A006809 -id:A006809 - cp's OEIS frontend

A006803 Percolation series for hexagonal lattice.

1, 0, 0, -1, 0, -3, 1, -9, 6, -29, 27, -99, 112, -351, 450, -1275, 1782, -4704, 6998, -17531, 27324, -65758, 106211, -247669, 411291, -935107, 1587391, -3535398, 6108103, -13373929, 23438144, -50592067, 89703467, -191306745, 342473589

Offset: 0

N. J. A. Sloane

sign

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Jensen, Table of n, a(n) for n = 0..51
J. Blease, Series expansions for the directed-bond percolation problem, J. Phys C vol 10 no 7 (1977), 917-924.
J. W. Essam, A. J. Guttmann and K. De'Bell, On two-dimensional directed percolation, J. Phys. A 21 (1988), 3815-3832.
I. Jensen, More terms
Iwan Jensen and Anthony J. Guttmann, Series expansions of the percolation probability for directed square and honeycomb lattices, arXiv:cond-mat/9509121, 1995; J. Phys. A 28 (1995), no. 17, 4813-4833.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A006809.

A006736 Series for first parallel moment of hexagonal lattice.

Original entry on oeis.org

0, 4, 24, 104, 384, 1284, 4012, 11924, 34100, 94584, 255852, 677850, 1764482, 4523924, 11447870, 28636218, 70907326, 173991368, 423469988, 1023162920, 2455645268, 5858183260, 13898041838, 32804047708, 77067740230

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Jensen, Table of n, a(n) for n = 0..90 (from link below)
J. W. Essam, A. J. Guttmann and K. De'Bell, On two-dimensional directed percolation, J. Phys. A 21 (1988), 3815-3832.
I. Jensen, More terms
Iwan Jensen, Anthony J. Guttmann, Series expansions of the percolation probability for directed square and honeycomb lattices, arXiv:cond-mat/9509121, 1995; J. Phys. A 28 (1995), no. 17, 4813-4833.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A006803, A006809, A006737.

A006737 Series for second parallel moment of hexagonal lattice.

Original entry on oeis.org

0, 6, 68, 442, 2218, 9528, 36834, 131856, 445000, 1433294, 4444006, 13349510, 39041224, 111583236, 312618368, 860662498, 2333112020, 6238124024, 16474149036, 43023953304, 111230237224, 284926172100, 723731637254

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Jensen, Table of n, a(n) for n = 0..90 (from link below)
I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions for the Spin-1 Ising Model, arXiv:hep-lat/9410005, 1994; J. Phys. A. 27 (1994) 6987-7006.
J. W. Essam, A. J. Guttmann and K. De'Bell, On two-dimensional directed percolation, J. Phys. A 21 (1988), 3815-3832.
I. Jensen, More terms
Iwan Jensen, Anthony J. Guttmann, Series expansions of the percolation probability for directed square and honeycomb lattices, arXiv:cond-mat/9509121, 1995; J. Phys. A 28 (1995), no. 17, 4813-4833.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A006803, A006809, A006736, A006738.

A006738 Series for second perpendicular moment of hexagonal lattice.

Original entry on oeis.org

0, 2, 12, 54, 206, 712, 2294, 7024, 20656, 58842, 163250, 443062, 1180156, 3092964, 7993116, 20401250, 51502616, 128748512, 319010540, 784179992, 1913668608, 4639155964, 11178566462, 26784974870, 63851541584

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Jensen, Table of n, a(n) for n = 0..90 (from link below)
I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions for the Spin-1 Ising Model, arXiv:hep-lat/9410005, 1994; J. Phys. A. 27 (1994) 6987-7006.
J. W. Essam, A. J. Guttmann and K. De'Bell, On two-dimensional directed percolation, J. Phys. A 21 (1988), 3815-3832.
I. Jensen, More terms
Iwan Jensen, Anthony J. Guttmann, Series expansions of the percolation probability for directed square and honeycomb lattices, arXiv:cond-mat/9509121, 1995; J. Phys. A 28 (1995), no. 17, 4813-4833.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A006803, A006809, A006736, A006737.

A006810 Bond percolation series for mean cluster size on directed cubic lattice.

Original entry on oeis.org

1, 3, 9, 27, 78, 225, 633, 1785, 4944, 13742, 37686, 103767, 282425, 772719

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Blease, Series expansions for the directed-bond percolation problem, J. Phys C vol 10 no 7 (1977), 917-924.
K. De'Bell and J. W. Essam, Directed percolation: series expansion for some three-dimensional lattices, J. Phys. A: Math. Gen., 16 (1983), 3553-3560.

Cf. different directed lattices: A006727 (square), A006809 (hexagonal), A006813 (hexagonal cyclic), A006811 (b.c.c.), A006812 (f.c.c.).

Cf. A006804 (percolation probability).

Name clarified and terms a(0), a(9)-a(13) added from De'Bell & Essam by Andrey Zabolotskiy, May 11 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

A006803 Percolation series for hexagonal lattice.

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A006736 Series for first parallel moment of hexagonal lattice.

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A006737 Series for second parallel moment of hexagonal lattice.

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A006738 Series for second perpendicular moment of hexagonal lattice.

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A006810 Bond percolation series for mean cluster size on directed cubic lattice.

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A370088 Decimal expansion of the two-dimensional backbone constant.

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