A377941 -id:A377941 - cp's OEIS frontend

A377942 Triangle read by rows: T(n,k) = number of free polyominoes with n cells, where the maximum number of collinear cell centers on any line in the plane is k.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 0, 9, 2, 1, 0, 0, 18, 13, 3, 1, 0, 0, 37, 48, 19, 3, 1, 0, 0, 62, 200, 77, 25, 4, 1, 0, 0, 86, 678, 369, 114, 33, 4, 1, 0, 0, 78, 2177, 1590, 593, 170, 41, 5, 1, 0, 0, 61, 6280, 6739, 2774, 928, 234, 51, 5, 1, 0, 0, 34, 17187, 27153, 12851, 4597, 1387, 323, 61, 6, 1

Offset: 1

Dave Budd, Nov 11 2024

The row sum is the number of free polyominoes with n cells.

			   |  k
n  |       1      2      3      4      5      6      7      8      9     10
----------------------------------------------------------------------------
 1 |       1
 2 |       0      1
 3 |       0      1      1
 4 |       0      2      2      1
 5 |       0      0      9      2      1
 6 |       0      0     18     13      3      1
 7 |       0      0     37     48     19      3      1
 8 |       0      0     62    200     77     25      4      1
 9 |       0      0     86    678    369    114     33      4      1
10 |       0      0     78   2177   1590    593    170     41      5      1
...
From _John Mason_, Feb 14 2025: (Start)
The first difference with A377941 occurs at n=5 when the following polyomino has maximum number of row or column cells = 2, but there are 3 cells on a 45 degree diagonal.
 O
 OO
  OO
(End)

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

Row sums are A000105.

Cf. A377941.

More terms from Pontus von Brömssen, Nov 12 2024

A378014 Triangle read by rows: T(n,k) = number of free hexagonal polyominoes with n cells, where the maximum number of cells on any lattice line is k. The term "lattice line" here means a line running through the cell centers and midpoints of their sides.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 3, 15, 3, 1, 0, 5, 50, 23, 3, 1, 0, 1, 171, 126, 30, 4, 1, 0, 1, 506, 710, 187, 39, 4, 1, 0, 1, 1459, 3520, 1268, 270, 48, 5, 1, 0, 1, 3792, 16617, 7703, 1948, 364, 59, 5, 1, 0, 1, 9292, 72870, 45099, 12885, 2840, 488, 70, 6, 1

Offset: 1

Dave Budd, Nov 14 2024

The row sums are the total number of free hexagonal 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      3     15      3      1                                             22
 6 |       0      5     50     23      3      1                                      82
 7 |       0      1    171    126     30      4      1                              333
 8 |       0      1    506    710    187     39      4      1                      1448
 9 |       0      1   1459   3520   1268    270     48      5      1               6572
10 |       0      1   3792  16617   7703   1948    364     59      5      1       30490
The T(4,2)=4 hexagon polyominoes are:
  #         #        #   #      # #
   # #       # #      # #      # #
      #     #

Dave Budd, Python code for a hex lattice

Row sums are A000228.

Cf. A377941, A378015.

cp's OEIS Frontend

A377942 Triangle read by rows: T(n,k) = number of free polyominoes with n cells, where the maximum number of collinear cell centers on any line in the plane is k.

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