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).
A197465
Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.
Original entry on oeis.org
1, 2, 2, 6, 8, 22, 42, 112, 252, 650, 1584, 4091, 10369, 26938, 69651, 182116, 476272, 1253067, 3302187, 8733551, 23142116, 61477564, 163612714, 436278921, 1165218495, 3117021788, 8349892686, 22397754046, 60153261611
Offset: 1
For n=3 there are 4 triaboloes. Of these, 2 conform to the tetrakis grid. Each of these 2 has a unique dissection into 6 tetrakis cells. - _George Sicherman_, Mar 25 2021
- Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
Analogous for other tilings:
A000105 (square),
A000228 (hexagonal),
A000577 (triangular),
A197156 (prismatic pentagonal),
A197159 (floret pentagonal),
A197459 (rhombille),
A197462 (kisrhombille),
A309159 (snub square),
A343398 (trihexagonal),
A343406 (truncated hexagonal),
A343577 (truncated square).
A343577
Number of generalized polyforms on the truncated square tiling with n cells.
Original entry on oeis.org
1, 2, 2, 7, 22, 93, 413, 2073, 10741, 57540, 312805, 1722483, 9564565, 53489304, 300840332, 1700347858, 9650975401
Offset: 0
Analogous for other tilings:
A000105 (square),
A000228 (hexagonal),
A000577 (triangular),
A197156 (prismatic pentagonal),
A197159 (floret pentagonal),
A197459 (rhombille),
A197462 (kisrhombille),
A197465 (tetrakis square),
A309159 (snub square),
A343398 (trihexagonal),
A343406 (truncated hexagonal).
A197159
Number of free poly-[3^4.6]-tiles (holes allowed) with n cells.
Original entry on oeis.org
1, 3, 8, 25, 80, 291, 1036, 3875, 14561, 55624, 213951, 830712, 3244355, 12747718, 50320659, 199491045, 793754027, 3168756843, 12687184463, 50932363171
Offset: 1
- Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
Analogous for other tilings:
A000105 (square),
A000228 (hexagonal),
A000577 (triangular),
A197156 (prismatic pentagonal),
A197459 (rhombille),
A197462 (kisrhombille),
A197465 (tetrakis square),
A309159 (snub square),
A343398 (trihexagonal),
A343406 (truncated hexagonal),
A343577 (truncated square).
A197459
Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).
Original entry on oeis.org
1, 1, 3, 4, 12, 27, 78, 208, 635, 1859, 5726, 17526, 54620, 170479, 536714, 1694567, 5376764, 17110286, 54631302, 174879997, 561229678, 1805022806, 5817191196, 18781911278, 60744460580
Offset: 1
- Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
Analogous for other tilings:
A000105 (square),
A000228 (hexagonal),
A000577 (triangular),
A197156 (prismatic pentagonal),
A197159 (floret pentagonal),
A197462 (kisrhombille),
A197465 (tetrakis square),
A309159 (snub square),
A343398 (trihexagonal),
A343406 (truncated hexagonal),
A343577 (truncated square).
A197156
Number of free poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.
Original entry on oeis.org
1, 3, 5, 20, 56, 225, 819, 3333, 13336, 55231, 229146, 963284, 4068503, 17301000, 73893082, 317013121, 1364917667, 5896350458, 25545737979, 110968732581
Offset: 1
- Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
Analogous for other tilings:
A000105 (square),
A000228 (hexagonal),
A000577 (triangular),
A197159 (floret pentagonal),
A197459 (rhombille),
A197462 (kisrhombille),
A197465 (tetrakis square),
A309159 (snub square),
A343398 (trihexagonal),
A343406 (truncated hexagonal),
A343577 (truncated square).
A343398
Number of generalized polyforms on the trihexagonal tiling with n cells.
Original entry on oeis.org
1, 2, 1, 4, 9, 30, 97, 373, 1405, 5630, 22672, 93045, 384403, 1602156, 6712128, 28268504, 119537113, 507375130, 2160476897, 9226446455, 39504435891
Offset: 0
Same but distinguishing mirror images:
A350739.
Analogous for other tilings:
A000105 (square),
A000228 (hexagonal),
A000577 (triangular),
A197156 (prismatic pentagonal),
A197159 (floret pentagonal),
A197459 (rhombille),
A197462 (kisrhombille),
A197465 (tetrakis square),
A309159 (snub square),
A343406 (truncated hexagonal),
A343577 (truncated square).
A343406
Number of generalized polyforms on the truncated hexagonal tiling with n cells.
Original entry on oeis.org
1, 2, 2, 9, 40, 218, 1377, 9285, 65039, 465888, 3385778, 24864272, 184115213, 1372589329, 10291503008, 77544953479
Offset: 0
Analogous for other tilings:
A000105 (square),
A000228 (hexagonal),
A000577 (triangular),
A197156 (prismatic pentagonal),
A197159 (floret pentagonal),
A197459 (rhombille),
A197462 (kisrhombille),
A197465 (tetrakis square),
A309159 (snub square),
A343398 (trihexagonal),
A343577 (truncated square).
A383496
Number of polyforms with n cells on the faces of a disdyakis triacontahedron up to rotation and reflection.
Original entry on oeis.org
1, 1, 3, 3, 9, 14, 38, 74, 185, 414, 1025, 2430, 6012, 14640, 36294, 89531, 222452, 552015, 1374367, 3421955, 8535077, 21296556, 53184718, 132846918, 331913842, 829176301, 2071018243, 5170617085, 12902719975, 32176288808
Offset: 0
A197464
Number of fixed poly-[4.6.12]-tiles (holes allowed) with n cells (division into triangles is significant).
Original entry on oeis.org
12, 18, 36, 75, 168, 386, 888, 2079, 4960, 11988, 29280, 72145, 179112, 447636, 1124956, 2840097, 7197780, 18301738, 46669404, 119308938, 305704584, 784938678, 2019335340, 5204309886, 13435321764, 34738762062, 89952855848
Offset: 1
- Branko Gruenbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
Showing 1-10 of 17 results.
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