A003051 -id:A003051 - cp's OEIS frontend

A054346 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other.

1, 1, 2, 2, 4, 3, 5, 3, 7, 5, 7, 4, 11, 5, 8, 8, 12, 6, 13, 6, 15, 10, 11, 7, 21, 9, 13, 12, 18, 9, 21, 9, 21, 14, 16, 13, 29, 11, 17, 16, 28, 12, 28, 12, 25, 21, 20, 13, 39, 16, 24, 20, 29, 15, 34, 18, 36, 22, 25, 16, 47, 17, 26, 29, 38, 21, 40, 18, 36, 26, 36, 19, 58, 20

Offset: 0

N. J. A. Sloane, May 06 2000

If we count sublattices as equivalent only if they are related by a rotation, we get A054345 instead of this sequence. If we only allow rotations and reflections that preserve the parent (square) lattice, we get A145393; the first discrepancy is at n = 25 (see illustration), the second is at n = 30. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145392. The analog for the hexagonal lattice is A300651. - Andrey Zabolotskiy, Mar 12 2018

			For n = 1, 2, 3, 4 the sublattices are generated by the rows of:
  [1 0] [2 0] [2 0] [3 0] [3 0] [4 0] [4 0] [2 0] [2 0]
  [0 1] [0 1] [1 1] [0 1] [1 1] [0 1] [1 1] [0 2] [1 2].

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.2 for the Dirichlet g.f. zeta^GL_{x^2+y^2}(s).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. - From _N. J. A. Sloane_, Feb 23 2009
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. A003051, A001615, A054345.

# See A159842 and A054345 for the definitions of functions used here
def a_GL(n):
    return (a_SL(n) + dc(fin(1, 0, 0, 1), u, u, f2)(n)) / 2
print([a_GL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

A187015 The number of different classes of 2-dimensional convex lattice polytopes having volume n/2 up to unimodular equivalence.

Original entry on oeis.org

1, 2, 3, 7, 6, 13, 13, 27, 26, 44, 43, 83, 81, 122, 136, 208, 215, 317, 341, 490, 542, 710, 778, 1073, 1186, 1519, 1708, 2178, 2405, 3042, 3408, 4247, 4785, 5782, 6438, 7870, 8833, 10560, 11857, 14131, 15733, 18636, 20773, 24381, 27353, 31764, 35284, 41081, 45791, 52762

Offset: 1

Jonathan Vos Post, Mar 01 2011

nonn

Lattice polytopes up to the equivalence relation used here are also called toric diagrams, see references below. - Andrey Zabolotskiy, May 10 2019

Liu & Zong give a(7) = 11, and others use their list, but their list lacks polygons No. 3 and 4 from Balletti's file 2-polytopes/v7.txt. - Andrey Zabolotskiy, Dec 28 2021

Günter Rote, Table of n, a(n) for n = 1..207
Gabriele Balletti, Dataset of "small" lattice polytopes. Beware that the vertices are not always listed in sorted order around the polygon boundary (clockwise or counterclockwise).
Gabriele Balletti, Enumeration of lattice polytopes by their volume, Discrete Comput. Geom., 65 (2021), 1087-1122; arXiv:1103.0103 [math.CO], 2018.
Sebastián Franco, Yang-Hui He, Chuang Sun and Yan Xiao, A comprehensive survey of brane tilings, Int. J. Mod. Phys. A, 32 (2017), 1750142, arXiv:1702.03958 [hep-th], 2017.
Heling Liu and Chuanming Zong, On the classification of convex lattice polytopes, Adv. Geom., 11 (2011), 711-729, arXiv:1103.0103 [math.MG], 2011. See table at p. 8.
Günter Rote, Number of lattice polygons of area at most 103.5, classified by the number k of vertices, the number B of lattice points on edges, and the number I of interior lattice points.
Yan Xiao, Quivers, Tilings and Branes, City, University of London, 2018. See Tables 3.2-3.7.

Cf. A126587, A003051 (triangles only), A322343, A366409.

