A145393 -id:A145393 - cp's OEIS frontend

A167156 Number of n-vertex 4-hedrites.

1, 0, 2, 0, 2, 0, 4, 0, 3, 0, 5, 0, 3, 0, 7, 0, 5, 0, 7, 0, 4, 0, 11, 0, 5, 0, 8, 0, 8, 0, 12, 0, 6, 0, 13, 0, 6, 0, 15, 0, 10, 0, 11, 0, 7, 0, 21, 0, 10, 0, 13, 0, 12, 0, 18, 0, 9, 0, 22, 0, 9, 0, 21, 0, 14, 0, 16, 0, 14

Offset: 2

Jonathan Vos Post, Oct 29 2009

nonn

Is this the same as A145393 alternating with zeros? - Andrey Zabolotskiy, Jul 05 2017

			Although every other term is zero, this sequence should be kept, contrary to the usual OEIS rules, to be analogous to the related sequences.

Mathieu Dutour Sikiric, Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.

Cf. A167157, A167158, A167159, A111361.

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.

Original entry on oeis.org

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

A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 4, 1, 1, 1, 4, 2, 1, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

After a(2), this matches A034380 except for n = 63, 65, 80, 85, ... - R. J. Mathar, Feb 27 2009 [Updated by Andrey Zabolotskiy, May 09 2018]

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
J. S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Act. Cryst. A65 (2009) 156-163, Table 5 symmetry *mm2.

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157231, A304182, A060594, A046072, A033948, A272592.

a[n_] := If[n <= 2, 0, Sum[Boole[Mod[k^2, n] == 1], {k, 1, n}]/2];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 12 2023 *)

From Andrey Zabolotskiy, Sep 30 2018: (Start)
a(n) = (A060594(n) - A019590(n))/2.
a(n) = 2^(A046072(n)-1) for n>2. Thus a(n) = 1 if n>2 is in A033948, a(n) = 2 if n is in A272592, etc. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 2, 2, 1, 3, 1, 4, 2, 3, 1, 4, 1, 3, 1, 4, 1, 6, 1, 2, 2, 3, 2, 4, 1, 3, 2, 4, 1, 6, 1, 4, 2, 3, 1, 4, 1, 3, 2, 4, 1, 3, 2, 4, 2, 3, 1, 8, 1, 3, 2, 2, 2, 6, 1, 4, 2, 6, 1, 4, 1, 3, 2, 4, 2, 6, 1, 4, 1, 3, 1, 8, 2, 3, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

Andrey Zabolotskiy's new formula confirms that a(n) indeed is a function of A305891(n). - Antti Karttunen, Oct 01 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
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 5.]

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157230, A157231, A304182, A007875, A029744.

A007875(n) = eulerphi(2^omega(n));
A157226(n) = if(n<=2,n-1,(A007875(n) + if(!(n%2),A007875(n/2)))); \\ Antti Karttunen, Oct 01 2018

From Andrey Zabolotskiy, Sep 30 2018: (Start)
Let b(n) = A007875(n) for n>1, b(1) = 0. Then
a(n) = b(n) for odd n,
a(n) = b(n) + b(n/2) for even n.
Thus the sorted list of all terms (except for a(1)=0) is A029744. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157231 Number of primitive inequivalent oblique sublattices of square lattice of index n (equivalence and symmetry of sublattices are determined using parent lattice symmetries).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 3, 7, 4, 6, 6, 7, 5, 8, 6, 8, 8, 9, 6, 14, 7, 10, 10, 11, 10, 15, 8, 13, 12, 14, 9, 20, 10, 15, 16, 16, 11, 20, 13, 20, 16, 18, 12, 25, 16, 20, 18, 20, 14, 30, 14, 22, 22, 22, 18, 32, 16, 24, 22, 32, 17, 32, 17

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
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 5.]

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157230, A001615, A304182.

a(n) = (A001615(n) - A019590(n) - 2 * (A157228(n) + A157226(n) + A157230(n))) / 4. - Andrey Zabolotskiy, May 09 2018

New name and more terms from Andrey Zabolotskiy, May 09 2018

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

A300784 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the tetragonal lattice of index n.

