OFFSET
0,2
COMMENTS
This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection, or a combination thereof) of the other.
LINKS
Peter Kagey, Animation of the a(4) = 9 polyforms with 4 cells.
Peter Kagey, Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb, Mathematics Stack Exchange.
Wikipedia, Tetrahedral-octahedral honeycomb
EXAMPLE
For n = 1, the a(1) = 2 polyforms are the tetrahedron and the octahedron.
For n = 2, the a(2) = 1 polyform is a tetrahedron and an octahedron connected at a face.
For n = 3, there are a(3) = 4 polyforms with 3 cells:
- 3 consisting of one octahedron with two tetrahedra, and
- 1 consisting of two octahedra and one tetrahedron.
For n = 4, there are a(4) = 9 polyforms with 4 cells:
- 3 with one octahedron and three tetrahedra,
- 5 with two octahedra and three octahedra, and
- 1 with three octahedra and one tetrahedron.
For n = 5, there are a(5) = 44 polyforms with 5 cells:
- 6 with one octahedron and four tetrahedra,
- 24 with two octahedra and three tetrahedra,
- 13 with three octahedra and two tetrahedra, and
- 1 with four octahedra and one tetrahedron.
CROSSREFS
Analogous for other honeycombs/tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A038119 (cubical), A068870 (tesseractic), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).
KEYWORD
nonn,hard,more
AUTHOR
Drake Thomas and Peter Kagey, May 03 2021
STATUS
approved