A003197 -id:A003197 - 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

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

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

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

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A003198 Cluster series for bond percolation problem on square lattice.

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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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A370088 Decimal expansion of the two-dimensional backbone constant.

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