A002909 Low temperature energy function for square lattice.

2, 0, -8, -24, -72, -240, -896, -3640, -15688, -70512, -326968, -1553288, -7523520, -37026704, -184677536, -931655064, -4746324296, -24387839056, -126257024696, -658011767016, -3449826712952, -18183760406080, -96309365029424, -512340286827272

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

References

• 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

• Robert Israel, Table of n, a(n) for n = 0..1000
• M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.

Programs

Maple

u:=v->((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):
S:= series(u((1-v)/(1+v))/((1-v)/(1+v))^2,v,101):
seq(coeff(S,v,j),j=0..100,2); # Sean A. Irvine, Nov 27 2017

Mathematica

Table[SeriesCoefficient[(1 + v)/(1 - v)^3 ((1 - v)^2 + 2/Pi (1 - 6 v + v^2) EllipticK[(16 v^2)/(1 - v)^4]), {v, 0, k}], {k, 0, 100}] (* Jan Mangaldan, Nov 28 2020 *)

Formula

G.f.: (1+x)/(1-x) + ((1-6*x+x^2)/(1-x^2))*Sum_{k>=0} (2*k)!^2 * (x*(1-x)^2/(1+x)^4)^k/k!^4. - Robert Israel, Nov 27 2017

a(n) ~ -2 * (1 + sqrt(2))^(2*n) / (Pi*n^2). - Vaclav Kotesovec, Nov 28 2017

Extensions

More terms from Sean A. Irvine, Nov 27 2017