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 1-6 of 6 results.

A366767 Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 2, 4, 2, 0, 1, 0, 2, 12, 6, 1, 0, 1, 0, 2, 38, 22, 0, 1, 0, 1, 0, 2, 126, 88, 0, 2, 1, 0, 1, 0, 2, 432, 372, 0, 6, 2, 1, 0, 1, 0, 2, 1520, 1628, 0, 19, 6, 4, 3, 0, 1, 0, 2, 5450, 7312, 0, 63, 19, 20, 0, 3
Offset: 1

Views

Author

Pontus von Brömssen, Oct 22 2023

Keywords

Comments

See A366766 (corresponding array for free polyominoids) for details.

Examples

			Array begins:
  n\k| 1  2  3   4   5    6     7      8      9      10       11        12
  ---+--------------------------------------------------------------------
   1 | 1  0  0   0   0    0     0      0      0       0        0         0
   2 | 1  1  1   1   1    1     1      1      1       1        1         1
   3 | 2  0  0   0   0    0     0      0      0       0        0         0
   4 | 2  2  2   2   2    2     2      2      2       2        2         2
   5 | 2  4 12  38 126  432  1520   5450  19820   72892   270536   1011722
   6 | 2  6 22  88 372 1628  7312  33466 155446  730534  3466170  16576874
   7 | 1  0  0   0   0    0     0      0      0       0        0         0
   8 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
   9 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
  10 | 1  4 20 110 638 3832 23592 147941 940982 6053180 39299408 257105146
  11 | 3  0  0   0   0    0     0      0      0       0        0         0
  12 | 3  3  3   3   3    3     3      3      3       3        3         3

Links

Crossrefs

Cf. A366766 (free), A366768.
The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C (see A366766). Some sequences occur in more than one row. Notation used in the table:
X: Allowed connection.
-: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).
.: Not applicable for (D,d) in this row.
!: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.
*: Whether a connection of type (g,h) is allowed or not is independent of h.
| | | connections |
| | | g:112223 |
n | D | d | h:010120 | sequence
----+---+---+-------------+----------
1 | 1 | 1 | * -..... | A063524
2 | 1 | 1 | * X..... | A000012
3 |!2 | 1 | * --.... | 2*A063524
4 |!2 | 1 | X-.... | 2*A000012
5 | 2 | 1 | -X.... | 2*A001168
6 | 2 | 1 | * XX.... | A096267
7 | 2 | 2 | * -.-... | A063524
8 | 2 | 2 | * X.-... | A001168
9 | 2 | 2 | * -.X... | A001168
10 | 2 | 2 | * X.X... | A006770
11 |!3 | 1 | * --.... | 3*A063524
12 |!3 | 1 | X-.... | 3*A000012
13 | 3 | 1 | -X.... | A365655
14 | 3 | 1 | * XX.... | A365560
15 |!3 | 2 | * ----.. | 3*A063524
16 |!3 | 2 | X---.. | 3*A001168
17 | 3 | 2 | -X--.. | A365655
18 | 3 | 2 | * XX--.. | A075678
19 |!3 | 2 | --X-.. | 3*A001168
20 |!3 | 2 | X-X-.. | 3*A006770
21 | 3 | 2 | -XX-.. | A365996
22 | 3 | 2 | XXX-.. | A365998
23 | 3 | 2 | ---X.. | A366000
24 | 3 | 2 | X--X.. | A366002
25 | 3 | 2 | -X-X.. | A366004
26 | 3 | 2 | XX-X.. | A366006
27 | 3 | 2 | * --XX.. | A365653
28 | 3 | 2 | X-XX.. | A366008
29 | 3 | 2 | -XXX.. | A366010
30 | 3 | 2 | * XXXX.. | A365651
31 | 3 | 3 | * -.-..- | A063524
32 | 3 | 3 | * X.-..- | A001931
33 | 3 | 3 | * -.X..- | A039742
34 | 3 | 3 | * X.X..- |
35 | 3 | 3 | * -.-..X | A039741
36 | 3 | 3 | * X.-..X |
37 | 3 | 3 | * -.X..X |
38 | 3 | 3 | * X.X..X |
39 |!4 | 1 | * --.... | 4*A063524
40 |!4 | 1 | X-.... | 4*A000012
41 | 4 | 1 | -X.... | A366341
42 | 4 | 1 | * XX.... | A365562
43 |!4 | 2 | * -----. | 6*A063524
44 |!4 | 2 | X----. | 6*A001168
45 | 4 | 2 | -X---. | A366339
46 | 4 | 2 | * XX---. | A366335
47 |!4 | 2 | --X--. | 6*A001168
48 |!4 | 2 | X-X--. | 6*A006770

A365654 Number of free n-polyominoids, allowing right-angled connections only ("hard" polyominoids).

