A047709 -id:A047709 - cp's OEIS frontend

A002911 Erroneous version of A047709.

Original entry on oeis.org

0, 0, 1, 0, 12, 4, 129, 122, 1332, 960, 10919, 11372, 132900, 126396, 1299851, 1349784

Offset: 2

dead

Previous name was: Susceptibility for hexagonal lattice.

The terms 122 and 10919 are incorrect (not supported by the work cited by the source of this sequence).

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)

A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

Original entry on oeis.org

0, 0, 1, 8, 60, 416, 2791, 18296, 118016, 752008, 4746341, 29727472, 185016612, 1145415208, 7059265827, 43338407712, 265168691392, 1617656173824, 9842665771649, 59748291677832, 361933688520940, 2188328005246304, 13208464812265559, 79600379336505560, 479025509574159232

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

The zero-field susceptibility per spin is 4m^2/kT * Sum_{n >= 0} a(n) * u^n, where u = exp(-4J/kT). (m is the magnetic moment of a single spin; this factor may be present or absent depending on the precise definition of the susceptibility.) The b-file has been obtained from the series by Guttmann and Jensen via the substitution r = u/(1-u)^2 and dividing by 4. - Andrey Zabolotskiy, Feb 11 2022

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1305
R. J. Baxter and I. G. Enting, Series expansions for corner transfer matrices: the square lattice Ising model, J. Stat. Physics 21 (1979) 103-123.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
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) 6987-7005.
J. W. Essam and M. E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys., 38 (1963), 802-812.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
Tony Guttmann, Homepage. See Numerical Data, Ising square lattice susceptibility series, Low temperature series.
Iwan Jensen, Series for the Ising model

Cf. A002906 (high-temperature), A002979 (antiferromagnetic susceptibility), A029872 (specific heat), A002928 (magnetization), A002890 (partition function), A047709 (hexagonal lattice), A002912 (honeycomb), A002926 (cubic lattice), A010115 (spin-1 Ising).

a(n) ~ c * n^(3/4) * (1 + sqrt(2))^(2*n), where c = 0.0187325517235678... - Vaclav Kotesovec, May 06 2024

Corrections and updates from Steven Finch

a(0) = a(1) = 0 prepended, terms a(20) and beyond added by Andrey Zabolotskiy, Feb 10 2022

A002920 High-temperature series in w = tanh(J/kT) for ferromagnetic susceptibility for the spin-1/2 Ising model on hexagonal lattice.

Original entry on oeis.org

1, 6, 30, 138, 606, 2586, 10818, 44574, 181542, 732678, 2935218, 11687202, 46296210, 182588850, 717395262, 2809372302, 10969820358, 42724062966, 166015496838, 643768299018, 2491738141314, 9628130289018, 37146098272266, 143110933254702, 550643544948090

Offset: 0

N. J. A. Sloane

Previous name was: Susceptibility series for hexagonal lattice.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
The actual susceptibility per spin is this series times m^2/kT. (m is the magnetic moment of a single spin; this factor may be present or absent depending on the precise definition of the susceptibility.)

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 380.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Y. Chan, A. J. Guttmann, B. G. Nickel, and J. H. H. Perk, The Ising Susceptibility Scaling Function, J Stat Phys 145 (2011), 549-590; arXiv:1012.5272 [cond-mat.stat-mech], 2010-2020. Gives 320 terms in the file Triangle_v319.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Michael E. Fisher, Transformations of Ising Models, Phys. Rev. 113 (1959), 969-981.
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.
M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, I. Two dimensional lattices, J. Phys. A 5 (1972) 624-639.

Cf. A002910, A128834, A047709, A002919.

G.f.: (h(v(w)) + h(-v(w))) / 2, where h(v) is the g.f. of A002910 and v(w)^2 = w*(1+w)/(1+w^3) [Fisher, p. 979]. - Andrey Zabolotskiy, Mar 01 2021

Edited and extended from Chan et al by Andrey Zabolotskiy, Mar 03 2021

A002978 Low-temperature series in y = exp(2J/kT) for antiferromagnetic susceptibility for the Ising model on honeycomb structure.

Original entry on oeis.org

0, 0, 4, 0, 12, 8, 48, 96, 320, 888, 2748, 8384, 26340, 83568, 268864, 873648, 2865216, 9470784, 31525524, 105594912, 355673804, 1204059144, 4094727168, 13983145888, 47932777680, 164881688088, 568990371212, 1969356192624, 6834965581764, 23782468159920

Offset: 1

N. J. A. Sloane

Previous name was: Susceptibility series for honeycomb.

