A002906 -id:A002906 - cp's OEIS frontend

A008547 Erroneous version of A002906.

Original entry on oeis.org

12, 100, 1972, 5172, 34876, 89764, 229628, 1486308, 3763460, 9497380, 36212402548

Offset: 0

Marek Karliner (marek(AT)proton.tau.ac.il)

dead

C. Domb, "Ising Model", in "Phase Transitions and Critical Phenomena", Vol. 3, "Series Expansions for Lattice Models", edited by C. Domb and M.S. Green, Academic Press (1974).

M. A. Samuel, G. Li, Estimating perturbative coefficients in quantum field theory and the ortho-positronium decay rate discrepancy, Physics Letters B, Volume 331, Issue 1-2, p. 114-118. In Table IX, [n,m] is A002906(n+m), while the particular pairs [n,m] have been chosen for illustrative reasons only.

A002931 Number of self-avoiding polygons of length 2n on square lattice (not allowing rotations).

Original entry on oeis.org

0, 1, 2, 7, 28, 124, 588, 2938, 15268, 81826, 449572, 2521270, 14385376, 83290424, 488384528, 2895432660, 17332874364, 104653427012, 636737003384, 3900770002646, 24045500114388, 149059814328236, 928782423033008, 5814401613289290, 36556766640745936

Offset: 1

N. J. A. Sloane

Translations are allowed, but not rotations or reflections.

a(n) is also the coefficient of n^2 in the sequence of quadratic polynomials giving the numbers of 2k-cycles in the n X n grid graph for n >= k-1 (see the example). - Eric W. Weisstein, Apr 05 2018

			At length 8 there are 7 polygons, consisting of the 2, 1, 4 resp. rotations of:
._. .___. .___.
| | | . | | ._|
| | |___| |_|
|_|
Let p(k,n) be the number of 2k-cycles in the n X n grid graph for n >= k-1.  p(k,n) are quadratic polynomials in n, with the first few given by:
p(1,n) = 0,
p(2,n) = 1 - 2*n + n^2,
p(3,n) = 4 - 6*n + 2*n^2,
p(4,n) = 26 - 28*n + 7*n^2,
p(5,n) = 164 - 140*n + 28*n^2,
p(6,n) = 1046 - 740*n + 124*n^2,
p(7,n) = 6672 - 4056*n + 588*n^2,
p(8,n) = 42790 - 22904*n + 2938*n^2,
p(9,n) = 275888 - 132344*n + 15268*n^2,
...
The quadratic coefficients give a(n), so the first few are 0, 1, 2, 7, 28, 124, .... - _Eric W. Weisstein_, Apr 05 2018

N. Clisby and I. Jensen: A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice, J. Phys. A: Math. Theor. 45 (2012). Also arXiv:1111.5877, 2011. [Extends sequence to a(65)]
I. G. Enting: Generating functions for enumerating self-avoiding rings on the square lattice, J. Phys. A: Math. Gen. 13 (1980). pp. 3713-3722. See Table 2.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 461.
I. Jensen: A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice, J. Phys. A: Math. Gen. 36 (2003). [Extends sequence to a(55)]
I. Jensen and A. J. Guttmann: Self-avoiding polygons on the square lattice, J. Phys. A: Math. Gen. 32 (1999). Also arXiv:cond-mat/9905291. [Extends sequence to a(45)]
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).

N. J. A. Sloane, Table of n, a(n) for n = 1..65 [Formed from tables in several references, the most recent being Clisby-Jensen, 2011/2012]
Jérôme Bastien, Construction and enumeration of circuits capable of guiding a miniature vehicle, arXiv:1603.08775 [math.CO], 2016. Cites this sequence.
Nathan Clisby, Lattice enumeration, Slides of talk at Enting fest, CSIRO, Aspendale, 2015; Lattice enumeration [Local copy].
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 364.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)
A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.
Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.
B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.
I. Jensen, A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice, Journal of Physics A, Vol. 36 (2003), pp. 5731-5745.
I. Jensen, More terms
G. S. Rushbrooke and J. Eve, On Noncrossing Lattice Polygons, Journal of Chemical Physics, 31 (1959), 1333-1334.
S. G. Whittington and J. P. Valleau, Figure eights on the square lattice: enumeration and Monte Carlo estimation, J. Phys. A 3 (1970), 21-27. See Table 2.

Cf. A056634, A036638, A036639. Equals A010566(n)/(4n).
Cf. A057730.
Cf. A302335 (constant coefficients in p(k,n)).
Cf. A302336 (linear coefficients in p(k,n)).

Updated by N. J. A. Sloane, Mar 18 2021

A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.

