A383735 -id:A383735 - cp's OEIS frontend

A003203 Cluster series for square lattice.

1, 4, 12, 24, 52, 108, 224, 412, 844, 1528, 3152, 5036, 11984, 15040, 46512, 34788, 197612, 4036, 929368, -702592, 4847552, -7033956, 27903296, -54403996, 170579740

Offset: 0

N. J. A. Sloane

The word "cluster" here essentially means polyomino or animal. This sequence can be computed based on a calculation of the perimeter polynomials of polyominoes. In particular, if P_n(x) is the perimeter polynomial for all fixed polyominoes of size n, then this sequence is the coefficients of x in Sum_{k>=1} k^2 * x^k * P_k(1-x). - Sean A. Irvine, Aug 15 2020

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Adler, Series Expansions, Computers in Physics, 8 (1994), 287-295.
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen., 28 (1995), 891-904. See Table 3.
Sean A. Irvine, Java program (github).
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A: Math. Gen., 9 (1976), 87-95.
Index entries for sequences related to cluster series.

Cf. A001168, A003202 (triangular net), A003204 (honeycomb net), A003198 (bond percolation), A338210 (perimeter polynomials).

Rows 5, 8, and 9 of A383735.

a(11)-a(14) from Sean A. Irvine, Aug 15 2020

a(15)-a(24) added from Conway & Guttmann by Andrey Zabolotskiy, Feb 01 2022

A003198 Cluster series for bond percolation problem on square lattice.

Original entry on oeis.org

1, 6, 18, 48, 126, 300, 762, 1668, 4216, 8668, 21988, 43058, 110832, 202432, 561020, 875382, 2881286, 3501056

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Daboul, Amnon Aharony, and Dietrich Stauffer, Universality of amplitude combinations in two-dimensional percolation, J. Phys. A: Math. Gen., 33 (2000), 1113-1137. See Table 1.
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A: Math. Gen., 9 (1976), 87-95.
Index entries for sequences related to cluster series.
Index entries for sequences related to square lattice.

Cf. A003197 (hexagonal), A003199 (honeycomb), A003207 (cubic), A003203 (site percolation).

Row 6 of A383735.

Name clarified, a(12)-a(14) from Sykes & Glen added by Andrey Zabolotskiy, Feb 02 2022

a(15)-a(17) from Daboul, Aharony & Stauffer added by Andrey Zabolotskiy, Nov 14 2024

A003207 Cluster series for bond percolation problem on cubic lattice.

Original entry on oeis.org

1, 10, 50, 238, 1114, 4998, 22562, 98174, 434894, 1855346, 8125390, 34149330

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stephan Mertens, Lattice Animals.
Stephan Mertens and Cristopher Moore, Series expansion of the percolation threshold on hypercubic lattices, J. Phys. A: Math. Theor., 51 (2018), 475001; arXiv:1805.02701 [cond-mat.stat-mech], 2018. See Appendix B.
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
Index entries for sequences related to cluster series.

Cf. A003211 (site percolation), A003198 (square lattice).

Row 14 of A383735.

Name clarified, a(10)-a(11) computed from Mertens & Moore's data added by Andrey Zabolotskiy, Feb 02 2022

A003209 Cluster series for f.c.c. lattice.

Original entry on oeis.org

1, 12, 84, 504, 3012, 17142, 96228, 532028, 2918388, 15763956

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
Index entries for sequences related to cluster series.
Index entries for sequences related to f.c.c. lattice.

Row 33 of A383735.

a(6)-a(9) from Sean A. Irvine, Aug 19 2020

A003210 Cluster series for b.c.c. lattice.

Original entry on oeis.org

1, 8, 56, 248, 1232, 5690, 26636, 113552, 532736, 2207108, 10385062

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
Index entries for sequences related to b.c.c. lattice.
Index entries for sequences related to cluster series.

Row 35 of A383735.

a(8)-a(10) from Sean A. Irvine, Aug 19 2020

A003211 Cluster series for cubic lattice.

