A343909
Number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.
Original entry on oeis.org
1, 2, 1, 4, 9, 44, 195, 1186, 7385, 49444, 337504, 2353664, 16608401, 118432965, 851396696, 6163949361, 44896941979
Offset: 0
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.
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).
A384486
Table read by rows: number of connected components of polyhedra in the quarter cubic honeycomb consisting of k tetrahedra and n-k truncated tetrahedra, up to translation, rotation, and reflection of the honeycomb, 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 3, 8, 8, 1, 0, 7, 31, 43, 14, 1, 0, 24, 126, 261, 152, 18, 0, 0, 88, 598, 1543, 1467, 369, 14, 0, 0, 385, 2986, 9276, 12161, 5661, 602, 8, 0, 0, 1713, 15467, 55426, 92723, 65892, 15251, 694, 3, 0, 0, 8112, 81217, 330821, 666705, 646974, 254615, 29830, 551, 1, 0, 0
Offset: 0
Table begins:
n\k| 0 1 2 3 4 5 6 7
----+------------------------------------
0 | 1;
1 | 1, 1;
2 | 1, 1, 0;
3 | 1, 3, 1, 0;
4 | 3, 8, 8, 1, 0;
5 | 7, 31, 43, 14, 1, 0;
6 | 24, 126, 261, 152, 18, 0, 0;
7 | 88, 598, 1543, 1467, 369, 14, 0, 0;
Cf.
A365970 (tetrahedral-octahedral honeycomb).
A384755
Triangle read by rows: T(n,k) is the number of face-connected components of polyhedra with k prisms and n-k truncated cuboctahedra in the omnitruncated cubic honeycomb up to rotation and reflection, 0 <= k <= n.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 7, 10, 2, 12, 41, 76, 46, 4, 61, 335, 809, 777, 232, 13, 407, 3065, 9512, 12863, 7186, 1206, 39, 3226, 30401, 114516, 204143, 172377, 60421, 6548, 155, 28335, 311782, 1381363, 3054599, 3507278, 1975767, 469525, 36081, 637, 262091, 3260971, 16569719, 43731912
Offset: 0
0 | 1;
1 | 1, 1;
2 | 1, 2, 1;
3 | 3, 7, 10, 2;
4 | 12, 41, 76, 46, 4;
5 | 61, 335, 809, 777, 232, 13;
6 | 407, 3065, 9512, 12863, 7186, 1206, 39;
Cf.
A365970 (tetrahedral-octahedral honeycomb),
A384486 (quarter cubic honeycomb),
A384782 (rectified cubic honeycomb).
A384782
Triangle read by rows: T(n,k) is the number of face-connected polyhedral components consisting of k cuboctahedra and n-k octahedra in the rectified cubic honeycomb up to translation, rotation, and reflection of the honeycomb, 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 0, 1, 1, 0, 3, 4, 2, 0, 3, 18, 12, 7, 0, 6, 60, 126, 75, 23, 0, 3, 165, 751, 1025, 473, 112, 0, 3, 346, 3784, 9414, 8936, 3539, 607, 0, 1, 565, 14112, 66503, 108739, 80531, 27027, 3811, 0, 1, 723, 42420, 362939, 994542, 1204093, 725795, 212122, 25413, 0, 0, 723, 101237, 1586479, 7065791, 13389295, 12792264, 6512671, 1678783, 178083
Offset: 0
Table begins:
0 | 1;
1 | 1, 1;
2 | 0, 1, 1;
3 | 0, 3, 4, 2;
4 | 0, 3, 18, 12, 7;
5 | 0, 6, 60, 126, 75, 23;
6 | 0, 3, 165, 751, 1025, 473, 112;
7 | 0, 3, 346, 3784, 9414, 8936, 3539, 607;
8 | 0, 1, 565, 14112, 66503, 108739, 80531, 27027, 3811;
9 | 0, 1, 723, 42420, 362939, 994542, 1204093, 725795, 212122, 25413;
Cf.
A365970 (tetrahedral-octahedral honeycomb),
A384486 (quarter cubic honeycomb),
A384755 (omnitruncated cubic honeycomb).
Showing 1-4 of 4 results.
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