A378015 -id:A378015 - cp's OEIS frontend

A377756 Number of free hexagonal polyominoes with n cells with at most 3 collinear cell centers on any line in the plane.

1, 1, 3, 6, 18, 55, 169, 477, 1245, 2750, 5380, 8989, 12674, 14741, 13928, 10297, 6185, 2910, 1012, 289, 69, 12, 2

Offset: 1

Dave Budd, Nov 06 2024

a(n) is the number of connected planer graphs with n nodes, where the nodes lie on a triangular lattice grid and no more than 3 nodes are collinear over the underlying plane.

a(n) is the sum of columns 1-3 in A378015, the n-th term = Sum(T(n,k)) for k<=3.

			For n=23, the 2 hexagon polyominoes are:
            @ @                      @
           @                    @     @
    @       @                    @ @   @
     @ @     @          @           @ @
@   @       @            @ @           @
 @   @       @          @   @           @
  @ @         @              @ @       @
     @     @ @                  @   @ @
      @ @ @                      @ @

Dave Budd, Collinear cells in hexagon polyominoes
Dave Budd, Optimized version for exhaustive proof of A377756

Cf. A000228, A378015.

Python
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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

A377756 Number of free hexagonal polyominoes with n cells with at most 3 collinear cell centers on any line in the plane.

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