A002918 -id:A002918 - cp's OEIS frontend

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

A002921 Susceptibility series for f.c.c. lattice.

Original entry on oeis.org

1, 12, 132, 1404, 14652, 151116, 1546332, 15734460, 159425580, 1609987708, 16215457188, 162961837500, 1634743178420, 16373484437340, 163778159931180, 1636328839130860

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, 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, Self-avoiding walks and the Ising and Heisenberg models, Solid State Physics = Journal of Physics C, 3 (1970) 256-284.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
D. S. Gaunt, High Temperature Series Analysis for the Three-Dimensional Ising Model: A Review of Some Recent Work, in: Phase Transitions: Cargèse 1980, eds. Maurice Lévy, Jean-Claude Le Guillou and Jean Zinn-Justin, Springer, Boston, MA, 1982.
Sati McKenzie, High-temperature reduced susceptibility of the Ising model, J. Phys. A: Math. Gen., 8 (1975), L102-L105.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
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.
Index entries for sequences related to f.c.c. lattice

Cf. A002913 (cubic), A002914 (b.c.c.); A002918 (specific heat).

Cf. Heisenberg model: A002166; A002170 (cubic), A002168 (b.c.c.); A002165 (specific heat).

a(13)-a(15) from McKenzie added by Andrey Zabolotskiy, Feb 05 2022

A047712 High-temperature coefficients for internal energy for spin-1/2 Ising model on f.c.c. lattice.

Original entry on oeis.org

6, 24, 132, 816, 5448, 38808, 290568, 2255232, 17981532, 146428728, 1213014960, 10192820592, 86687830596, 744919762584

Offset: 1

N. J. A. Sloane

M. E. Fisher and M. F. Sykes, Antiferromagnetic susceptibilities of the simple cubic and body-centered cubic Ising lattices, Physica, 28 (1962), 939-956.
S. McKenzie, Extended high-temperature series expansions for the spin-s Ising model, J. Phys. A: Math. Gen., 16 (1983), 2875-2880. See table 1, column c(n); divide by 2 to get a(n).
M. F. Sykes, J. L. Martin and D. L. Hunter, Specific heat of a three-dimensional Ising ferromagnet above the Curie temperature, Proc. Phys. Soc., 91 (1967), 671-677.
Index entries for sequences related to f.c.c. lattice

Cf. A010571 (cubic), A047711 (b.c.c.), A001407 (partition function), A002918 (specific heat).

G.f. 6*v + 24*v^2 + 132*v^3 + 816*v^4 ... = v*q/2 + (1-v^2) * f'(v) / f(v), where f(v) = 1 + 8*v^3 + 33*v^4 + ... is the g.f. of A001407 and q = 12 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 14 2022

a(9)-a(11) from Sykes, Martin & Hunter added by Andrey Zabolotskiy, Feb 05 2022
Name clarified and a(12)-a(13) added by Andrey Zabolotskiy, Feb 14 2022
a(14) from McKenzie added by Andrey Zabolotskiy, Jan 18 2023

A002917 High temperature series for spin-1/2 Ising specific heat on 3-dimensional b.c.c. lattice.

Original entry on oeis.org

4, 140, 4056, 129360, 4381848, 153700408, 5519859080, 201714989064

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).

G. A. Baker, Further application of the Padé approximant method to the Ising and Heisenberg models, Phys. Rev. 129 (1963) 99-102.
Index entries for sequences related to b.c.c. lattice
Index entries for sequences related to specific heat

Cf. A002916 (cubic), A002918 (f.c.c.), A001406 (partition function), A047711 (internal energy), A002914 (susceptibility), A002167 (Heisenberg model).

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} A001406(n) * v^(2*n) and q = 8 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 15 2022

Better description from Steven Finch

a(5)-a(7) from Andrey Zabolotskiy, Feb 15 2022

cp's OEIS Frontend

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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A002921 Susceptibility series for f.c.c. lattice.

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A047712 High-temperature coefficients for internal energy for spin-1/2 Ising model on f.c.c. lattice.

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A002917 High temperature series for spin-1/2 Ising specific heat on 3-dimensional b.c.c. lattice.

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