A007206 -id:A007206 - cp's OEIS frontend

A002928 Magnetization for square lattice.

1, 0, -2, -8, -34, -152, -714, -3472, -17318, -88048, -454378, -2373048, -12515634, -66551016, -356345666, -1919453984, -10392792766, -56527200992, -308691183938, -1691769619240, -9301374102034

Offset: 0

N. J. A. Sloane, Simon Plouffe

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).
J. M. Yeomans, Statistical mechanics of phase transitions, Oxford Univ. Press, 1992, p. 93.

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
David Park, A summation method for crystal statistics, Physica 22 (1956), 932-940.

Cf. other structures: A007206, A007207, A002929, A002930, A003193, A003196.
Cf. higher spins: A010102, A010105/A030120, A010103, A010106/A030121, A010104.
Cf. Potts model: A057374, A057378.
Cf. A002927 (susceptibility).

series((1+x)^(1/4)*(1-6*x+x^2)^(1/8)/(1-x)^(1/2),x,40).

CoefficientList[Series[(1+x)^(1/4)*(1-6*x+x^2)^(1/8)/(1-x)^(1/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 27 2024 *)

n*a(n) + 6*(-n+1)*a(n-1) + 4*a(n-2) + 6*(n-3)*a(n-3) + (-n+4)*a(n-4) = 0. - R. J. Mathar, Mar 08 2013

a(n) ~ -Gamma(1/8) * (1 + sqrt(2))^(2*n - 1/2) / (Pi * 2^(57/16) * n^(9/8)). - Vaclav Kotesovec, Apr 27 2024

A007207 Magnetization for hexagonal lattice.

Original entry on oeis.org

1, 0, 0, -2, 0, -12, 2, -78, 24, -548, 228, -4050, 2030, -30960, 17670, -242402, 152520, -1932000, 1312844, -15612150, 11297052, -127551884, 97291026, -1051478274, 838994486, -8732657724, 7246304736, -72983051674, 62686156026, -613243234224, 543146222970

Offset: 0

Simon Plouffe

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Shigeo Naya, On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices, Progress of Theoretical Physics, 11 (1954), 53-62.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A002928, A007206.

CoefficientList[Series[(1 - 16 * x^3 / ((1+3*x) * (1-x)^3))^(1/8), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 27 2024 *)

G.f.: (1 - 16 * x^3 / ((1+3*x) * (1-x)^3))^(1/8) [Shigeo Naya]. - Andrey Zabolotskiy, Jun 01 2022

a(n) ~ (-1)^n * 3^n / (Gamma(1/8) * 2^(1/4) * n^(7/8)) * (1 - (-1)^n * sqrt(sqrt(2) - 1) * Gamma(1/8)^2 / (2^(13/4) * Pi * n^(1/4))). - Vaclav Kotesovec, Apr 27 2024

Offset changed, signs of terms changed, and more terms added by Andrey Zabolotskiy, Jun 01 2022

cp's OEIS Frontend

A002928 Magnetization for square lattice.

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