A003204 -id:A003204 - 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

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

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

cp's OEIS Frontend

A003203 Cluster series for square lattice.

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

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A003202 Cluster series for hexagonal lattice.

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A003199 Cluster series for bond percolation problem on honeycomb.

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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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