Python
```
# See the Python program for A366409.
```

a(8) from Yan Xiao added by Andrey Zabolotskiy, May 10 2019

Name edited, a(7) corrected, a(9)-a(50) added using Balletti's data by Andrey Zabolotskiy, Dec 28 2021

A054384 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated to give the other.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 3, 5, 5, 6, 4, 10, 5, 7, 8, 11, 6, 13, 7, 14, 10, 12, 8, 20, 11, 13, 14, 17, 10, 24, 11, 21, 16, 18, 14, 31, 13, 19, 18, 30, 14, 28, 15, 28, 26, 24, 16, 42, 17, 31, 24, 31, 18, 40, 24, 35, 26, 30, 20, 56, 21, 31, 31, 43, 26, 48, 23, 42, 32, 42, 24, 65

Offset: 0

N. J. A. Sloane, May 08 2000

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

If reflections are allowed, we get A300651. If only rotations that preserve the parent hexagonal lattice are allowed, we get A145394. The analog for square lattice is A054345. - Andrey Zabolotskiy, Mar 10 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.1 for the Dirichlet g.f. zeta^SL_{x^2+xy+y^2}(s).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163 [see Table 2]. - From _N. J. A. Sloane_, Feb 23 2009
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003051, A054345, A054346, A145394, A300651.

# see A159842 for the definitions of dc, fin, u, N
def gg(m, k1, minus = True):
    def f(n):
        if n == 1: return 1
        r = 1
        for (p, k) in factor(n):
            if p % 3 != m or k1 and k > 1: return 0
            if minus: r *= (-1)**k
        return r
    return f
g1, g2, g3 = gg(1, True), gg(1, True, False), gg(2, False)
def a_SL(n):
    return (dc(u, N, g1)(n) + 2 * dc(u, g3)(n)) / 3
print([a_SL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

A300651 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if they are related by any rotation or reflection.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 17, 9, 17, 16, 13, 9, 28, 11, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 20, 27, 16, 26, 13, 24, 18, 24, 13, 40

Offset: 1

Andrey Zabolotskiy, Mar 10 2018

nonn

If we count sublattices as equivalent only if they are related by a rotation, we get A054384 instead of this sequence. If we only allow rotations and reflections that preserve the parent (hexagonal) lattice, we get A003051; the first discrepancy is at n = 42 (see illustration), the second is at n = 49. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145394. The analog for square lattice is A054346.

Although A003051 has its counterpart A003050 which counts primitive sublattices only, this sequence has no such counterpart sequence because a primitive sublattice can turn to a non-primitive one via a non-parent-lattice-preserving rotation, so the straightforward definition of primitiveness does not work in this case.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.1 for the Dirichlet g.f. zeta^GL_{x^2+xy+y^2}(s).
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003050, A003051, A054384, A145394, A054346.

# See A159842 and A054384 for the definitions of functions used here
def a_GL(n):
    return (a_SL(n) + dc(fin(1, -1, 0, 2), u, u, g2)(n)) / 2
print([a_GL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

A048259 Number of distinct solutions to x + y + z = 0 (mod n), where two solutions are equivalent if one can be obtained from the other by multiplying by units in Z/nZ and permuting x,y,z.

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 7, 4, 8, 6, 8, 4, 15, 5, 10, 11, 14, 5, 17, 6, 18, 14, 12, 6, 31, 9, 14, 13, 22, 7, 33, 8, 24, 16, 16, 16, 39, 9, 18, 19, 38, 9, 41, 10, 28, 28, 20, 10, 57, 15, 30, 21, 32, 11, 43, 20, 46, 24, 24, 12, 77, 13, 26, 35, 42, 23, 53, 14, 38, 26, 52, 14, 83

Offset: 0

N. J. A. Sloane

nonn

			For n=6 the 7 solutions are (x,y,z) = (0,0,0), (5,1,0), (4,2,0), (4,1,1), (3,3,0), (3,2,1), (2,2,2).

Sean A. Irvine, Java program (github)

Cf. A007997, A007998, A003050, A003051.

iscanon(n,v)={for(d=1, n-1, if(gcd(n,d)==1 && lex(v,vecsort(v*d%n))>0, return(0))); 1}
a(n)={if(n==0, 1, sum(x=0, n-1, sum(y=x, n-1, my(z=(-x-y)%n); y<=z && iscanon(n,[x,y,z]) )))} \\ Andrew Howroyd, Jun 11 2021

a(42) onward corrected by Sean A. Irvine, Jun 10 2021

A300782 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the simple cubic lattice of index n.