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

Y. Chan, A. J. Guttmann, B. G. Nickel, J. H. H. Perk, The Ising Susceptibility Scaling Function, J Stat Phys 145 (2011), 549-590; arXiv:1012.5272 [cond-mat.stat-mech], 2010-2020. Gives 641 term in the file Honeycomb_y641_af_.txt.
M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.
M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, I. Two dimensional lattices, J. Phys. A 5 (1972) 624-639.

Cf. A002910, A002912, A007214-A007218, A057391, A057395, A047709, A128834.

From Andrey Zabolotskiy, Mar 03 2021: (Start)
a(n) = 4*A007214(n-3).
G.f.: 8*t(u(y)) - 4*h(y), where t(u) is the g.f. of A047709, h(y) is the g.f. of A002912, and u(y) = y/(1-y+y^2) [Sykes & Fisher, p. 934-935]. (End)

New name from and more terms from Chan et al added by Andrey Zabolotskiy, Mar 03 2021

A005399 E.g.f.: high-temperature series in J/2kT for ferromagnetic susceptibility for the spin-1/2 Heisenberg model on hexagonal lattice.

Original entry on oeis.org

1, 6, 48, 408, 3600, 42336, 781728, 13646016, 90893568, -1798204416, 70794720768, 7538546211840, 63813109782528, -12977417912045568, -320549902414196736, 33016479733605777408, 1709506241695601983488

Offset: 0

N. J. A. Sloane

Previous name was: Susceptibility series for hexagonal lattice.

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).

G. A. Baker Jr., H. E. Gilbert, J. Eve, and G. S. Rushbrooke, On the two-dimensional, spin-1/2 Heisenberg ferromagnetic models, Phys. Lett., 25A (1967), 207-209.
N. Elstner, R. R. P. Singh and A. P. Young, Finite temperature properties of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice, Phys. Rev. Lett., 71 (1993), 1629-1632.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
J. Oitmaa and E. Bornilla, High-temperature-series study of the spin-1/2 Heisenberg ferromagnet, Phys. Rev. B, 53 (1996), 14228.
Laurent Pierre, Bernard Bernu and Laura Messio, High temperature series expansions of S = 1/2 Heisenberg spin models: Algorithm to include the magnetic field with optimized complexity, SciPost Phys. 17, 105 (2024); arXiv:2404.02271 [cond-mat.str-el], 2024. See the supporting file Triangle_16_16.py; multiply pol1[1] by 2 to get this sequence.

Cf. A002920 (Ising high-temperature), A047709 (Ising low-temperature), A005400 (series for specific heat, or free energy), A005401 (square lattice), A005402 (specific heat for square lattice).

New name from Andrey Zabolotskiy, Mar 03 2021
a(10)-a(12) added from Oitmaa and Bornilla by Andrey Zabolotskiy, Oct 20 2021
a(0) and a(13) using data from Elstner et al. (see Table I for the values -(-1)^n*n*a(n-1)) added by Andrey Zabolotskiy, Jun 17 2022
a(14)-a(16) using Pierre, Bernu & Messio's data added by Andrey Zabolotskiy, Nov 25 2024

A002919 High-temperature series for susceptibility for the spin-1/2 Ising model on hexagonal lattice.

Original entry on oeis.org

1, 6, 24, 90, 318, 1098, 3696, 12270, 40224, 130650, 421176, 1348998, 4299018, 13635630, 43092888, 135698970, 426144654, 1334488074, 4170038328, 13001153910, 40464412482, 125706293478, 389962873920, 1207855307874, 3736709089176, 11544946664622, 35633199126576

Offset: 0

N. J. A. Sloane

Previous name was: Susceptibility for hexagonal lattice.

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, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, I. Two dimensional lattices, J. Phys. A 5 (1972) 624-639.

Cf. A002910, A002920, A047709.

a(n) = A002910(2*n), cf. A002920. - Andrey Zabolotskiy, Mar 01 2021

New name and more terms using A002920 from Andrey Zabolotskiy, Mar 03 2021

cp's OEIS Frontend

A002911 Erroneous version of A047709.

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A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

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A002920 High-temperature series in w = tanh(J/kT) for ferromagnetic susceptibility for the spin-1/2 Ising model on hexagonal lattice.

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A002978 Low-temperature series in y = exp(2J/kT) for antiferromagnetic susceptibility for the Ising model on honeycomb structure.

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A005399 E.g.f.: high-temperature series in J/2kT for ferromagnetic susceptibility for the spin-1/2 Heisenberg model on hexagonal lattice.

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A002919 High-temperature series for susceptibility for the spin-1/2 Ising model on hexagonal lattice.

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