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 1-8 of 8 results.

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

Views

Author

Joseph Myers, Oct 15 2011

Keywords

Comments

[3.6.3.6] refers to the face configuration of the rhombille tiling. - Peter Kagey, Mar 01 2020
If we draw the short diagonals of each tile in the rhombille tiling, we get a subset of edges of the regular hexagonal grid; two edges are adjacent if and only if the corresponding rhombi are adjacent. These are polyedges where angles are constrained to 120 degrees. So there is a 1-to-1 correspondence with the subset of polyedges counted in A159867 after removing polyedges with angles of 60 and/or 180 degrees. - Joseph Myers, Jul 12 2020
These are also known as polytwigs, by association with their representation as polyedges. - Aaron N. Siegel, May 15 2022

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. A019988, A159867, A197460, A197461.
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).

Extensions

a(18)-a(22) from Aaron N. Siegel, May 15 2022
a(23)-a(25) from Bert Dobbelaere, Jun 04 2025

A197460 Number of one-sided poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

Original entry on oeis.org

1, 1, 4, 6, 19, 49, 143, 403, 1235, 3681, 11354, 34944, 108956, 340635, 1072593, 3388161, 10751029, 34217608, 109255023, 349750874, 1122436127, 3610017340
Offset: 1

Views

Author

Joseph Myers, Oct 15 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.

Crossrefs

Cf. A197459, A197461.

Extensions

a(18)-a(22) from Aaron N. Siegel, May 15 2022

A378344 Number of fixed site animals with n nodes on the nodes of the prismatic pentagonal tiling.

Original entry on oeis.org

3, 5, 12, 35, 106, 332, 1062, 3466, 11496, 38621, 131042, 448146, 1542548, 5338641, 18563680, 64814950, 227117365, 798387748, 2814618634
Offset: 1

Views

Author

Johann Peters, Nov 23 2024

Keywords

Comments

Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Dual to the polyhouses, AKA the site animals on the nodes of the elongated triangular tiling, counted by A197158, insofar as the tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(19) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttman reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.

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

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

Formula

It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.

A378362 Number of fixed site animals containing n nodes on the nodes of the cairo pentagonal tiling.

Original entry on oeis.org

6, 10, 24, 68, 198, 594, 1816, 5650, 17824, 56836, 182788, 592060, 1929676, 6323418, 20819284, 68828316, 228372578, 760188362, 2537770576, 8494004948
Offset: 1

Views

Author

Johann Peters, Nov 23 2024

Keywords

Comments

Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Is dual to the polycairos counted by A196991, AKA site animals on the nodes of the snub square tiling, insofar that these tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(20) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttman reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.

Examples

			There are six translationally distinct nodes in the cairo pentagonal tiling, so a(1)=6.

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

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

Formula

It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.

A378416 Number of fixed site animals with n nodes on the nodes of the rhombille tiling.

Original entry on oeis.org

3, 6, 21, 73, 273, 1049, 4117, 16416, 66263, 270211, 1111443, 4605575, 19204920, 80515734, 339137432, 1434319849
Offset: 1

Views

Author

Johann Peters, Nov 25 2024

Keywords

Comments

Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Dual to the site animals on the nodes of the trihexagonal (AKA kagome) tiling, counted by A197461, insofar as the tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(16) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttman reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.

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

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

Formula

It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.

A379052 Number of fixed site animals with n nodes on the nodes of the floret pentagonal tiling.

Original entry on oeis.org

9, 15, 39, 124, 405, 1344, 4548, 15765, 55763, 199928, 723468, 2637378, 9677509, 35714337, 132445734, 493209254, 1843263534, 6910868397
Offset: 1

Views

Author

Johann Peters, Dec 17 2024

Keywords

Comments

Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Dual to the site animals on the nodes of the snub hexagonal tiling, counted by A197160, insofar as the tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(18) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttman reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.

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

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

Formula

It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.

A379163 Number of fixed site animals with n nodes on the nodes of the tetrakis square tiling.

Original entry on oeis.org

2, 6, 26, 121, 597, 3040, 15876, 84520, 456584, 2494906, 13759902, 76475067, 427805198, 2406492158, 13602178244, 77206507977
Offset: 1

Views

Author

Johann Peters, Dec 17 2024

Keywords

Comments

Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Dual to the site animals on the nodes of the truncated square tiling, counted by A197467, insofar as the tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n->oo} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(16) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttman reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.

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

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

Formula

It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.

Extensions

a(16) from Michael Bartmann, Jul 16 2025

A379178 Number of fixed site animals with n nodes on the nodes of the kisrhombille tiling.

Original entry on oeis.org

6, 18, 90, 479, 2718, 16126, 97885, 603741, 3771287, 23792622, 151342506, 969465873, 6248109573
Offset: 1

Views

Author

Johann Peters, Dec 17 2024

Keywords

Comments

Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Dual to the site animals on the nodes of the truncated trihexagonal tiling, counted by A197464, insofar as the tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(13) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttman reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.

Examples

			There are 6 translationally distinct sites in the kisrhombille lattice, so a(1)=6.

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

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

Formula

It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.
Showing 1-8 of 8 results.

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