OFFSET
0,4
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
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
Andrey Zabolotskiy, Table of n, a(n) for n = 0..500
P. D. Beale, Exact distribution of energies in the two-dimensional Ising model, Phys. Rev. Lett. 76 (1996) 78-81
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
G. Siudem, A. Fronczak, and P. Fronczak, Exact low-temperature series expansion for the partition function of the two-dimensional zero-field s= 1/2 Ising model on the infinite square lattice, arXiv preprint arXiv:1410.7963 [math-ph], 2014-2015.
Gandhimohan M. Viswanathan, The hypergeometric series for the partition function of the 2-D Ising model arXiv:1411.2495 [cond-mat.stat-mech], 2014-2015.
Gandhimohan M. Viswanathan, The double hypergeometric series for the partition function of the 2D anisotropic Ising model, arXiv:2104.03430 [cond-mat.stat-mech], 2021.
FORMULA
a(n) ~ exp(2*G/Pi) * (1 + sqrt(2))^(2*n-1) / (Pi*sqrt(2)*n^3), where G is the Catalan's constant A006752. - Vaclav Kotesovec, May 02 2024
MATHEMATICA
(* For 25 terms, a PC computation lasts less than half an hour *) m = 48 (* max y exponent *); coes = CoefficientList[ Series[ Log[(1 + y^2)^2 - 2*y*(1 - y^2)*(Cos[2*Pi*u] + Cos[2*Pi*v])], {y, 0, m}], y] // Rest; nint[f_, {n_}] := If[n == 2 || OddQ[n], 0, Print[n]; Integrate[ Integrate[f, {u, 0, 1}], {v, 0, 1}]]; fy = MapIndexed[nint, coes].Table[y^k, {k, 1, m}]; CoefficientList[ Series[ Exp[fy/2], {y, 0, m}] , y^2] (* Jean-François Alcover, Mar 19 2013 *)
CoefficientList[(1+u) Exp[-x HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16x] /. {x -> (u (1 - u)^2)/(1 + u)^4}] + O[u]^50, u] (* Andrey Zabolotskiy, Feb 12 2022, using the g. f. from Gandhimohan M. Viswanathan, 2014-2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Corrections and updates from Steven Finch
"Free energy" changed back to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 11 2022
STATUS
approved