login
This site is supported by donations to The OEIS Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002909 Low temperature energy function for square lattice.
(Formerly M0035 N0009)
3
2, 0, -8, -24, -72, -240, -896, -3640, -15688, -70512, -326968, -1553288, -7523520, -37026704, -184677536, -931655064, -4746324296, -24387839056, -126257024696, -658011767016, -3449826712952, -18183760406080, -96309365029424, -512340286827272 (list; graph; refs; listen; history; text; internal format)
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).

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..1000

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

CROSSREFS

Sequence in context: A319196 A241682 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 17 23:15 EST 2019. Contains 319251 sequences. (Running on oeis4.)