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.

User: Dave Budd

Dave Budd's wiki page.

Dave Budd has authored 8 sequences.

A380991 a(n) = largest number of cells of a polyomino with at most n collinear cell centers.

Original entry on oeis.org

1, 4, 15, 48
Offset: 1

Views

Author

Dave Budd, Feb 11 2025

Keywords

Crossrefs

Cf. A378169, A380990.

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

Views

Author

Dave Budd, Feb 11 2025

Keywords

Links

Crossrefs

Cf. A000105, A377942, A378169.

Programs

  • Python
    # See links

Formula

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

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

Views

Author

Dave Budd, Nov 18 2024

Keywords

Examples

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

Links

Crossrefs

Cf. A000105, A377756, A377942.

Programs

  • Python
    # See link

Formula

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

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

Original entry on oeis.org

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

Views

Author

Dave Budd, Nov 06 2024

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A000228, A378015.

Programs

  • Python
    # See links

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

Views

Author

Dave Budd, Nov 14 2024

Keywords

Comments

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

Examples

			   |  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:
  #         #        #   #      # #
   # #       # #      # #      # #
      #     #

Links

Crossrefs

Row sums are A000228.
Cf. A377941, A378015.

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

Views

Author

Dave Budd, Nov 14 2024

Keywords

Comments

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

Examples

			   |  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:
 #          #   #
#   #        # #
 # #        #

Links

Crossrefs

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).

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.

Original entry on oeis.org

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

Views

Author

Dave Budd, Nov 11 2024

Keywords

Comments

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

Examples

			   | 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:
  * *      * *
  * *        * *

Links

Crossrefs

Row sums are A000105.
Cf. A377942.

Extensions

More terms from Pontus von Brömssen, Nov 12 2024

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.

Original entry on oeis.org

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

Views

Author

Dave Budd, Nov 11 2024

Keywords

Comments

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

Examples

			   |  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)

Links

Crossrefs

Row sums are A000105.
Cf. A377941.

Extensions

More terms from Pontus von Brömssen, Nov 12 2024

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