A003198 -id:A003198 - 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

A003200 Cluster series for site percolation problem on honeycomb matching lattice (honeycomb structure with 1st, 2nd and 3rd neighbors connected).

Original entry on oeis.org

1, 12, 66, 312, 1368, 5685, 23034, 90288, 350124

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).

Keith Malcolm Gwilym, Theory of percolation processes, PhD Thesis, University of London, 1972. See Table 3.1 on p. 55.
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.

Cf. cluster series for site percolation problem: A003201, A003202, A003203, A003204, A003209, A003210, A003211, A003212, A036392, A036394, A036395, A036396, A036397, A036398, A036400, A036401, A036402 and for bond percolation problem: A003197, A003198, A003199, A003205, A003206, A003207, A003208.

Name clarified by Andrey Zabolotskiy, Mar 04 2021

a(6)-a(8) from Gwilym added by Andrey Zabolotskiy, Apr 13 2023

A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 0, 1, 0, 2, 0, 2, 1, 0, 2, 0, 2, 4, 1, 0, 2, 0, 2, 12, 6, 1, 0, 2, 0, 2, 24, 18, 0, 1, 0, 2, 0, 2, 52, 48, 0, 4, 1, 0, 2, 0, 2, 108, 126, 0, 12, 4, 1, 0, 2, 0, 2, 224, 300, 0, 24, 12, 8, 1, 0, 2, 0, 2, 412, 762, 0, 52, 24, 32, 0, 1

Offset: 1

Pontus von Brömssen, May 10 2025

T(n,k) is the coefficient of p^(k+1), k >= 0, in the power series expansion of the expected finite size of the cluster containing a given cell for percolation with probability p on the polyominoid cells corresponding to row n of A366766. If the given cell is not open, its cluster is empty. Equivalently, T(n,k) can be taken to be the coefficient of p^k if we condition on the event that the given cell is open.

See A366766 for details on how the polyominoids are specified and on the ordering of the rows.

			Array begins:
  n\k| 0  1  2   3   4    5    6    7     8     9     10     11      12
  ---+-----------------------------------------------------------------
   1 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   2 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   3 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   4 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   5 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   6 | 1  6 18  48 126  300  762 1668  4216  8668  21988  43058  110832
   7 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   8 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   9 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
  10 | 1  8 32 108 348 1068 3180 9216 26452 73708 206872 563200 1555460
  11 | 1  0  0   0   0    0    0    0     0     0      0      0       0
  12 | 1  2  2   2   2    2    2    2     2     2      2      2       2

Index entries for sequences related to cluster series.

Cf. A366766, A366767, A366768.
Rows include:
   n | sequence for row n
  ---+-------------------
| A000007
| A040000
| A000007
| A040000
| A003203
| A003198
| A000007
| A003203
| A003203
| A003201
| A000007
| A040000
| A383737
| A003207
| A000007
| A003203
| A383737
| A383736
| A003203
| A003201
  ...
| A000007
| A003211
| A003209
| A036396
| A003210
  ...
| A036402
| A000007
| A040000
  ...
| A000007
| A003203
  ...
| A003203
| A003201

T(n,k) = [p^k] Sum_P m^2*p^(m-1)*(1-p)^j / binomial(D,d) = Sum_P m^2*(-1)^(k-m+1)*binomial(j,k-m+1) / binomial(D,d), where the sum is over all fixed polyominoids P (corresponding to row n of A366766), m is the number of cells of P, and j is the number of cells that are not in P but are adjacent to a cell in P; d is the dimension of the cells and D is the dimension of the ambient space. It is sufficient to take the sums over those P that have at most k+1 cells.

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

A003197 Cluster series for bond percolation problem on hexagonal lattice.

Original entry on oeis.org

1, 10, 46, 186, 706, 2568, 9004, 30894, 103832, 343006, 1123770, 3623234, 11630150

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

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).

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
D. F. Styer, M. D. Edwards and E. A. Andrews, The size function in two-dimensional bond percolation: a series analysis, J. Phys. A: Math. Gen., 21 (1988), L1153-L1156. 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 A2 = hexagonal = triangular lattice

Cf. A003198 (square), A003199 (honeycomb), A003202 (site percolation).

Name clarified, a(10) from Sykes & Glen added by Andrey Zabolotskiy, Feb 02 2022

a(11)-a(12) from Styer, Edwards & Andrews added by Andrey Zabolotskiy, Nov 14 2024

A003199 Cluster series for bond percolation problem on honeycomb.

Original entry on oeis.org

1, 4, 8, 16, 32, 54, 100, 182, 328, 494, 984, 1572, 2656, 4212, 8162, 11176, 21704, 30994, 60548

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.
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 honeycomb

Cf. A003197 (hexagonal), A003198 (square), A003204 (site percolation).

Name clarified, a(15)-a(18) from Sykes & Glen 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

A370088 Decimal expansion of the two-dimensional backbone constant.

Original entry on oeis.org

3, 5, 6, 6, 6, 6, 8, 3, 6, 7, 1, 2, 8, 8, 9, 5, 8, 2, 8, 3, 7, 3, 0, 7, 3, 8, 1, 0, 0, 1, 2, 6, 6, 2, 6, 9, 9, 0, 3, 8, 7, 0, 1, 5, 3, 4, 0, 7, 6, 2, 4, 4, 1, 3, 9, 9, 0, 6, 0, 9, 7, 3, 7, 6, 3, 7, 3, 6, 1, 3, 8, 4, 2, 0, 8, 8, 8, 5, 5, 4, 8, 5, 1, 9, 6, 7, 2

Offset: 0

Charles R Greathouse IV, Feb 08 2024

This constant is the negative of the exponent of the growth rate of the probability of Bernoulli percolation on the 2-dimensional triangular lattice at criticality (p = 1/2). It is transcendental (Theorem 1.2 in Nolin, Qian, Sun, & Zhuang).

			0.35666683671288958283730738100126626990387015340762441399060973763736138420....

Pierre Nolin, Wei Qian, Xin Sun, and Zijie Zhuang, Backbone exponent for two-dimensional percolation, arXiv preprint (2023). arXiv:2309.05050 [math.PR], 2023-2024.
Leila Sloman, Maze proof establishes a 'backbone' for statistical mechanics, Quanta Magazine (2024).
Index entries for transcendental numbers

Cf. A003197, A003198, A003202, A006809.

t=sqrt(3)/4; u=2*Pi/3; solve(x=.3,.4, my(s=sqrt(12*x+1)); sin(s*u)+s*t)

This is the unique constant 1/4 < x < 2/3 with sqrt(36*x+3)/4 + sin(2*Pi*sqrt(12*x+1)/3) = 0.

cp's OEIS Frontend

A003203 Cluster series for square lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A003200 Cluster series for site percolation problem on honeycomb matching lattice (honeycomb structure with 1st, 2nd and 3rd neighbors connected).

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A003207 Cluster series for bond percolation problem on cubic lattice.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A003197 Cluster series for bond percolation problem on hexagonal lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A003199 Cluster series for bond percolation problem on honeycomb.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A370088 Decimal expansion of the two-dimensional backbone constant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula