A002928 -id:A002928 - cp's OEIS frontend

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

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

A010102 Spontaneous magnetization coefficients for square lattice spin 1 Ising model.

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, -4, 3, 0, -30, 48, -52, -120, 368, -612, -254, 2524, -6216, 4040, 11805, -49400, 68268, 14928, -332511, 734508, -568038, -1641320, 6202774, -9239676, -2503162, 42749908, -99021392, 72255812, 215763902

Offset: 0

N. J. A. Sloane, Simon Plouffe

sign

I. Jensen, Table of n, a(n) for n = 0..113 (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.
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.

Cf. A002928, A010103, A010104, A010105, A010106, A010107, A010111, A010115.

Link corrected by Ralf Stephan, Aug 04 2013

A136264 Expansion of g.f. (1+x)^2(x^2-6x+1)/(x-1)^4.

Original entry on oeis.org

1, 0, -16, -64, -160, -320, -560, -896, -1344, -1920, -2640, -3520, -4576, -5824, -7280, -8960, -10880, -13056, -15504, -18240, -21280, -24640, -28336, -32384, -36800, -41600, -46800, -52416, -58464, -64960, -71920, -79360, -87296, -95744, -104720, -114240, -124320, -134976, -146224, -158080

Offset: 0

Roger L. Bagula, Apr 07 2008

This g.f. is the eighth power of the spontaneous magnetization series for the two-dimensional square lattice in the parameter x = exp(-4J/kT), cf. A002928.

Terrel L. Hill, Statistical Mechanics: Principles and Selected Applications, Dover, New York, 1956, page 331. See eq. 44.12 for the g.f. with x replaced by x^2.

M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Essentially the same as A102860. Cf. A115046, A002928.

CoefficientList[Series[(1+x)^2(x^2-6x+1)/(x-1)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{1,0,-16,-64,-160},40] (* Harvey P. Dale, Mar 15 2020 *)

Vec((1+x)^2*(x^2-6*x+1)/(x-1)^4 + O(x^100)) \\ Altug Alkan, Oct 26 2015

a(n) = 8*n*(1 - n^2)/3, n>0. - R. J. Mathar, Mar 09 2009

E.g.f.: 1 - 8*exp(x)*x^2*(3 + x)/3. - Stefano Spezia, Oct 11 2023

A007206 Magnetization for honeycomb lattice.

Original entry on oeis.org

1, 0, 0, -2, -6, -18, -54, -168, -534, -1732, -5706, -19038, -64176, -218190, -747180, -2574488, -8918070, -31036560, -108457488, -380390574, -1338495492, -4723664566, -16714545822, -59286878556, -210755970528, -750721297056, -2679075662922, -9577156141654, -34290858526926, -122959225609518

Offset: 0

Simon Plouffe

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)
Shigeo Naya, On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices, Progress of Theoretical Physics, 11 (1954), 53-62.

Cf. A002928, A007207.

CoefficientList[Series[(1 - 16 * x^3 * (1+x^3) / ((1-x)^3 * (1-x^2)^3))^(1/8), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 27 2024 *)

G.f.: (1 - 16 * z^3 * (1+z^3) / ((1-z)^3 * (1-z^2)^3))^(1/8) [Shigeo Naya]. - Andrey Zabolotskiy, Jun 01 2022

a(n) ~ -Gamma(1/8) * sqrt(sqrt(2) - 1) * (2 + sqrt(3))^n / (2^(27/8) * 3^(1/16) * Pi * n^(9/8)). - Vaclav Kotesovec, Apr 27 2024

Offset changed, signs of terms changed, and more terms added by Andrey Zabolotskiy, Jun 01 2022

A007207 Magnetization for hexagonal lattice.

Original entry on oeis.org

1, 0, 0, -2, 0, -12, 2, -78, 24, -548, 228, -4050, 2030, -30960, 17670, -242402, 152520, -1932000, 1312844, -15612150, 11297052, -127551884, 97291026, -1051478274, 838994486, -8732657724, 7246304736, -72983051674, 62686156026, -613243234224, 543146222970

Offset: 0

Simon Plouffe

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. 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)
Shigeo Naya, On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices, Progress of Theoretical Physics, 11 (1954), 53-62.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A002928, A007206.

CoefficientList[Series[(1 - 16 * x^3 / ((1+3*x) * (1-x)^3))^(1/8), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 27 2024 *)

G.f.: (1 - 16 * x^3 / ((1+3*x) * (1-x)^3))^(1/8) [Shigeo Naya]. - Andrey Zabolotskiy, Jun 01 2022

a(n) ~ (-1)^n * 3^n / (Gamma(1/8) * 2^(1/4) * n^(7/8)) * (1 - (-1)^n * sqrt(sqrt(2) - 1) * Gamma(1/8)^2 / (2^(13/4) * Pi * n^(1/4))). - Vaclav Kotesovec, Apr 27 2024

Offset changed, signs of terms changed, and more terms added by Andrey Zabolotskiy, Jun 01 2022

cp's OEIS Frontend

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

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A010102 Spontaneous magnetization coefficients for square lattice spin 1 Ising model.

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A136264 Expansion of g.f. (1+x)^2*(x^2-6*x+1)/(x-1)^4.

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A007206 Magnetization for honeycomb lattice.

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A007207 Magnetization for hexagonal lattice.

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A136264 Expansion of g.f. (1+x)^2(x^2-6x+1)/(x-1)^4.