A001393 -id:A001393 - cp's OEIS frontend

A046743 Erroneous version of A001393.

1, 0, 3, 22, 192, 2070, 24943

Offset: 0

dead

A002913 High temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

1, 6, 30, 150, 726, 3510, 16710, 79494, 375174, 1769686, 8306862, 38975286, 182265822, 852063558, 3973784886, 18527532310, 86228667894, 401225368086, 1864308847838, 8660961643254, 40190947325670, 186475398518726, 864404776466406, 4006394107568934, 18554916271112254, 85923704942057238

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.
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..32 (terms a(24), a(25) taken from the Campostrini et al. 2002 article by _Per H. Lundow_, terms a(26)-a(32) taken from the Toshiaki Fujiwara and Hiroaki Arisue's slides)
P. Butera and M. Comi, N-vector spin models on the simple-cubic and the body-centered-cubic lattices: A study of the critical behavior of the susceptibility and of the correlation length by high-temperature series extended to order beta^21, Phys. Rev. B 56 (1997) 8212-8240; arXiv:hep-lat/9703018, 1997.
P. Butera and M. Comi, Extension to order b23 of the high-temperature expansions for the spin-1/2 Ising model on the simple-cubic and the body-centered-cubic lattices, BICOCCA/FT-00-09 (June 2000). Phys. Rev. B62 (2000) 14837-14843.
M. Campostrini, Linked-Cluster Expansion of the Ising Model, Journal of Statistical Physics, 103 (2001), 369-394.
M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice, Phys. Rev. E, 65 (2002), 66-127.
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.
Toshiaki Fujiwara and Hiroaki Arisue (presenter), ３次元イジング模型の高温展開 (High-temperature expansion for the 3D Ising model), Computational Physics with CP-PACS 2002 Workshop [in Japanese].
Toshiaki Fujiwara and Hiroaki Arisue (presenter), New algorithm of the high-temperature expansion for the Ising model in three dimensions, Asia-Pacific Mini-Workshop on Lattice QCD, Center for Computational Physics, University of Tsukuba, 2003: abstract, slides, source.
D. S. Gaunt, High Temperature Series Analysis for the Three-Dimensional Ising Model: A Review of Some Recent Work, pp. 217-246 in: Phase Transitions: Cargèse 1980, eds. Maurice Lévy, Jean-Claude Le Guillou and Jean Zinn-Justin, Springer, Boston, MA, 1982.
M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, II. Three dimensional lattices, J. Phys. A 5 (1972) 640-652.

Cf. other quantities: A001393 (partition function), A010571 (internal energy), A002916 (specific heat), A003490 (surface susceptibility), A007287 (layer susceptibility), A010040, A010043, A010046.
Cf. other structures: A002906 (square), A002920 (hexagonal), A002910 (honeycomb), A002914 (b.c.c.), A002921 (f.c.c.), A003119 (diamond), A010556 (4D cubic), A010579 (5D cubic), A010580 (6D cubic), A030008 (7D cubic).
Cf. low-temperature series: A002926 (ferromagnetic), A002915 (antiferromagnetic).
Cf. other models: A002170 (Heisenberg), A003279 (spherical).

Corrections and updates from Steven Finch
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 01 2008
Several errors in the sequence were corrected by Per H. Lundow, Jan 17 2011

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

A002891 Low temperature series for spin-1/2 Ising partition function on 3-dimensional simple cubic lattice.

Original entry on oeis.org

1, 0, 0, 1, 0, 3, -3, 15, -30, 101, -261, 807, -2308, 7065, -21171, 65337, -200934, 627249, -1962034, 6192066, -19610346, 62482527, -199807110, 641837193, -2068695927, 6691611633, -21710041944, 70645706963, -230488840446, 753903842400, -2471624380458, 8120879664294, -26736570257010

Offset: 0

N. J. A. Sloane, C. Vohwinkel

sign

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).

Daniel Andrén, Series expansion for the density of states of the Ising and Potts models, arXiv:0706.3116 [cond-mat.str-el], 2007.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
A. J. Guttmann and I. G. Enting, Series studies of the Potts model: I. The simple cubic Ising model, J. Phys. A 26 (1993) 807-821; arXiv:hep-lat/9212032, 1992.
A. J. Wakefield, Statistics of the simple cubic lattice, Proc. Cambridge Philos. Soc. 47 (1951) 419-435 and 799-810.

Cf. A002926 (ferromagnetic susceptibility), A002915 (antiferromagnetic susceptibility), A001393 (high-temperature), A002890 (square lattice), A002892 (f.c.c. lattice), A030045 (4D cubic), A030047 (5D cubic).

Corrections and updates from Steven Finch
"Free energy" changed back to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 12 2022
a(28)-a(32) added by Andrey Zabolotskiy, Jun 30 2022 using Andrén's data (see his Table 2, column a_n for the coefficients of the expansion of the logarithm of the g.f. of this sequence)

A001406 High temperature series for partition function for spin-1/2 Ising model on b.c.c. lattice.

Original entry on oeis.org

1, 0, 12, 148, 2568, 53944, 1283270, 33268932, 918420570

Offset: 0

N. J. A. Sloane

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).

