A003203 -id:A003203 - cp's OEIS frontend

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

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

Offset: 0

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.

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

A003202 Cluster series for hexagonal lattice.

Original entry on oeis.org

1, 6, 18, 48, 126, 300, 750, 1686, 4074, 8868, 20892, 44634, 103392, 216348, 499908, 1017780, 2383596, 4648470, 11271102, 20763036, 52671018, 91377918

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).

S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58 (1990) 1095-1108.
Stephan Mertens and Markus E. Lautenbacher, Counting lattice animals: A parallel attack, J. Stat. Phys., 66 (1992), 669-678.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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.
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.

Cf. A003203 (square net), A003204 (honeycomb net), A003197 (bond percolation).

a(10)-a(11) from Sean A. Irvine, Aug 16 2020
a(12)-a(18) added from Mertens by Andrey Zabolotskiy, Feb 01 2022
a(19)-a(21) from Mertens & Lautenbacher added by Andrey Zabolotskiy, Jan 28 2023

A003204 Cluster series for honeycomb.

Original entry on oeis.org

1, 3, 6, 12, 24, 33, 60, 99, 156, 276, 438, 597, 1134, 1404, 2904, 3522, 6876, 7548, 16680, 18153, 39846, 41805

Offset: 0

N. J. A. Sloane

The word "cluster" here essentially means polyiamond. This sequence can be computed based on a calculation of the perimeter polynomials of polyiamonds. In particular, if P_n(x) is the perimeter polynomial for all fixed polyiamonds 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 16 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).

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.

Cf. A001420, A003202 (triangular net), A003203 (square net), A003199 (bond percolation).

a(12)-a(18) from Sean A. Irvine, Aug 16 2020

a(19)-a(21) added from Sykes & Glen by Andrey Zabolotskiy, Feb 01 2022

A338210 Triangle of coefficients of perimeter polynomials for fixed polyominoes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 8, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 28, 12, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 54, 80, 60, 16, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 136, 252, 228, 100, 20, 2

Offset: 0

Sean A. Irvine, Oct 16 2020

Considered as a triangle, T(n,k) is the number of polyominoes of n cells having a (cell) perimeter of k.

			Polynomials begin:
  1;
  x^4;
  2*x^6;
  4*x^7 + 2*x^8;
  9*x^8 + 8*x^9 + 2*x^10;
  ...

Sean A. Irvine, Table of n, a(n) for n = 0..437; rows 0..19 flattened
Sean A. Irvine, Java program (github)

Cf. A001168 (row sums), A338211 (free equivalent), A338212 (sprawl), A003203.

A001168(n) = Sum_{k=0..2*n+2} T(n,k).

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

A036392 Coefficients of cluster series for site percolation problem on triangular lattice with 1st and 2nd neighbor bonds.

Original entry on oeis.org

1, 12, 72, 342, 1632, 6780

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.

Name clarified by Andrey Zabolotskiy, Mar 04 2021

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

A036394 Coefficients of cluster series for site percolation problem on f.c.c. lattice with 1st and 2nd neighbor bonds.

Original entry on oeis.org

1, 18, 186, 1710, 14862, 124950

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.

Name clarified by Andrey Zabolotskiy, Mar 04 2021

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

cp's OEIS Frontend

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

A003198 Cluster series for bond percolation problem on square lattice.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A003202 Cluster series for hexagonal lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A003204 Cluster series for honeycomb.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A338210 Triangle of coefficients of perimeter polynomials for fixed polyominoes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

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

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A036392 Coefficients of cluster series for site percolation problem on triangular lattice with 1st and 2nd neighbor bonds.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A036394 Coefficients of cluster series for site percolation problem on f.c.c. lattice with 1st and 2nd neighbor bonds.

Views

Author