A057374 -id:A057374 - cp's OEIS frontend

A057405 Low-temperature partition function expansion for Kagome net (Potts model, q=4).

1, 0, 0, 0, 9, 0, 24, 36, 90, 228, 468, 1296, 2946, 7272, 20808, 48612, 143028, 362700, 977562, 2702592, 7084962, 20179368, 54981234, 154896552, 438387402, 1224515736, 3507218586, 9902866092, 28373581638, 81503463852

Offset: 0

N. J. A. Sloane, Aug 30 2000

nonn

I. Jensen, Table of n, a(n) for n = 0..58 (from link below)
I. Jensen, More terms

Cf. A057374-A057404.

A002928 Magnetization for square lattice.

Original entry on oeis.org

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

A057375 Low-temperature susceptibility expansion for square lattice (Potts model, q=3).

Original entry on oeis.org

2, 0, 16, 16, 100, 216, 844, 1552, 7844, 12112, 60268, 118944, 424072, 1081392, 3201728, 8670688, 25713154, 67206560, 203077760, 532881432, 1558159918, 4250639632, 11956293152, 33296697848, 92820406096, 257249275776

Offset: 0

N. J. A. Sloane, Aug 29 2000

nonn

I. Jensen, Table of n, a(n) for n = 0..67 (from link below)
Ian G. Enting and Anthony J. Guttmann, Susceptibility amplitudes for the three- and four-state Potts models, Physica A, 321 (2003), 90-107.
I. Jensen, More terms

Cf. A057374-A057405.

A057378 Low-temperature magnetization expansion for square lattice (Potts model, q=4).

Original entry on oeis.org

1, 0, 0, 0, -4, 0, -16, -32, -28, -288, -400, -1024, -5268, -5920, -32160, -82720, -163020, -737568, -1482784, -4644992, -15095436, -33307648, -117747376, -312435552, -842726356, -2747491616, -7020371952, -21348043296

Offset: 0

N. J. A. Sloane, Aug 30 2000

sign

I. Jensen, Table of n, a(n) for n = 0..59 (from link below)
I. Jensen, More terms

Cf. A057374-A057405.

A057382 Low-temperature magnetization expansion for hexagonal lattice (Potts model, q=3).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, -3, 0, 0, 0, -18, -18, 24, 0, -171, -162, 153, 252, -1704, -2106, 1998, 2586, -14364, -28098, 19008, 43020, -147024, -317304, 125775, 612954, -1370868, -3909528, 907209, 7487136, -11849868, -46762686, 252159

Offset: 0

N. J. A. Sloane, Aug 30 2000

sign

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

I. Jensen, Table of n, a(n) for n = 0..69 (from link below)
I. Jensen, More terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A057374-A057405.

A057383 Low-temperature susceptibility expansion for hexagonal lattice (Potts model, q=3).

Original entry on oeis.org

2, 0, 0, 0, 24, 24, -20, 0, 366, 324, -42, -312, 4788, 6036, -1356, -1820, 54036, 99252, -3024, -53352, 686988, 1382336, 285870, -926172, 7988984, 19975392, 6245886, -12161464, 89970804, 273568968, 134393334, -181279824

Offset: 0

N. J. A. Sloane, Aug 30 2000

sign

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

I. Jensen, Table of n, a(n) for n = 0..62 (from link below)
I. Jensen, More terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A057374-A057405.

A057387 Low-temperature susceptibility expansion for hexagonal lattice (Potts model, q=4).

Original entry on oeis.org

3, 0, 0, 0, 36, 72, -72, 0, 711, 1080, 144, -2556, 12852, 23004, -504, -21192, 122877, 525996, 69366, -531576, 1970154, 7833756, 6613164, -12953124, 24243261, 137623572, 130318974, -138059232, 115953372, 2338653528

Offset: 0

N. J. A. Sloane, Aug 30 2000

sign

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

I. Jensen, Table of n, a(n) for n = 0..53 (from link below)
I. Jensen, More terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A057374-A057405.

A057390 Low-temperature magnetization expansion for honeycomb net (Potts model, q=3).

Original entry on oeis.org

1, 0, 0, -3, -9, -36, -123, -450, -1764, -6690, -26649, -104112, -421248, -1688337, -6888978, -28063296, -115459524, -475617330, -1970737233, -8184006855, -34118533647, -142565353488, -597406140090

Offset: 0

N. J. A. Sloane, Aug 30 2000

sign

I. Jensen, Table of n, a(n) for n = 0..35 (from link below)
I. Jensen, More terms

Cf. A057374-A057405.

A057391 Low-temperature susceptibility expansion for honeycomb net (Potts model, q=3).

Original entry on oeis.org

4, 24, 132, 672, 3192, 15996, 74396, 354936, 1639764, 7669876, 35282064, 162809928, 745459776, 3413032716, 15560103924, 70861321612, 321879751956, 1460223461700, 6612700085376, 29909912167920

Offset: 0

N. J. A. Sloane, Aug 30 2000

nonn

I. Jensen, Table of n, a(n) for n = 0..31 (from link below)
I. Jensen, More terms

Cf. A057374-A057405.

A057395 Low-temperature susceptibility expansion for honeycomb net (Potts model, q=4).

Original entry on oeis.org

6, 36, 234, 1284, 6804, 38160, 198912, 1070316, 5499054, 29005692, 149318838, 776570508, 3987307152, 20560750344, 105345948384, 540305120844, 2761471319562, 14111436147228, 71964766006350, 366780011157360

Offset: 0

N. J. A. Sloane, Aug 30 2000

nonn

I. Jensen, Table of n, a(n) for n = 0..27 (from link below)
I. Jensen, More terms

Cf. A057374-A057405.

cp's OEIS Frontend

A057405 Low-temperature partition function expansion for Kagome net (Potts model, q=4).

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A002928 Magnetization for square lattice.

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A057375 Low-temperature susceptibility expansion for square lattice (Potts model, q=3).

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A057378 Low-temperature magnetization expansion for square lattice (Potts model, q=4).

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A057382 Low-temperature magnetization expansion for hexagonal lattice (Potts model, q=3).

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A057383 Low-temperature susceptibility expansion for hexagonal lattice (Potts model, q=3).

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A057387 Low-temperature susceptibility expansion for hexagonal lattice (Potts model, q=4).

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A057390 Low-temperature magnetization expansion for honeycomb net (Potts model, q=3).

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A057391 Low-temperature susceptibility expansion for honeycomb net (Potts model, q=3).

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A057395 Low-temperature susceptibility expansion for honeycomb net (Potts model, q=4).

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