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
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
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms and new name from Andrey Zabolotskiy, Oct 19 2017
STATUS
approved