A002908 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 square lattice with nearest-neighbor interaction J at temperature T. (Formerly M1161 N0444)

2, 4, 8, 24, 84, 328, 1372, 6024, 27412, 128228, 613160, 2985116, 14751592, 73825416, 373488764, 1907334616, 9820757380, 50934592820, 265877371160, 1395907472968, 7366966846564, 39062802311672, 208015460898924, 1112050252939612, 5966352507546872

Offset

1,1

Comments

Previous name was: Energy function for square lattice.

References

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, 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).

Links

Table of n, a(n) for n=1..25.

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 D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.

Lars Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117 (1944).

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

Formula

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

Maple

series((1+v^2)*(1-(2/Pi)*(1-6*v^2+v^4)*EllipticK(4*v*(1-v^2)/(1+v^2)^2)/(1+v^2)^2)/2*v, v, 50); # Sean A. Irvine, Nov 26 2017

Mathematica

u[h_]:=Coth[2h](1+(2/Pi)(2Tanh[2h]^2-1)EllipticK[(2Sinh[2h]/Cosh[2h]^2)^2]);
Table[SeriesCoefficient[u[ArcTanh[v]], {v, 0, 2n-1}], {n, 10}]
(* Andrey Zabolotskiy, Sep 12 2017; see Onsager's eq. (116) *)
Rest[CoefficientList[Series[(1+x)/2 - (1 - 6*x + x^2)*EllipticK[(16*(-1 + x)^2*x)/(1 + x)^4] / (Pi*(1+x)), {x, 0, 25}], x]] (* Vaclav Kotesovec, Apr 27 2024 *)

Crossrefs

Cf. A002906-A002930, A010571, A010572, A010573, A010574.

Sequence in context: A179190 A291482 A065654 * A004528 A066535 A134455

Adjacent sequences: A002905 A002906 A002907 * A002909 A002910 A002911

Keyword

nonn,nice

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms and new name from Andrey Zabolotskiy, Oct 19 2017

Status

approved