Original entry on oeis.org

1, 6, 30, 138, 618, 2730, 11946, 51882, 224130, 964134, 4133166, 17668938, 75355206, 320734686, 1362791250, 5781765582, 24497330322, 103673967882, 438296739594, 1851231376374, 7812439620678, 32944292555934, 138825972053046

Offset: 0

N. J. A. Sloane

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

A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 459.
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).

I. Jensen, Table of n, a(n) for n = 0..40 (from the Jensen link below)
Sergio Caracciolo, Anthony J. Guttmann, Iwan Jensen, Andrea Pelissetto, Andrew N. Rogers, Alan D. Sokal, Correction-to-Scaling Exponents for Two-Dimensional Self-Avoiding Walks, Journal of Statistical Physics, September 2005, Volume 120, Issue 5, pp. 1037-1100.
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)
A. J. Guttmann and J. Wang, The extension of self-avoiding random walk series in two dimensions, J. Phys. A 24 (1991), 3107-3109.
B. D. Hughes, Random Walks and Random Environments, vol. 1, Oxford 1995, Tables and references for self-avoiding walks counts [Annotated scanned copy of several pages of a preprint or a draft of chapter 7 "The self-avoiding walk"]
I. Jensen, Series Expansions for Self-Avoiding Walks
J. L. Martin, M. F. Sykes and F. T. Hioe, Probability of initial ring closure for self-avoiding walks on the face-centered cubic and triangular lattices, J. Chem. Phys., 46 (1967), 3478-3481.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
D. C. Rapaport, End-to-end distance of linear polymers in two dimensions: a reassessment, J. Phys. A 18 (1985), L201.
S. Redner, Distribution functions in the interior of polymer chains, J. Phys. A 13 (1980), 3525-3541, doi:10.1088/0305-4470/13/11/023.
M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.
Joris van der Hoeven, On asymptotic extrapolation, Journal of symbolic computation, 2009, p. 1010.

Cf. A001411, A001412, A001336, A001666, A001335, A036418, A192208.

mo={{2, 0},{-1, 1},{-1, -1},{-2, 0},{1, -1},{1, 1}}; a[0]=1;
a[tg_, p_:{{0, 0}}] := Block[{e, mv = Complement[Last[p]+# & /@ mo, p]}, If[tg == 1, Length@mv, Sum[a[tg-1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 6]
(* Robert FERREOL, Nov 28 2018; after the program of Giovanni Resta in A001411 *)

def add(L,x):
    M=[y for y in L];M.append(x)
    return(M)
plus=lambda L,M : [x+y for x,y in zip(L,M)]
mo=[[2,0],[-1,1],[-1, -1],[-2,0],[1,-1],[1, 1]]
def a(n,P=[[0, 0]]):
    if n==0: return(1)
    mv1 = [plus(P[-1],x) for x in mo]
    mv2=[x for x in mv1 if x not in P]
    if n==1: return(len(mv2))
    else: return(sum(a(n-1,add(P,x)) for x in mv2))
[a(n) for n in range(11)]
# Robert FERREOL, Dec 11 2018

A002913 High temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

Original entry on oeis.org

1, 6, 30, 150, 726, 3510, 16710, 79494, 375174, 1769686, 8306862, 38975286, 182265822, 852063558, 3973784886, 18527532310, 86228667894, 401225368086, 1864308847838, 8660961643254, 40190947325670, 186475398518726, 864404776466406, 4006394107568934, 18554916271112254, 85923704942057238

Offset: 0

N. J. A. Sloane

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 381.
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).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..32 (terms a(24), a(25) taken from the Campostrini et al. 2002 article by _Per H. Lundow_, terms a(26)-a(32) taken from the Toshiaki Fujiwara and Hiroaki Arisue's slides)
P. Butera and M. Comi, N-vector spin models on the simple-cubic and the body-centered-cubic lattices: A study of the critical behavior of the susceptibility and of the correlation length by high-temperature series extended to order beta^21, Phys. Rev. B 56 (1997) 8212-8240; arXiv:hep-lat/9703018, 1997.
P. Butera and M. Comi, Extension to order b23 of the high-temperature expansions for the spin-1/2 Ising model on the simple-cubic and the body-centered-cubic lattices, BICOCCA/FT-00-09 (June 2000). Phys. Rev. B62 (2000) 14837-14843.
M. Campostrini, Linked-Cluster Expansion of the Ising Model, Journal of Statistical Physics, 103 (2001), 369-394.
M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice, Phys. Rev. E, 65 (2002), 66-127.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and R. J. Burford, Theory of critical point scattering and correlations I: the Ising model, Phys. Rev. 156 (1967), 583-621.
Toshiaki Fujiwara and Hiroaki Arisue (presenter), ３次元イジング模型の高温展開 (High-temperature expansion for the 3D Ising model), Computational Physics with CP-PACS 2002 Workshop [in Japanese].
Toshiaki Fujiwara and Hiroaki Arisue (presenter), New algorithm of the high-temperature expansion for the Ising model in three dimensions, Asia-Pacific Mini-Workshop on Lattice QCD, Center for Computational Physics, University of Tsukuba, 2003: abstract, slides, source.
D. S. Gaunt, High Temperature Series Analysis for the Three-Dimensional Ising Model: A Review of Some Recent Work, pp. 217-246 in: Phase Transitions: Cargèse 1980, eds. Maurice Lévy, Jean-Claude Le Guillou and Jean Zinn-Justin, Springer, Boston, MA, 1982.
M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, II. Three dimensional lattices, J. Phys. A 5 (1972) 640-652.

