A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

1, 2, 9, 66, 691, 9216, 134325

Offset: 1

Author

Pontus von Brömssen, Sep 26 2023

Keywords

• nonn
• hard
• more

Comments

This sequence and the related sequences A365650-A365655 and A365996-A366010 count polyominoids (A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.

Links

• Pontus von Brömssen, Illustration for sizes 1..3.
• Wikipedia, Polyominoid.
• Index entries for sequences related to polyominoes.

Crossrefs

Cf. A365996 (fixed).

21st row of A366766.

The following table lists counting sequences for free, fixed, and one-sided polyominoids with different sets of allowed connections. "|" means flat connections and "L" means right-angled connections.

corner-connections | edge-connections | free | fixed | 1-sided

-------------------+------------------+---------+---------+--------

none | | | A000105 |3*A001168| A000105

none | L | A365654 | A365655 |

none | |L | A075679 | A075678 | A056846

| | none | A000105 |3*A001168| A000105

| | | | A030222 |3*A006770| A030222

| | L | A365995 | A365996 |

| | |L | A365997 | A365998 |

L | none | A365999 | A366000 |

L | | | A366001 | A366002 |

L | L | A366003 | A366004 |

L | |L | A366005 | A366006 |

|L | none | A365652 | A365653 |

|L | | | A366007 | A366008 |

|L | L | A366009 | A366010 |

|L | |L | A365650 | A365651 |

Extensions

a(7) from Pontus von Brömssen, Mar 03 2025