Original entry on oeis.org

1, 3, 3, 9, 5, 13, 7, 24, 14, 23, 11, 49, 15, 33, 31, 66, 21, 70, 25, 89, 49, 61, 33, 162, 50, 81, 75, 137, 49, 177, 55, 193, 97, 123, 99, 296, 75, 147, 129, 312, 89, 291, 97, 269, 218, 203, 113, 534, 146, 302, 203, 357, 141, 451, 207, 508, 247, 307, 171, 789

Offset: 1

Andrey Zabolotskiy, Mar 12 2018

nonn

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
Matt DeCross, Lattice Polytopes and Orbifolds, 2015.
Matt DeCross, Lattice Polytopes and Orbifolds in Quiver Gauge Theories, 2015. See slides 18-21.
Gus L. W. Hart and Rodney W. Forcade, Algorithm for generating derivative structures, Phys. Rev. B 77, 224115 (2008), DOI: 10.1103/PhysRevB.77.224115 [see Table IV].
Materials Simulation Group, Derivative structure enumeration library
Index entries for sequences related to sublattices
Index entries for sequences related to cubic lattice

Cf. A159842, A300783, A300784, A003051, A145393, A001001, A128119, A160870, A145396, A145398.

# see A159842 for the definition of dc, fin, per, u, N, N2
def a(n): # from DeCross's slides
    return (dc(u, N, N2)(n) + 6*dc(fin(1, -1, 0, 4), u, u, N)(n)
      + 3*dc(fin(1, 3), u, u, N)(n)
      + 8*dc(fin(1, 0, -1, 0, 0, 0, 0, 0, 3), u, u, per(0, 1, -1))(n)
      + 6*dc(fin(1, 1), u, u, per(0, 1, 0, -1))(n))//24
print([a(n) for n in range(1, 300)])
# Andrey Zabolotskiy, Sep 02 2019

Terms a(11) and beyond from Andrey Zabolotskiy, Sep 02 2019

A300783 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the 3D hexagonal lattice of index n.

Original entry on oeis.org

1, 3, 5, 11, 7, 19, 11, 34, 23, 33, 19, 77, 25, 53, 55, 104, 37, 115, 45, 143, 91, 105, 61, 272, 90, 139, 137, 235, 91, 309, 103, 331, 183, 219, 185, 516, 141, 267, 245, 544, 169, 529, 185, 485, 411, 375, 217, 952, 278, 550, 389, 647, 271, 829, 397, 922, 477

Offset: 1

Andrey Zabolotskiy, Mar 12 2018

nonn

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
Gus L. W. Hart and Rodney W. Forcade, Algorithm for generating derivative structures, Phys. Rev. B 77, 224115 (2008), DOI: 10.1103/PhysRevB.77.224115 [see Table IV].
Materials Simulation Group, Derivative structure enumeration library
Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda and Isao Tanaka, Enumeration of nonequivalent substitutional structures using advanced data structure of binary decision diagram, J. Chem. Phys. 153, 104109 (2020); preprint: Derivative structure enumeration using binary decision diagram, arXiv:2002.12603 [physics.comp-ph], 2020.
Index entries for sequences related to sublattices

Cf. A159842, A300782, A300784, A003051, A145393.

# see A159842 for the definitions of dc, fin, per, u, N, N2
def a(n):
    return (dc(u, N, N2)(n) + 6*dc(fin(1, -1, 0, 4), u, u, N)(n)
            + dc(fin(1, 3), u, u, N)(n)
            + 4*dc(fin(1, 0, 1), u, u, per(0, 1, -1))(n)) // 12
print([a(n) for n in range(1, 100)])
# Andrey Zabolotskiy, Feb 03 2020

Terms a(11) and beyond from Andrey Zabolotskiy, Feb 03 2020

A006984 Greatest minimal norm of sublattice of index n in hexagonal lattice.

Original entry on oeis.org

1, 1, 3, 4, 3, 4, 7, 7, 9, 7, 7, 12, 13, 12, 13, 16, 13, 13, 19, 16, 21, 19, 19, 21, 25, 21, 27, 28, 21, 27, 31, 28, 27, 28, 31, 36, 37, 31, 39, 37, 37, 36, 43, 39, 39, 39, 39, 48, 49, 43, 43

Offset: 1

N. J. A. Sloane, Mira Bernstein

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

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 = 1..500
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
N. J. A. Sloane, Computer printout with notes, Mar. 1994.

Cf. A003051, A003050, A001615.

A173824 Number of four-dimensional simplical toric diagrams with hypervolume n.

Original entry on oeis.org

1, 2, 4, 10, 8, 19, 13, 45, 33, 47, 30, 129, 43, 96, 108, 226, 78, 264, 102, 357, 226, 277, 163, 813, 260, 425, 436, 780, 297, 1092, 355, 1281, 678, 856, 712, 2215, 569, 1155, 1050, 2537, 752, 2544, 856, 2447, 2048, 1944, 1093, 5388, 1447, 3083, 2150, 3827

Offset: 1

Rak-Kyeong Seong (rak-kyeong.seong(AT)imperial.ac.uk), Feb 25 2010

nonn

Also gives the number of distinct abelian orbifolds of C^5/Gamma, Gamma in SU(5).

