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A002908 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A002908 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A002908" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA002908">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A002908 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A002908 </div> <div class=seqname> 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. <br><font size=-1>(Formerly M1161 N0444)</font> </div> </div> <div class=scorerefs> 6 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>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</div> <div class=seqdatalinks> (<a href="/A002908/list">list</a>; <a href="/A002908/graph">graph</a>; <a href="/search?q=A002908+-id:A002908">refs</a>; <a href="/A002908/listen">listen</a>; <a href="/history?seq=A002908">history</a>; <a href="/search?q=id:A002908&fmt=text">text</a>; <a href="/A002908/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Previous name was: Energy function for square lattice.</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 386.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A002908/b002908.txt">Table of n, a(n) for n=1..25.</a></div> <div class=sectline>C. Domb, <a href="/A007239/a007239.pdf">Ising model</a>, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)</div> <div class=sectline>M. E. Fisher and D. S. Gaunt, <a href="https://doi.org/10.1103/PhysRev.133.A224">Ising model and self-avoiding walks on hypercubical lattices and high density expansions</a>, Phys. Rev. 133 (1964) A224-A239.</div> <div class=sectline>Lars Onsager, <a href="https://doi.org/10.1103/PhysRev.65.117">Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition</a>, Phys. Rev. 65, 117 (1944).</div> <div class=sectline>M. F. Sykes and M. E. Fisher, <a href="https://doi.org/10.1016/0031-8914(62)90080-0">Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices</a>, Physica, 28 (1962), 919-938.</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) ~ 2 * (1 + sqrt(2))^(2*n-1) / (Pi * n^2). - <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Apr 27 2024</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>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); # <a href="/wiki/User:Sean_A._Irvine">Sean A. Irvine</a>, Nov 26 2017</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>u[h_]:=Coth[2h](1+(2/Pi)(2Tanh[2h]^2-1)EllipticK[(2Sinh[2h]/Cosh[2h]^2)^2]);</div> <div class=sectline>Table[SeriesCoefficient[u[ArcTanh[v]], {v, 0, 2n-1}], {n, 10}]</div> <div class=sectline>(* <a href="/wiki/User:Andrey_Zabolotskiy">Andrey Zabolotskiy</a>, Sep 12 2017; see Onsager's eq. (116) *)</div> <div class=sectline>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]] (* <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Apr 27 2024 *)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A002906" title="High temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.">A002906</a>-<a href="/A002930" title="Magnetization series for diamond.">A002930</a>, <a href="/A010571" title="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 o...">A010571</a>, <a href="/A010572" title="High-temperature coefficients for the internal energy for spin-1/2 Ising model on 4-d cubic lattice.">A010572</a>, <a href="/A010573" title="High-temperature coefficients for 5-d cubic lattice.">A010573</a>, <a href="/A010574" title="High-temperature coefficients for 6-d cubic lattice.">A010574</a>.</div> <div class=sectline>Sequence in context: <a href="/A179190" title="Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4*x - 4*x^2).">A179190</a> <a href="/A291482" title="Expansion of e.g.f. arcsin(x)*exp(x).">A291482</a> <a href="/A065654" title="Fixed points for A065652, a permutation of the natural numbers.">A065654</a> * <a href="/A004528" title="Ratios of successive terms are 1,2,2,2,3,4,4,4,5,6,6,6,7,...">A004528</a> <a href="/A066535" title="Number of ways of writing n as a sum of n squares.">A066535</a> <a href="/A134455" title="a(0) = a(1) = 1, a(2) = 2; a(n) = 2*a(n-2) + a(n-1)*a(n-3).">A134455</a></div> <div class=sectline>Adjacent sequences: <a href="/A002905" title="Number of connected graphs with n edges.">A002905</a> <a href="/A002906" title="High temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.">A002906</a> <a href="/A002907" title="High temperature series in v = tanh(J/kT) for residual correlation function (correction to susceptibility) for the spin-1/2 ...">A002907</a> * <a href="/A002909" title="Low temperature energy function for square lattice.">A002909</a> <a href="/A002910" title="High-temperature series in v = tanh(J/kT) for susceptibility for the Ising model on honeycomb structure.">A002910</a> <a href="/A002911" title="Erroneous version of A047709.">A002911</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="an exceptionally nice sequence">nice</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>More terms and new name from <a href="/wiki/User:Andrey_Zabolotskiy">Andrey Zabolotskiy</a>, Oct 19 2017</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified April 10 14:57 EDT 2025. Contains 382673 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>