A001420 -id:A001420 - cp's OEIS frontend

A364684 Number of achiral triangular polyominoes with 6n cells and sixfold rotational symmetry.

1, 1, 1, 1, 3, 4, 7, 9, 16, 22, 46, 63, 121, 167, 455, 912, 1263, 2535, 3514, 7099, 9873, 20043, 27956, 56807, 79397, 161736, 226559, 462482, 649100, 1327165, 1865833, 3820605, 5379507, 11028753, 15550459, 31913892, 45057416, 92557088, 130837407, 268988726

Offset: 1

Robert A. Russell, Aug 02 2023

nonn

These are polyominoes of the regular tiling with Schläfli symbol {3,6}. Their center is a vertex. As their symmetry group of order 12 is maximal, we can consider them either fixed, oriented (one-sided), or unoriented (free).

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                /\         /\  /\  /\        /\  /\  /\
    ____   ____/__\____   /__\/__\/__\      /__\/__\/__\
   /\  /\  \  /\  /\  /  /\  /    \  /\    /\  /\  /\  /\
  /__\/__\  \/__\/__\/  /__\/      \/__\  /__\/__\/__\/__\
  \  /\  /  /\  /\  /\  \  /\      /\  /  \  /\  /\  /\  /
   \/__\/  /__\/__\/__\  \/__\____/__\/    \/__\/__\/__\/
               \  /       \  /\  /\  /      \  /\  /\  /
                \/         \/__\/__\/        \/__\/__\/

Robert A. Russell, Table of n, a(n) for n = 1..53

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A030223 (achiral), A001420 (fixed).

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

Johann Peters, Nov 23 2024

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.

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

Anthony J. Guttman (Ed.), Polygons, Polyominoes, and Polycubes. Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
N. Madras, A pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
N. Madras and G. Slade, The Self-Avoiding Walk. Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttman, On the number of hexagonal polyominoes. Theoretical Computer Science, 307 (2003), 433-453.

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.

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

Johann Peters, Nov 23 2024

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.

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

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

Anthony J. Guttman (Ed.), Polygons, Polyominoes, and Polycubes, Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
N. Madras, A pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
N. Madras and G. Slade, The Self-Avoiding Walk, Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttman, On the number of hexagonal polyominoes, Theoretical Computer Science, 307 (2003), 433-453.

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.

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

Johann Peters, Nov 25 2024

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.

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

Anthony J. Guttman (Ed.), Polygons, Polyominoes, and Polycubes. Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
N. Madras, A pattern theorem for lattice clustersA pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
N. Madras and G. Slade, The Self-Avoiding Walk. Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttman, On the number of hexagonal polyominoes. Theoretical Computer Science, 307 (2003), 433-453.

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.

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

Johann Peters, Dec 17 2024

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.

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

Anthony J. Guttman (Ed.), Polygons, Polyominoes, and Polycubes, Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
N. Madras, A pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
N. Madras and G. Slade, The Self-Avoiding Walk, Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttman, On the number of hexagonal polyominoes, Theoretical Computer Science, 307 (2003), 433-453.

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.

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

Johann Peters, Dec 17 2024

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.

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

Anthony J. Guttman (Ed.), Polygons, Polyominoes, and Polycubes, Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
N. Madras, A pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
N. Madras and G. Slade, The Self-Avoiding Walk, Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttman, On the number of hexagonal polyominoes, Theoretical Computer Science, 307 (2003), 433-453.

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.

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.

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

Johann Peters, Dec 17 2024

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.

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

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

Anthony J. Guttman (Ed.), Polygons, Polyominoes, and Polycubes, Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
N. Madras, A pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
N. Madras and G. Slade, The Self-Avoiding Walk, Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttman, On the number of hexagonal polyominoes. Theoretical Computer Science, 307 (2003), 433-453.

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.

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.

cp's OEIS Frontend

A364684 Number of achiral triangular polyominoes with 6n cells and sixfold rotational symmetry.

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

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

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

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

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

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

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