cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 17 results. Next

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

Views

Author

Drake Thomas and Peter Kagey, May 03 2021

Keywords

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.

Examples

			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.

Links

Crossrefs

Row sums of A365970.
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).

Extensions

a(11)-a(16) from Bert Dobbelaere, Jun 10 2025

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

Views

Author

Joseph Myers, Oct 15 2011

Keywords

Comments

See the link below for a definition of the tetrakis square tiling. When a square grid cell is divided into triangles, it must be divided dexter (\) or sinister (/) according to the parity of the grid cell.

Examples

			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

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Links

Crossrefs

Cf. A197466, A197467, A006074.
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).

Extensions

Name clarified by George Sicherman, Mar 25 2021
a(21)-a(26) from Aaron N. Siegel, May 18 2022
a(27)-a(29) from Bert Dobbelaere, Jun 04 2025

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

Views

Author

Peter Kagey, Apr 20 2021

Keywords

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 glide reflection) of the other.
a(n) >= A343417(n), the number of (n-k)-polyominoes with k distinguished vertices.

Links

Crossrefs

Cf. A121197 (one-sided).
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).

Extensions

a(11) from Drake Thomas, May 02 2021
a(12)-a(16) from John Mason, Mar 20 2022

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

Views

Author

Joseph Myers, Oct 10 2011

Keywords

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Links

Crossrefs

Cf. A197160.
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).

Extensions

a(16)-a(20) from Bert Dobbelaere, Jun 02 2025

A197462 Number of free poly-[4.6.12]-tiles (holes allowed) with n cells (division into triangles is significant).

Original entry on oeis.org

1, 3, 3, 9, 14, 38, 74, 185, 414, 1026, 2440, 6077, 14926, 37454, 93749, 237035, 599815, 1526020, 3889117, 9944523, 25475398, 65416733, 168277945, 433705325, 1119610147, 2894928713
Offset: 1

Views

Author

Joseph Myers, Oct 15 2011

Keywords

Comments

[4.6.12] refers to the face configuration of the kisrhombille tiling. - Peter Kagey, May 10 2021

References

  • Branko Gruenbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Links

Crossrefs

Cf. A197463, A197464.
Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Extensions

a(20)-a(26) from Aaron N. Siegel, Jun 03 2022

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

Views

Author

Joseph Myers, Oct 10 2011

Keywords

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Links

Crossrefs

Cf. A197157, A197158.
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).

Extensions

a(16)-a(20) from Bert Dobbelaere, Jun 02 2025

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

Views

Author

Peter Kagey, Apr 13 2021

Keywords

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 glide reflection) of the other.

Links

Crossrefs

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).

Extensions

a(12)-a(15) from John Mason, Mar 04 2022
a(16)-a(20) from Bert Dobbelaere, Jun 06 2025

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

Views

Author

Peter Kagey, Apr 14 2021

Keywords

Comments

Equivalently, the number of polyhexes with n-k cells and k distinguished vertices.
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 glide reflection) of the other.

Links

Crossrefs

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).

Extensions

a(10)-a(15) from Bert Dobbelaere, Jun 06 2025

A197461 Number of fixed poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

Original entry on oeis.org

3, 6, 14, 36, 99, 281, 816, 2415, 7260, 22074, 67743, 209552, 652566, 2043672, 6431715, 20328504, 64494678, 205303537, 655493697, 2098500135, 6734498816, 21660087087, 69805491342
Offset: 1

Views

Author

Joseph Myers, Oct 15 2011

Keywords

References

  • Branko Gruenbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Crossrefs

Cf. A197459, A197460.

Extensions

a(18)-a(22) from Aaron N. Siegel, May 16 2022
a(23) from Johann Peters, Dec 16 2024

A344211 Number of generalized polyforms on the rhombitrihexagonal tiling with n cells.

Original entry on oeis.org

1, 3, 2, 7, 16, 60, 201, 838, 3407, 14767, 64200, 284676, 1269981, 5715325, 25854146, 117576949, 536918541, 2461230475, 11319321354, 52212782646
Offset: 0

Views

Author

Peter Kagey, May 11 2021

Keywords

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 glide reflection) of the other.

Links

Crossrefs

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), A343577 (truncated square).

Extensions

a(11)-a(19) from Bert Dobbelaere, Jun 05 2025
Showing 1-10 of 17 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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