cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A365566 Triangle read by rows: T(n,d) is the number of inequivalent properly d-dimensional n-polysticks (or polyedges), 1 <= d <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 15, 12, 3, 1, 54, 105, 39, 6, 1, 221, 863, 566, 117, 11
Offset: 1

Views

Author

Pontus von Brömssen, Sep 09 2023

Keywords

Examples

			Triangle begins:
  n\d | 1   2   3   4   5  6
  ----+---------------------
   1  | 1
   2  | 1   1
   3  | 1   4   2
   4  | 1  15  12   3
   5  | 1  54 105  39   6
   6  | 1 221 863 566 117 11

Links

Crossrefs

Cf. A000055, A049430 (polyominoes), A365565 (row sums), A385582 (fixed), A385583.

Formula

T(n,n) = A000055(n+1).
T(n,d) = A385583(n,d) - A385583(n,d-1) (with A385583(n,0) = 0). - Pontus von Brömssen, Jul 13 2025

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

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Jul 04 2025

Keywords

Comments

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).

Examples

			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

Links

Crossrefs

Cf. A000384 (row n=2), A385291 (polyominoes), A385582, A385583 (free).
Columns: A096267 (d=2), A365560 (d=3), A365562 (d=4), A365564 (d=5).

Formula

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

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.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Aug 14 2025

Keywords

Comments

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.

Examples

			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           ?           ?          ?        ?  ?  ?  ?  ?

Links

Crossrefs

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

Formula

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).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.