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.

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice

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

• 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.

Crossrefs

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

Programs

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 *)

Formula

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

Extensions

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