A377942 -id:A377942 - cp's OEIS frontend

A377941 Triangle read by rows: T(n,k) = number of free polyominoes with n cells, where the maximum number of cells in any row or column is k.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 8, 2, 1, 0, 1, 17, 13, 3, 1, 0, 1, 39, 45, 19, 3, 1, 0, 1, 79, 182, 77, 25, 4, 1, 0, 1, 162, 607, 363, 114, 33, 4, 1, 0, 1, 301, 2004, 1539, 593, 170, 41, 5, 1, 0, 1, 589, 6139, 6361, 2764, 928, 234, 51, 5, 1, 0, 1, 1141, 18278, 25072, 12733, 4597, 1387, 323, 61, 6, 1

Offset: 1

Dave Budd, Nov 11 2024

The row sum are the total number of polyominoes with n cells.

			   | k
n  |       1      2      3      4      5      6      7      8      9     10  Total
----------------------------------------------------------------------------------
 1 |       1                                                                     1
 2 |       0      1                                                              1
 3 |       0      1      1                                                       2
 4 |       0      2      2      1                                                5
 5 |       0      1      8      2      1                                        12
 6 |       0      1     17     13      3      1                                 35
 7 |       0      1     39     45     19      3      1                         108
 8 |       0      1     79    182     77     25      4      1                  369
 9 |       0      1    162    607    363    114     33      4      1          1285
10 |       0      1    301   2004   1539    593    170     41      5      1   4655
 ...
The T(4,2)=2 polyominoes are:
  * *      * *
  * *        * *

John Mason, Table of n, a(n) for n = 1..153 (17 rows)
Dave Budd, Python code for a square lattice

Row sums are A000105.

Cf. A377942.

More terms from Pontus von Brömssen, Nov 12 2024

A378169 Number of free polyominoes with n cells with at most 3 collinear cell centers on any line in the plane.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 37, 62, 86, 78, 61, 34, 14, 4, 1

Offset: 1

Dave Budd, Nov 18 2024

			For n=15, the only polyomino is
    ###
  ###
###
  ## #
   ###

Dave Budd, Exhaustive proof of A378169

Cf. A000105, A377756, A377942.

Python
```
# See link
```

a(n) = Sum_{k=1..3} A377942(n,k).

A378015 Triangle read by rows: T(n,k) = number of free hexagonal polyominoes with n cells, where the maximum number of collinear cell centers on any line in the plane is k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 2, 16, 3, 1, 0, 3, 52, 23, 3, 1, 0, 0, 169, 129, 30, 4, 1, 0, 0, 477, 740, 187, 39, 4, 1, 0, 0, 1245, 3729, 1274, 270, 48, 5, 1, 0, 0, 2750, 17578, 7785, 1948, 364, 59, 5, 1, 0, 0, 5380, 75827, 46045, 12895, 2840, 488, 70, 6, 1

Offset: 1

Dave Budd, Nov 14 2024

The row sums are the total number of free hexagon polyominoes with n cells.

			   |  k
 n |       1      2      3      4      5      6      7      8      9     10       Total
---------------------------------------------------------------------------------------
 1 |       1                                                                          1
 2 |       0      1                                                                   1
 3 |       0      2      1                                                            3
 4 |       0      4      2      1                                                     7
 5 |       0      2     16      3      1                                             22
 6 |       0      3     52     23      3      1                                      82
 7 |       0      0    169    129     30      4      1                              333
 8 |       0      0    477    740    187     39      4      1                      1448
 9 |       0      0   1245   3729   1274    270     48      5      1               6572
10 |       0      0   2750  17578   7785   1948    364     59      5      1       30490
The T(5,2)=2 hexagon polyominoes are:
 #          #   #
#   #        # #
 # #        #

Dave Budd, Python code for a hex lattice.

Cf. A000228 (row sums).
Cf. A377942 (similar collinear cell constraint for square polyominoes).
Cf. A377756 (specific case for the cumulative value for k<=3 i.e. T(n,1)+T(n,2)+T(n,3) ).
Cf. A378014 (collinear cell constraint applied only to cells on lattice lines).

A380990 Number of free polyominoes with n cells with at most 4 collinear cell centers on any line in the plane.

Original entry on oeis.org

1, 1, 2, 5, 11, 31, 85, 262, 764, 2255, 6341, 17221, 43994, 106205, 239367, 502611, 977791, 1771624, 2989373, 4687803, 6819069, 9234529, 11622453, 13527854, 14571011, 14643347, 13747913, 12041014, 9905945, 7763985, 5805906, 4139266, 2858796, 1971455, 1368967, 942226, 618148, 368480, 186275, 73649, 20236, 3476, 400, 96, 27, 12, 2, 1

Offset: 1

Dave Budd, Feb 11 2025

Dave Budd, python code

Cf. A000105, A377942, A378169.

Python
```
# See links
```

a(n) = Sum_{k=1..4} A377942(n,k).

cp's OEIS Frontend

A377941 Triangle read by rows: T(n,k) = number of free polyominoes with n cells, where the maximum number of cells in any row or column is k.

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A378169 Number of free polyominoes with n cells with at most 3 collinear cell centers on any line in the plane.

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A378015 Triangle read by rows: T(n,k) = number of free hexagonal polyominoes with n cells, where the maximum number of collinear cell centers on any line in the plane is k.

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A380990 Number of free polyominoes with n cells with at most 4 collinear cell centers on any line in the plane.

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Python

Formula