the number of fixed d-dimensional polysticks of size n.

1, 2, 1, 3, 6, 1, 4, 15, 22, 1, 5, 28, 95, 88, 1, 6, 45, 252, 681, 372, 1, 7, 66, 525, 2600, 5277, 1628, 1, 8, 91, 946, 7065, 29248, 43086, 7312, 1, 9, 120, 1547, 15696, 104097, 349132, 365313, 33466, 1, 10, 153, 2360, 30513, 285828, 1632915, 4351944, 3186444, 155446, 1

Offset: 1

Pontus von Brömssen, Jul 04 2025

The first 17 antidiagonals are from Mertens and Moore (2018), either directly from Table 1 or computed using the perimeter polynomials in Appendix A. T(14,5) is the only unknown value in the 18th antidiagonal.

T(13,6) = 14054816418877200 (Mertens and Moore).

			Table begins:
  n\d| 1     2       3        4         5          6          7           8
  ---+---------------------------------------------------------------------
  1  | 1     2       3        4         5          6          7           8
  2  | 1     6      15       28        45         66         91         120
  3  | 1    22      95      252       525        946       1547        2360
  4  | 1    88     681     2600      7065      15696      30513       53936
  5  | 1   372    5277    29248    104097     285828     661549     1356384
  6  | 1  1628   43086   349132   1632915    5551480   15314936    36449288
  7  | 1  7312  365313  4351944  26817465  113045832  372033993  1028383408
  8  | 1 33466 3186444 56062681 456137580 2386821009 9377038237 30118187174

Pontus von Brömssen, Table of n, a(n) for n = 1..153 (first 17 antidiagonals)
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. A000384 (row n=2), A385291 (polyominoes), A385582, A385583 (free).

Columns: A096267 (d=2), A365560 (d=3), A365562 (d=4), A365564 (d=5).

T(n,d) = Sum_{k=1..d} binomial(n,k)*A385582(n,k) (with A385582(n,k) = 0 if d > n).

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

A385581 Square array read by antidiagonals: T(n,d) is the number of fixed d-dimensional polysticks of size n.

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