A002908 -id:A002908 - cp's OEIS frontend

A007239 Energy function for hexagonal lattice.

3, 6, 12, 24, 54, 138, 378, 1080, 3186, 9642, 29784, 93552, 297966, 960294, 3126408, 10268688, 33989388, 113277582, 379833906, 1280618784, 4339003044, 14767407522, 50464951224, 173099580168, 595786322292, 2057106617226, 7123467773790, 24734460619704

Offset: 1

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.

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 386.
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)
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.

Cf. A002908.

See Eq. (32) of Sykes for the g.f. U(v). - Andrey Zabolotskiy, Feb 14 2022

Terms a(21) and beyond from Andrey Zabolotskiy, Feb 14 2022

A002906 High temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

Original entry on oeis.org

1, 4, 12, 36, 100, 276, 740, 1972, 5172, 13492, 34876, 89764, 229628, 585508, 1486308, 3763460, 9497380, 23918708, 60080156, 150660388, 377009364, 942106116, 2350157268, 5855734740, 14569318492, 36212402548, 89896870204

Offset: 0

N. J. A. Sloane

The zero-field susceptibility per spin is m^2/kT * Sum_{n >= 0} a(n) * v^n, where v = tanh(J/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 t = v/(1-v^2). - 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. 380.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.
B. G. Nickel, personal communication.
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..2043 (terms up to n = 116 from Fred Hucht)
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
S. Gartenhaus and W. S. McCullough, Higher order corrections for the quadratic Ising lattice susceptibility at criticality, Phys. Rev. B 38 (1988) 11688-11703.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)
Tony Guttmann, Homepage. See Numerical Data, Ising square lattice susceptibility series, High temperature series.
Iwan Jensen, Series for the Ising model
B. Nickel, On the singularity structure of the 2D Ising model susceptibility, Journal of Physics A, Math. Gen. 32, 3889 (1999); Addendum, 33, 1693 (2000).
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.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
Peter Young, Coefficients in the series expansions

Cf. A002927 (low-temperature), A002908 (energy), A002920 (hexagonal lattice), A002910 (honeycomb), A002913 (cubic lattice), A005401 (Heisenberg).

a(n) ~ c * n^(3/4) * (1 + sqrt(2))^n, where c = 0.839697019... - Vaclav Kotesovec, May 04 2024

Corrections and updates from Steven Finch

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 01 2008

A010571 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on cubic lattice with nearest-neighbor interaction J at temperature T.

Original entry on oeis.org

3, 12, 120, 1368, 18300, 268728, 4179852, 67767744, 1133826324, 19443072084, 340085761968, 6046276240668, 108970501777080, 1986820814551056, 36587507853481908, 679619087721892176, 12720247240214281860, 239685390231729125004, 4543441582487318876664

Offset: 1

N. J. A. Sloane

M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev., 133 (1964), A224-A239. See Table II for correct a(1)-a(6) but incorrect a(7).
M. E. Fisher and M. F. Sykes, Antiferromagnetic susceptibilities of the simple cubic and body-centered cubic Ising lattices, Physica, 28 (1962), 939-956.

Cf. A001393, A002908, A007239, A010572-A010574, A047711, A047712.

Sum_{n>=1} a(n) * v^(2*n-1) = v*q/2 + (1-v^2) * f'(v) / f(v), where f(v) = Sum_{n>=0} A001393(n) * v^(2*n) and q = 6 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 14 2022

New name from Andrey Zabolotskiy, Jan 14 2019

a(7) corrected and more terms added by Andrey Zabolotskiy, Feb 14 2022

A370953 Numerators of coefficients of the partition function per spin, lambda (divided by 2), in the very high temperature region, expressed as a power series in the parameter K^2, for the spin-1/2 Ising model on square lattice.

Original entry on oeis.org

1, 1, 4, 77, 1009, 101627, 1302779, 2513121979, 11291682179, 1354947005798, 23064317580681848, 20189102649892270054, 776220757551441546419, 641273428219629914673014, 5433381672262390009892530636, 1399751922597075578762073697769

Offset: 0

N. J. A. Sloane, Mar 10 2024

Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60 (1941), 252-262.
Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev. 60 (1941), 263-276. See (41), p. 263.
Hendrik A. Kramers and Gregory H. Wannier, Extract from page 263 of Part II.
Gandhimohan M. Viswanathan, The hypergeometric series for the partition function of the 2D Ising model, J. Stat. Mech. (2015) P07004; arXiv:1411.2495 [cond-mat.stat-mech], 2014-2015.
Wikipedia, Square lattice Ising model.