C. Domb and M. F. Sykes, The calculation of lattice constants in crystal statistics, Phil. Mag., 2 (1957), 733ff.
M. F. Sykes, D. L. Hunter, D. S. McKenzie and B. R. Heap, Specific heat of a three dimensional Ising ferromagnet above the Curie temperature. II, J. Phys. A: Gen. Phys., 5 (1972), 667-673. The power-series part of Eq. (1.2) is the logarithm of the g.f. for this sequence (even powers only).
Index entries for sequences related to b.c.c. lattice

Cf. A001393, A001407.

a(6)-a(8) added, name clarified by Andrey Zabolotskiy, Feb 14 2022

A001407 High temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

Original entry on oeis.org

1, 0, 0, 8, 33, 168, 962, 5928, 38907, 268056, 1918938, 14169360, 107333498, 830660688, 6546655404, 52410001448

Offset: 0

N. J. A. Sloane

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).

C. Domb and M. F. Sykes, The calculation of lattice constants in crystal statistics, Phil. Mag., 2 (1957), 733ff.
M. F. Sykes, D. L. Hunter, D. S. McKenzie and B. R. Heap, Specific heat of a three dimensional Ising ferromagnet above the Curie temperature. II, J. Phys. A: Gen. Phys., 5 (1972), 667-673. The power-series part of Eq. (1.3) is the logarithm of the g.f. for this sequence.
Index entries for sequences related to f.c.c. lattice

Cf. A001393, A001406, A002892, A047712, A002918.

a(10)-a(14) added and name clarified by Andrey Zabolotskiy, Feb 14 2022

a(15) added using data from A047712 by Andrey Zabolotskiy, Jan 18 2023

A002916 High temperature series for spin-1/2 Ising specific heat on 3-dimensional simple cubic lattice.

Original entry on oeis.org

3, 33, 564, 8976, 155124, 2791308, 51382068, 962178084, 18258531348, 350143322088, 6772382631732, 131922552534036, 2585198190891636, 50919899448451512, 1007393565758096820, 20007153991627682124, 398699967207692643924, 7969220499183448073760, 159718349893920279061428

Offset: 0

N. J. A. Sloane

nonn

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).

G. A. Baker, Further application of the Padé approximant method to the Ising and Heisenberg models, Phys. Rev. 129 (1963) 99-102.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
A. J. Guttmann and I. G. Enting, The high-temperature specific heat exponent of the 3-dimensional Ising model, J. Phys. A 27 (1994) 8007-8010.
Index entries for sequences related to specific heat

Equals 3*A001408.

Cf. A002917 (b.c.c.), A002918 (f.c.c.), A001393 (partition function), A010571 (internal energy), A002913 (susceptibility), A002169 (Heisenberg model), A029872 (square, low-temperature).

3 Cases[Import["https://oeis.org/A001408/b001408.txt", "Table"], {, }][[All, 2]] (* Jean-François Alcover, Jan 17 2020 *)

Sum_{n>=0} a(n) * v^(2*n) = (v^2-1) * (-q/2*f(v)^2 - (v^2-1) * f'(v)^2 + f(v) * (2*v*f'(v) + (v^2-1)*f''(v))) / f(v)^2, where f(v) = Sum_{n>=0} A001393(n) * v^(2*n) and q = 6 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 15 2022

Corrections and updates from Steven Finch

Terms a(13) and beyond from Andrey Zabolotskiy, Feb 15 2022

A030048 High temperature series for spin-1/2 Ising partition function on 5D simple cubic lattice.

Original entry on oeis.org

1, 0, 10, 180, 5075, 180496

Offset: 0

Steven Finch

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

Daniel Andrén, Series expansion for the density of states of the Ising and Potts models, arXiv:0706.3116 [cond-mat.str-el], 2007.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
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 p. A226, in particular Eq. (2.11) (together with Eq. (4.10)), for the power series with coefficients g_n, which is the logarithm of this power series, and its physical and combinatorial interpretations.

Cf. A010579 (susceptibility), A010573 (internal energy), A001393 (3D cubic lattice), A030044 (4D cubic lattice).

"Free energy" corrected to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 12 2022

a(5) added by Andrey Zabolotskiy, Jun 30 2022 using Andrén's data (see his Table 4, column b_n^5 for the coefficients of the expansion of the logarithm of the g.f. of this sequence)

A030049 High temperature series for spin-1/2 Ising partition function on 6D simple cubic lattice.

Original entry on oeis.org

1, 0, 15, 350, 13380, 664566

Offset: 0

Steven Finch

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
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.

Cf. A010574 (internal energy), A010580 (susceptibility); A001393 (3D cubic lattice), A030044 (4D), A030048 (5D).

"Free energy" corrected to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 12 2022

a(5) added using data from A010574 by Andrey Zabolotskiy, Jan 20 2023

cp's OEIS Frontend

A046743 Erroneous version of A001393.

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A002913 High temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic 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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A002891 Low temperature series for spin-1/2 Ising partition function on 3-dimensional simple cubic lattice.

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A001406 High temperature series for partition function for spin-1/2 Ising model on b.c.c. lattice.

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A001407 High temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

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A002916 High temperature series for spin-1/2 Ising specific heat on 3-dimensional simple cubic lattice.

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A030048 High temperature series for spin-1/2 Ising partition function on 5D simple cubic lattice.

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A030049 High temperature series for spin-1/2 Ising partition function on 6D simple cubic lattice.

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