A007239 -id:A007239 - cp's OEIS frontend

A003119 High-temperature series for spin-1/2 Ising magnetic susceptibility on diamond structure.

1, 4, 12, 36, 108, 324, 948, 2772, 8076, 23508, 67980, 196548, 566820, 1633956, 4697412, 13501492, 38742652, 111146820, 318390684, 911904996, 2608952940, 7463042916, 21328259716

Offset: 0

N. J. A. Sloane

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 381.
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)
J. W. Essam and M. F. Sykes, The crystal statistics of the diamond lattice, Physica, 29 (1963), 378-388.
D. S. Gaunt and M. F. Sykes, Estimation of Critical Indices for the Three-Dimensional Ising Model, J. Phys. A 6 (1973), 1517-1526.
Peter Young, Coefficients in the series expansions.

Cf. low-temperature: A003220 (ferromagnetic), A007216 (antiferromagnetic); A002913 (cubic structure); A002923 (Heisenberg model).

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

Name clarified by Andrey Zabolotskiy, Nov 14 2024

A003220 Low-temperature series for spin-1/2 Ising ferromagnetic susceptibility on diamond.

Original entry on oeis.org

0, 0, 1, 8, 44, 208, 984, 4584, 21314, 98292, 448850, 2038968, 9220346, 41545564, 186796388, 838623100

Offset: 0

N. J. A. Sloane, 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. [has a typo in a(15)]
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)
M. F. Sykes, D. S. Gaunt, J. W. Essam and C. J. Elliott, Derivation of low-temperature expansions for Ising model. VI. Three-dimensional lattices-temperature grouping, J. Phys. A: Math. Nucl. Gen., 6 (1973), 1507-1516.

Cf. A002924 (f.c.c.), A002925 (b.c.c.), A002926 (cubic); A007216 (antiferromagnetic), A002930 (magnetization), A003119 (high-temperature).

Name clarified, a(15) corrected by Andrey Zabolotskiy, Nov 14 2024

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

A007214 Low temperature antiferromagnetic susceptibility for honeycomb lattice.

Original entry on oeis.org

1, 0, 3, 2, 12, 24, 80, 222, 687, 2096, 6585, 20892, 67216, 218412, 716304, 2367696, 7881381, 26398728, 88918451, 301014786, 1023681792, 3495786472, 11983194420, 41220422022, 142247592803, 492339048156, 1708741395441, 5945617039980, 20737219258440, 72488490389592

Offset: 0

Simon Plouffe

nonn

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).
L. J. De Jongh (Ed.), Magnetic Properties of Layered Transition Metal Compounds, Kluwer Academic, Dordrecht, 1989, page 175. [Gives incorrect a(14)-a(18): they actually belong to A002978, i.e. differ by a factor of 4.]

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

Cf. A002978.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 07 2010

a(14)-a(18) corrected and more terms added by Andrey Zabolotskiy, Mar 03 2021

A007215 Low temperature antiferromagnetic susceptibility for square lattice.

Original entry on oeis.org

1, 0, 4, 8, 39, 152, 672, 3016, 13989, 66664

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)

A007217 Low temperature antiferromagnetic susceptibility for cubic lattice.

Original entry on oeis.org

1, 0, 0, -2, 15, -36, 104, -312, 1050, -3312, 10734, -34518, 113210, -370236, 1220922, -4028696, 13364424, -44409312

Offset: 0

Simon Plouffe

sign

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)

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)

cp's OEIS Frontend

A003119 High-temperature series for spin-1/2 Ising magnetic susceptibility on diamond structure.

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A003220 Low-temperature series for spin-1/2 Ising ferromagnetic susceptibility on diamond.

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A007206 Magnetization for honeycomb lattice.

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A007207 Magnetization for hexagonal lattice.

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A007214 Low temperature antiferromagnetic susceptibility for honeycomb lattice.

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A007215 Low temperature antiferromagnetic susceptibility for square lattice.

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A007217 Low temperature antiferromagnetic susceptibility for cubic lattice.

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A002911 Erroneous version of A047709.

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