J. Davey, A. Hanany and R. K. Seong, Counting Orbifolds, J. High Energ. Phys. (2010) 2010: 10; arXiv:1002.3609 [hep-th], 2010.
A. Hanany and R. K. Seong, Symmetries of abelian orbifolds, J. High Energ. Phys. (2011) 2011: 27; arXiv:1009.3017 [hep-th], 2010-2011. Table 5 gives a(1)-a(80), but the terms a(36) and a(65) there are apparently erroneous.
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.

Cf. A003051 (No. of two-dimensional triangular toric diagrams of area n), A045790 (No. of three-dimensional tetrahedral toric diagrams of volume n), A173877, A173878.

# see Python in A159842 for the definition of dc, fin, per, u, N, N2
def fin_d(d):
    return fin(*(d.get(n+1, 0) for n in range(max(d))))
def a(n): # see Hanany & Seong 2011, Table 1 row D=5 and Table 9
    return (dc(u, N, N2, lambda n: n**3)(n) +
        10 * dc(u, u, N, N2, fin(1, -1, 0, 8))(n) +
        15 * dc(u, u, N, N, fin_d({1: 1, 2: -3, 4: 14, 8: -12, 16: 16}))(n) +
        20 * dc(u, u, N, per(0, 1, -1), fin(1, 0, -1, 0, 0, 0, 0, 0, 9))(n) +
        20 * dc(u, u, u, per(0, 1, -1), fin(1, -1, 0, 2), fin(1, 0, -1, 0, 0, 0, 0, 0, 3))(n) +
        30 * dc(u, u, u, per(0, 1, 0, -1), fin_d({1: 1, 2: -2, 4: 3, 16: 6, 32: -8, 64: 8}))(n) +
        24 * dc(u, per(0, 1, -1, -1, 1), per(0, 1, I, -I, -1), per(0, 1, -I, I, -1))(n)) / 120
print([a(n) for n in range(1, 100)])

a(16) corrected, terms a(31) and beyond added from Hanany & Seong 2011 by Andrey Zabolotskiy, Jun 30 2019

a(36) corrected from 2202 to 2215 by Andrey Zabolotskiy, Sep 20 2022

A173877 Number of five-dimensional simplical toric diagrams with hypervolume n.

Original entry on oeis.org

1, 3, 6, 17, 13, 40, 27, 106, 78, 127, 79, 391, 129, 321, 358, 832, 285, 1070, 409, 1549

Offset: 1

Rak-Kyeong Seong (rak-kyeong.seong(AT)imperial.ac.uk), Mar 01 2010

Also gives the number of distinct abelian orbifolds of C^6/Gamma, Gamma in SU(6).

J. Davey, A. Hanany and R. K. Seong, Counting Orbifolds, J. High Energ. Phys. (2010) 2010: 10; arXiv:1002.3609 [hep-th], 2010.
A. Hanany and R. K. Seong, Symmetries of abelian orbifolds, J. High Energ. Phys. (2011) 2011: 27; arXiv:1009.3017 [hep-th], 2010-2011.
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.

Cf. A003051 (No. of two-dimensional triangular toric diagrams of area n), A045790 (No. of three-dimensional tetrahedral toric diagrams of volume n), A173824 (No. of four-dimensional simplical toric diagrams of hypervolume n), A173878.

a(9) corrected, a(15)-a(20) added from Hanany & Seong 2011 by Andrey Zabolotskiy, Jun 30 2019

cp's OEIS Frontend

A054346 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other.

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A187015 The number of different classes of 2-dimensional convex lattice polytopes having volume n/2 up to unimodular equivalence.

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A054384 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated to give the other.

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A300651 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if they are related by any rotation or reflection.

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A048259 Number of distinct solutions to x + y + z = 0 (mod n), where two solutions are equivalent if one can be obtained from the other by multiplying by units in Z/nZ and permuting x,y,z.

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A300782 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the simple cubic lattice of index n.

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A300783 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the 3D hexagonal lattice of index n.

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Python

Extensions

A006984 Greatest minimal norm of sublattice of index n in hexagonal lattice.

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A173824 Number of four-dimensional simplical toric diagrams with hypervolume n.

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