Cf. other quantities: A001393 (partition function), A010571 (internal energy), A002916 (specific heat), A003490 (surface susceptibility), A007287 (layer susceptibility), A010040, A010043, A010046.
Cf. other structures: A002906 (square), A002920 (hexagonal), A002910 (honeycomb), A002914 (b.c.c.), A002921 (f.c.c.), A003119 (diamond), A010556 (4D cubic), A010579 (5D cubic), A010580 (6D cubic), A030008 (7D cubic).
Cf. low-temperature series: A002926 (ferromagnetic), A002915 (antiferromagnetic).
Cf. other models: A002170 (Heisenberg), A003279 (spherical).

Corrections and updates from Steven Finch
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 01 2008
Several errors in the sequence were corrected by Per H. Lundow, Jan 17 2011

A002908 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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Simon Plouffe

Previous name was: Energy function for square lattice.

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 386.
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).

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.
Lars Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117 (1944).
M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.

Cf. A002906-A002930, A010571, A010572, A010573, A010574.

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); # Sean A. Irvine, Nov 26 2017

u[h_]:=Coth[2h](1+(2/Pi)(2Tanh[2h]^2-1)EllipticK[(2Sinh[2h]/Cosh[2h]^2)^2]);
Table[SeriesCoefficient[u[ArcTanh[v]],{v,0,2n-1}],{n,10}]
(* Andrey Zabolotskiy, Sep 12 2017; see Onsager's eq. (116) *)
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]] (* Vaclav Kotesovec, Apr 27 2024 *)

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

More terms and new name from Andrey Zabolotskiy, Oct 19 2017

A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

Original entry on oeis.org

0, 0, 1, 8, 60, 416, 2791, 18296, 118016, 752008, 4746341, 29727472, 185016612, 1145415208, 7059265827, 43338407712, 265168691392, 1617656173824, 9842665771649, 59748291677832, 361933688520940, 2188328005246304, 13208464812265559, 79600379336505560, 479025509574159232

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

The zero-field susceptibility per spin is 4m^2/kT * Sum_{n >= 0} a(n) * u^n, where u = exp(-4J/kT). (m is the magnetic moment of a single spin; this factor may be present or absent depending on the precise definition of the susceptibility.) The b-file has been obtained from the series by Guttmann and Jensen via the substitution r = u/(1-u)^2 and dividing by 4. - Andrey Zabolotskiy, Feb 11 2022

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
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).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1305
R. J. Baxter and I. G. Enting, Series expansions for corner transfer matrices: the square lattice Ising model, J. Stat. Physics 21 (1979) 103-123.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions for the Spin-1 Ising Model, arXiv:hep-lat/9410005, 1994; J. Phys. A. 27 (1994) 6987-7005.
J. W. Essam and M. E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys., 38 (1963), 802-812.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
Tony Guttmann, Homepage. See Numerical Data, Ising square lattice susceptibility series, Low temperature series.
Iwan Jensen, Series for the Ising model

Cf. A002906 (high-temperature), A002979 (antiferromagnetic susceptibility), A029872 (specific heat), A002928 (magnetization), A002890 (partition function), A047709 (hexagonal lattice), A002912 (honeycomb), A002926 (cubic lattice), A010115 (spin-1 Ising).

a(n) ~ c * n^(3/4) * (1 + sqrt(2))^(2*n), where c = 0.0187325517235678... - Vaclav Kotesovec, May 06 2024

Corrections and updates from Steven Finch

a(0) = a(1) = 0 prepended, terms a(20) and beyond added by Andrey Zabolotskiy, Feb 10 2022

A010556 High temperature series for spin-1/2 Ising magnetic susceptibility on 4D simple cubic lattice.