Original entry on oeis.org

1, 1, 5, 16, 90, 537, 3826, 28655, 225534
Offset: 1

Views

Author

Pontus von Brömssen, Sep 17 2023

Keywords

Comments

Two squares are allowed to meet in a straight 180-degree connection, but the structure must be connected through right-angled ("hard") connections only. This seems to be in agreement with the definition of "hard" polyominoids in the Mireles Jasso link (the number of fixed hard hexominoids given by the "sample report" linked from that web-page agrees with A365655(6) = 22417), but differs from the definition in the Wikipedia article. The smallest example of a polyominoid that is included here but is not hard according to Wikipedia consists of two squares between (0,0,1) and (2,1,1), two between (0,0,1) and (2,0,2), and one between (1,0,0) and (1,1,1) (a "one-legged sofa", see illustration in the Mireles Jasso link). This explains why a(5) = 90, while the number of hard pentominoids is 89 according to the Wikipedia article.
Equivalently, number of n-polysticks in 3 dimensions, connected through right-angled connections.
Also, the number of face-connected polyhedral components in the square bipyramidal honeycomb up to translation, rotation, and reflection of the honeycomb. - Peter Kagey, Jun 10 2025

Links

Crossrefs

13th and 17th row of A366766.
Cf. A075679 (polyominoids), A365559 (polysticks in 3 dimensions), A365655 (fixed).

Extensions

a(9) from Pontus von Brömssen, Mar 03 2025

A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

Original entry on oeis.org

1, 2, 9, 66, 691, 9216, 134325
Offset: 1

Views

Author

Pontus von Brömssen, Sep 26 2023

Keywords

Comments

This sequence and the related sequences A365650-A365655 and A365996-A366010 count polyominoids (A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.

Links

Crossrefs

Cf. A365996 (fixed).
21st row of A366766.
The following table lists counting sequences for free, fixed, and one-sided polyominoids with different sets of allowed connections. "|" means flat connections and "L" means right-angled connections.
corner-connections | edge-connections | free | fixed | 1-sided
-------------------+------------------+---------+---------+--------
none | | | A000105 |3*A001168| A000105
none | L | A365654 | A365655 |
none | |L | A075679 | A075678 | A056846
| | none | A000105 |3*A001168| A000105
| | | | A030222 |3*A006770| A030222
| | L | A365995 | A365996 |
| | |L | A365997 | A365998 |
L | none | A365999 | A366000 |
L | | | A366001 | A366002 |
L | L | A366003 | A366004 |
L | |L | A366005 | A366006 |
|L | none | A365652 | A365653 |
|L | | | A366007 | A366008 |
|L | L | A366009 | A366010 |
|L | |L | A365650 | A365651 |

Extensions

a(7) from Pontus von Brömssen, Mar 03 2025

A366339 Number of fixed (4,2)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

6, 48, 592, 8664, 140088, 2414056
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

Two squares sharing an edge have a right-angled connection if they do not lie in the same plane.
Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.

Links

Crossrefs

Cf. A075678, A365655, A366335, A366338 (free), A366341.
45th row of A366767.

A366341 Number of fixed (4,3)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

4, 24, 200, 1924, 20228, 225788, 2631672
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

Two cubes sharing a face have a right-angled connection if they do not lie in the same 3-dimensional affine subspace.
Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.
Also, number of fixed polysticks in 4 dimensions with right-angled connections.

Links

Crossrefs

Cf. A075678, A365655, A366337, A366339, A366340 (free).
41st and 77th row of A366767.

A383737 Cluster series for percolation on polyominoid cells, with connections only between orthogonal cells ("hard" polyominoids).

Original entry on oeis.org

1, 8, 40, 168, 720, 2886, 11684, 46536, 181328
Offset: 0

Views

Author

Pontus von Brömssen, May 10 2025

Keywords

Comments

Equivalently, cluster series for percolation on polystick cells in 3 dimensions, with connections only between orthogonal cells.

Links

Crossrefs

Rows 13 and 17 of A383735.
Cf. A299279 (coordination sequence for hard polyominoid cells), A365654, A365655, A383736.
Showing 1-6 of 6 results.

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

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