Original entry on oeis.org

1, 5, 5, 17, 9, 29, 13, 51, 28, 53, 25, 115, 33, 81, 73, 153, 51, 176, 61, 219, 121, 161, 85, 403, 126, 213, 188, 353, 129, 473, 145, 487, 257, 335, 261, 776, 201, 405, 345, 815, 243, 801, 265, 731, 584, 569, 313, 1407, 398, 838, 559, 975, 393, 1256, 573, 1375

Offset: 1

Andrey Zabolotskiy, Mar 12 2018

nonn

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

Cf. A159842, A300782, A300783, A003051, A145393.

# see A159842 for the definition of dc, fin, per, u, N, N2
def a(n):
    return (dc(u, N, N2)(n) + 2*dc(fin(1, -1, 0, 4), u, u, N)(n)
      + 3*dc(fin(1, 3), u, u, N)(n)
      + 2*dc(fin(1, 1), u, u, per(0, 1, 0, -1))(n)) // 8
print([a(n) for n in range(1, 300)])
# Andrey Zabolotskiy, Jan 31 2020

Terms a(11) and beyond from Andrey Zabolotskiy, Jan 31 2020

A350871 Number of well-rounded sublattices of index n in square lattice.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 2, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 4, 3, 2, 0, 0, 2, 2, 0, 1, 0, 2, 2, 1, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 8, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 0, 1, 2, 2, 4, 0, 2, 0, 0, 6, 1, 2, 0, 2, 4, 0, 0

Offset: 1

Andrey Zabolotskiy, Jan 20 2022

nonn

A sublattice is well-rounded if the linear span of its vectors of minimal length is the whole space.
A sublattice of the square lattice is well-rounded when it is square or centered rectangular (rhombic) with not too oblong unit cell: the angles of the rhombus should be at least Pi/3.
In this sequence, any two sublattices differing by any isometry are counted as distinct.

			a(4) = 1 well-rounded index-4 sublattice has basis (2, 0), (0, 2).
a(5) = 2 w.-r. index-5 sublattices have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).
At index 12, for the first time centered rectangular sublattices occur, there are a(12) = 2 of them with the bases (3, 2), (3, -2) and (2, 3), (-2, 3).

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Michael Baake and Peter Zeiner, Geometric enumeration problems for lattices and embedded Z-modules, arXiv:1709.07317 [math.MG], 2017; in: Aperiodic Order, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172. See table "Some counts of the enumeration problems for Z^2"; beware of the typo in the 60th term.
Peter Zeiner, Coincidence Site Lattices and Coincidence Site Modules, Thesis, Universität Bielefeld, 2015.
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. enumeration of other classes of sublattices of Z^2: A000203 (all sublattices), A002654 (square sublattices), A000089 (primitive square sublattices), A350872 (coincidence sublattices), A145393 (all sublattices up to isometries of the parent lattice).

Cf. A097584.

fa[s_] := Count[Divisors[s], _?(#^2 < (s/#)^2 < 3 #^2 &)];
f0[s_] := If[OddQ[s], 0, 2 fa[s/2]];
f1[s_] := With[{e2 = IntegerExponent[s, 2]}, 2 (-1)^e2 fa[s/2^e2]];
pr[s_] := Count[Range[s], _?(Mod[#^2 + 1, s] == 0 &)]; (*A000089*)
sq[n_] := Sum[pr[n/d^2], {d, Select[Range[n], Mod[n, #^2] == 0 &]}]; (*A002654*)
a[n_] := sq[n] + Sum[pr[n/d] (f0[d] + f1[d]), {d, Divisors[n]}];
Array[a, 87]

See Zeiner (2015), Theorem 7.3.1. [Note that the formula from Baake & Zeiner (2017) contains an error.]

cp's OEIS Frontend

A167156 Number of n-vertex 4-hedrites.

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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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A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

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A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

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A157231 Number of primitive inequivalent oblique sublattices of square lattice of index n (equivalence and symmetry of sublattices are determined using parent lattice symmetries).

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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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Python

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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

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A300784 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the tetragonal lattice of index n.

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Crossrefs

Programs

Python

Extensions

A350871 Number of well-rounded sublattices of index n in square lattice.

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