See A370954 for denominators.

Cf. A370955, A002908, A002890.

CoefficientList[With[{nmax = 7}, Exp[-Log[2]/2 + 1/(2 Pi) Integrate[Log[Cosh[2k]^2 + Sqrt[Sinh[2k]^4 + 1 - 2 Sinh[2k]^2 Cos[2\[Theta]] + O[k]^(2nmax+1)]], {\[Theta], 0, Pi}] + O[k]^(2nmax+1)]], k][[;; ;; 2]] // Numerator (* Andrey Zabolotskiy, Mar 10 2024 *)
CoefficientList[Cosh[2k] Exp[-x HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16x] /. {x -> (Sinh[2k]/(2Cosh[2k]^2))^2}] + O[k]^32, k][[;; ;; 2]] // Numerator (* Andrey Zabolotskiy, Mar 13 2024, using the g. f. from Gandhimohan M. Viswanathan *)

a(n) / A370954(n) ~ c * 2^(2*n) / (n^3 * log(1 + sqrt(2))^(2*n)), where c = 0.15662885... - Vaclav Kotesovec, May 02 2024

Terms a(5) and beyond from Andrey Zabolotskiy, Mar 10 2024

A370954 Denominators of coefficients of the partition function per spin, lambda (divided by 2), in the very high temperature region, expressed as a power series in the parameter K^2, for the spin-1/2 Ising model on square lattice.

Original entry on oeis.org

1, 1, 3, 45, 315, 14175, 66825, 42567525, 58046625, 1993723875, 9280784638125, 2143861251406875, 21132346621010625, 4370553505709015625, 9086380738369043484375, 564653660170076273671875

Offset: 0

N. J. A. Sloane, Mar 10 2024

Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60 (1941), 252-262.
Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev. 60 (1941), 263-276. See (41), p. 263.
Hendrik A. Kramers and Gregory H. Wannier, Extract from page 263 of Part II.
Wikipedia, Square lattice Ising model.

See A370953 for numerators.

Cf. A370955, A002908.

Terms a(5) and beyond from Andrey Zabolotskiy, Mar 10 2024

A002907 High temperature series in v = tanh(J/kT) for residual correlation function (correction to susceptibility) for the spin-1/2 Ising model on square lattice.

Original entry on oeis.org

2, 2, 20, 38, 146, 368, 1070, 2824, 7680, 19996, 53024, 136350, 355254, 906254, 2331416, 5909810, 15067236, 37992680, 96210436, 241564514, 608469654, 1522388638, 3818281784, 9525139886, 23806217352, 59237754234, 147621207142, 366533832540, 911151508282

Offset: 0

N. J. A. Sloane

nonn

Previous name was: Susceptibility for square 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).

M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.

Cf. A002906, A002908.

G.f.: ((1-3*v)^2*xi(v) - (1-v)^2 + 2*v*u(v)) / (8*v^7*(1+v)^2), where xi(v) is the g.f. of A002906 and u(v) is the g.f. of A002908 (odd powers only!); the actual "residual correlation function" is the numerator of this expression [Sykes & Fisher]. - Andrey Zabolotskiy, Feb 28 2021

New name and terms a(10) and beyond from Andrey Zabolotskiy, Feb 28 2021

cp's OEIS Frontend

A007239 Energy function for hexagonal lattice.

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A002906 High temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

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A010571 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on cubic lattice with nearest-neighbor interaction J at temperature T.

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A370953 Numerators of coefficients of the partition function per spin, lambda (divided by 2), in the very high temperature region, expressed as a power series in the parameter K^2, for the spin-1/2 Ising model on square lattice.

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Formula

Extensions

A370954 Denominators of coefficients of the partition function per spin, lambda (divided by 2), in the very high temperature region, expressed as a power series in the parameter K^2, for the spin-1/2 Ising model on square lattice.

Views

Author

Keywords

Links

Crossrefs

Extensions

A002907 High temperature series in v = tanh(J/kT) for residual correlation function (correction to susceptibility) for the spin-1/2 Ising model on square lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions