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A343909 Number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.

%I A343909 #33 Jun 10 2025 13:53:49
%S A343909 1,2,1,4,9,44,195,1186,7385,49444,337504,2353664,16608401,118432965,
%T A343909 851396696,6163949361,44896941979
%N A343909 Number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.
%C A343909 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.
%H A343909 Peter Kagey, <a href="/A343909/a343909.gif">Animation of the a(4) = 9 polyforms with 4 cells</a>.
%H A343909 Peter Kagey, <a href="https://math.stackexchange.com/q/4128528/121988">Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb</a>, Mathematics Stack Exchange.
%H A343909 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycomb">Tetrahedral-octahedral honeycomb</a>
%e A343909 For n = 1, the a(1) = 2 polyforms are the tetrahedron and the octahedron.
%e A343909 For n = 2, the a(2) = 1 polyform is a tetrahedron and an octahedron connected at a face.
%e A343909 For n = 3, there are a(3) = 4 polyforms with 3 cells:
%e A343909   - 3 consisting of one octahedron with two tetrahedra, and
%e A343909   - 1 consisting of two octahedra and one tetrahedron.
%e A343909 For n = 4, there are a(4) = 9 polyforms with 4 cells:
%e A343909   - 3 with one octahedron and three tetrahedra,
%e A343909   - 5 with two octahedra and three octahedra, and
%e A343909   - 1 with three octahedra and one tetrahedron.
%e A343909 For n = 5, there are a(5) = 44 polyforms with 5 cells:
%e A343909   - 6 with one octahedron and four tetrahedra,
%e A343909   - 24 with two octahedra and three tetrahedra,
%e A343909   - 13 with three octahedra and two tetrahedra, and
%e A343909   - 1 with four octahedra and one tetrahedron.
%Y A343909 Row sums of A365970.
%Y A343909 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).
%K A343909 nonn,hard,more
%O A343909 0,2
%A A343909 _Drake Thomas_ and _Peter Kagey_, May 03 2021
%E A343909 a(11)-a(16) from _Bert Dobbelaere_, Jun 10 2025