A365862 Number of 4-fold rotationally symmetric pentomino tilings of the 5n X 5n square such that no two tiles can be replaced with two pentomominoes in a new configuration.

1, 0, 0, 0, 2, 0, 20, 54, 6464

Offset: 0

Author

Jamie Tucker-Foltz, Sep 29 2023

Keywords

• nonn
• more

Comments

a(n) is even for all n > 0 since no pentomino tiling can have both rotation and reflection symmetry.

Examples

			a(4) = 2:
._______________________________________.
|_. | ._. | ._|_. |_________|_________| |
| | |_| |_| |_. | ._|_. | ._. | ._| .___|
| |___| |___| | |_| ._| |_| |_| | |_|_. |
| | |_____| ._| | ._| |___| |___| .___| |
| |___. |___| |_|_| |_. |_____| |_| |___|
|_| ._|_| |_______| ._|___| | |_. |___. |
| | |___. |_. | |___|_______| ._|___| |_|
| |___| |_| | |_______| | ._| |_______| |
| | .___| ._| |_. | .___| | |_|_. |_. ._|
| |_| |___| |_| | |_|_. | |_. ._|_. |_| |
|_| |_. |___. | |___| | |_| |_| ._|_| | |
| .___|___| |_| | .___|___| | ._| .___| |
|_| ._____| |___|_|_____. | |_| |_|_. | |
| |_|_. |_. | ._____| ._|_|___| .___| |_|
|___. |_. |_|_|_. |_. |___. ._|_| |___| |
| ._|_| |_|_. ._|_. |_| | |_| ._|___. | |
| |___. | ._| |_. |_| ._| | ._|_. ._|_| |
|___| |_| | |_| | |___| | |_| ._| |_. | |
| .___|___|_____|___|_. |___| | |_| | |_|
|_|_________|_________|___|___|_____|___| (*2)
		

Links

• Jamie Tucker-Foltz, Locked Polyomino Tilings, arXiv:2307.15996 [math.CO], 2023.
• Jamie Tucker-Foltz, GitHub repository with Python code and pictures of all tilings.
• Wikipedia, Pentomino.

Extensions

a(8) from Jamie Tucker-Foltz, Oct 30 2023