Original entry on oeis.org

1, 6, 30, 114, 438, 1542, 5754, 19574, 71958, 233574, 870666, 2696274, 10375770, 30198116, 122634404, 327024444, 1460721616, 3347244554, 17795165832

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Adler, Series Expansions, Computers in Physics, 8 (1994), 287-295.
S. Luther and S. Mertens, Counting lattice animals in high dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (9), 546-565; arXiv:1106.1078 [cond-mat.stat-mech], 2011. See Table 5.
Stephan Mertens, Lattice Animals
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
Index entries for sequences related to cluster series.

Cf. A003209 (f.c.c.), A003210 (b.c.c.), A003212 (diamond), A003207 (bond percolation).

Row 32 of A383735.

a(9)-a(12) from Sean A. Irvine, Aug 19 2020

a(13)-a(18) from Luther & Mertens added by Andrey Zabolotskiy, Feb 02 2022

A003201 Cluster series for site percolation problem on square matching lattice (square lattice with 1st and 2nd neighbors connected).

Original entry on oeis.org

1, 8, 32, 108, 348, 1068, 3180, 9216, 26452, 73708, 206872, 563200, 1555460, 4124568, 11450284

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58 (1990) 1095-1108 (Table II, column nnSquare).
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
M. F. Sykes and Sylvia Flesia, Lattice animals: Supplementation of perimeter polynomial data by graph-theoretic methods, Journal of Statistical Physics, 63 (1991), 487-489.
Index entries for sequences related to cluster series.

Cf. cluster series for site percolation problem: A003200, A003202, A003203, A003204, A003209, A003210, A003211, A003212, A036392, A036394-A036402 and for bond percolation problem: A003197, A003198, A003199, A003205, A003206, A003207, A003208.

Row 10 of A383735.

Name clarified by Andrey Zabolotskiy, Mar 04 2021
a(8)-a(13) from Mertens added by Andrey Zabolotskiy, Feb 01 2022
a(14) from Sykes & Flesia added by Andrey Zabolotskiy, Jan 28 2023

A036402 Coefficients of cluster series for site percolation problem on simple cubic lattice with 1st, 2nd and 3rd neighbor bonds.

Original entry on oeis.org

1, 26, 386, 4886, 59534

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.

Index entries for sequences related to cluster series.

Cf. A003200, A003203.

Row 38 of A383735.

Name clarified by Andrey Zabolotskiy, Mar 04 2021

a(0) from Pontus von Brömssen, May 21 2025

A036396 Coefficients of cluster series for site percolation problem on simple cubic lattice with 1st and 2nd neighbor bonds.

Original entry on oeis.org

1, 18, 186, 1638, 14398, 117744

Offset: 0

N. J. A. Sloane

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.

Index entries for sequences related to cluster series.

Cf. A003200, A003203.

Row 34 of A383735.

Name clarified by Andrey Zabolotskiy, Mar 04 2021

a(0) from Pontus von Brömssen, May 21 2025

A383736 Cluster series for percolation on polyominoid cells.

Original entry on oeis.org

1, 12, 92, 604, 3732, 22766, 136564

Offset: 0

Pontus von Brömssen, May 10 2025

The coordination sequence for polyominoid cells (row 18 of A366768) is A005914, except that the term for distance 1 should be 12.

Index entries for sequences related to cluster series.

Row 18 of A383735.

Cf. A005914, A075678, A075679, A366768, A383737.

cp's OEIS Frontend

A003203 Cluster series for square lattice.

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A003198 Cluster series for bond percolation problem on square lattice.

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A003207 Cluster series for bond percolation problem on cubic lattice.

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A003209 Cluster series for f.c.c. lattice.

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A003210 Cluster series for b.c.c. lattice.

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A003211 Cluster series for cubic lattice.

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A003201 Cluster series for site percolation problem on square matching lattice (square lattice with 1st and 2nd neighbors connected).

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A036402 Coefficients of cluster series for site percolation problem on simple cubic lattice with 1st, 2nd and 3rd neighbor bonds.

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A036396 Coefficients of cluster series for site percolation problem on simple cubic lattice with 1st and 2nd neighbor bonds.

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A383736 Cluster series for percolation on polyominoid cells.

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