A002909 Low temperature energy function for square lattice. (Formerly M0035 N0009)

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

0,1

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.

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

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 *)

Crossrefs

Sequence in context: A241682 A076687 A062134 * A118437 A134185 A013489

Adjacent sequences: A002906 A002907 A002908 * A002910 A002911 A002912

Keyword

sign

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 27 2017

Status

approved