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 1-5 of 5 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

A366334 Number of free (4,2)-polyominoids with n cells.

Original entry on oeis.org

1, 2, 12, 95, 1267, 22349
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes with integer coordinates in D-dimensional space, where two cubes are connected if they share a (d-1)-dimensional facet. For example, (3,2)-polyominoids are normal polyominoids (A075679), (D,D)-polyominoids are D-dimensional polyominoes (A000105, A038119, A068870, ...), and (D,1)-polyominoids are polysticks in D dimensions (A019988, A365559, A365561, ...).

Links

Crossrefs

46th row of A366766.
Cf. A366335 (fixed).
Free (D,d)-polyominoids:
D\d| 1 2 3 4
---+--------------------------------
1 | A000012
2 | A019988 A000105
3 | A365559 A075679 A038119
4 | A365561 A366334 A366336 A068870

A366337 Number of fixed (4,3)-polyominoids with n cells.

Original entry on oeis.org

4, 36, 500, 8364, 155300, 3082380
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

See A366335 for definitions.

Links

Crossrefs

Cf. A366335, A366336 (free).

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.

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

Views

Author

John Mason, Jul 07 2025

Keywords

Examples

			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

Links

Crossrefs

Rows: A001168 (n=2), A075678 (n=3), A366335 (n=4).
Columns: A000217 (k=1), A213820 (k=2).
Cf. A385291 (polyominoes), A385581 (polysticks).
Showing 1-5 of 5 results.

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