Original entry on oeis.org

1, 8, 56, 392, 2696, 18536, 126536, 863720, 5873768, 39942184, 271009112, 1838725896, 12457092504, 84392312392, 571140732808, 3865210690888, 26138072412040, 176752645426600, 1194553221342296, 8073068110703880, 54534614510976680

Offset: 0

N. J. A. Sloane

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012); arXiv:1209.3592 [hep-lat], 2012.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
D. S. Gaunt, M. F. Sykes and S. McKenzie, Susceptibility and fourth-field derivative of the spin-1/2 Ising model for T > T_c and d = 4, J. Phys. A 12 (1979), 871-877.
M. A. Moore, Critical behavior of the four-dimensional Ising ferromagnet and the breakdown of scaling, Phys. Rev. B 1 (1970), 2238-2240.

Cf. A002906 (2D), A002913 (3D), A010579 (5D), A010580 (6D), A030008 (7D).

Cf. A030046, A010041, A010044, A010047.

a(17) corrected (was 176752645540264), a(18)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 08 2022

A010579 High temperature series for spin-1/2 Ising magnetic susceptibility on 5D simple cubic lattice.

Original entry on oeis.org

1, 10, 90, 810, 7210, 64170, 568970, 5044810, 44649930, 395180650, 3494051130, 30893156970, 272971707930, 2411975074570, 21302972395370, 188151452434090, 1661273238131050, 14668124524584170, 129481802727508250, 1142991284620073450, 10087904498275867530

Offset: 0

N. J. A. Sloane

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012).
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
Misha Gofman, Joan Adler, Amnon Aharony, A. B. Harris and Dietrich Stauffer, Series and Monte Carlo study of high-dimensional Ising models, J. Stat. Phys. 71, 1221-1230 (1993).

Cf. A002906 (2D), A002913 (3D), A010556 (4D), A010580 (6D), A030008 (7D).

Corrections and updates from Steven Finch

Terms a(16)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 09 2022

A010580 High temperature series for spin-1/2 Ising magnetic susceptibility on 6D simple cubic lattice.

Original entry on oeis.org

1, 12, 132, 1452, 15852, 173052, 1884972, 20532252, 223437852, 2431526492, 26447593812, 287669976492, 3128064123732, 34013987172972, 369792173040492, 4020299656610636, 43702216875039660, 475060467524653980, 5163624600479230260, 56125562454502452780, 610010748386503122684

Offset: 0

N. J. A. Sloane

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012).
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
Misha Gofman, Joan Adler, Amnon Aharony, A. B. Harris and Dietrich Stauffer, Series and Monte Carlo study of high-dimensional Ising models, J. Stat. Phys. 71, 1221-1230 (1993).

Cf. A002906 (2D), A002913 (3D), A010556 (4D), A010579 (5D), A030008 (7D).

Corrections and updates from Steven Finch

Terms a(16)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 09 2022

A030008 High temperature series for spin-1/2 Ising magnetic susceptibility on 7D simple cubic lattice.

Original entry on oeis.org

1, 14, 182, 2366, 30590, 395486, 5105870, 65919182, 850586702, 10975573182, 141586912166, 1826501185054, 23558885899318, 303871575267918, 3919114007263518, 50545912921275198, 651868436561980638, 8406864950367314046, 108415583649894484278, 1398136240855886669662, 18029995593288775967598

Offset: 0

Steven Finch

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

P. Butera and M. Pernici, High-temperature expansions of the higher susceptibilities for the Ising model in general dimension d, Phys. Rev. E 86, 011139 (2012).
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964), A224-A239.
Misha Gofman, Joan Adler, Amnon Aharony, A. B. Harris and Dietrich Stauffer, Series and Monte Carlo study of high-dimensional Ising models, J. Stat. Phys. 71, 1221-1230 (1993).

Cf. A002906 (2D), A002913 (3D), A010556 (4D), A010579 (5D), A010580 (6D).

Terms a(16)-a(20) added using Butera & Pernici's formulas by Andrey Zabolotskiy, Aug 09 2022

cp's OEIS Frontend

A008547 Erroneous version of A002906.

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A002931 Number of self-avoiding polygons of length 2n on square lattice (not allowing rotations).

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A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.

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A002913 High temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

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A002908 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.

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A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

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A010556 High temperature series for spin-1/2 Ising magnetic susceptibility on 4D simple cubic lattice.

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A010579 High temperature series for spin-1/2 Ising magnetic susceptibility on 5D simple cubic lattice.

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A010580 High temperature series for spin-1/2 Ising magnetic susceptibility on 6D simple cubic lattice.

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