A157235 -id:A157235 - cp's OEIS frontend

A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.

1, 3, 2, 4, 2, 6, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 2, 6, 2, 8, 4, 6, 2, 8, 2, 6, 2, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 8, 4, 6, 2, 8, 2, 6, 4, 8, 2, 6, 4, 8, 4, 6, 2, 16, 2, 6, 4, 4, 4, 12, 2, 8, 4, 12, 2, 8, 2, 6, 4, 8, 4, 12, 2, 8, 2, 6, 2, 16, 4

Offset: 1

Andrey Zabolotskiy, May 07 2018

			There are 6 = A001615(4) lattices in Z^2 whose quotient group is C_4. The reflection through an axis relates <(4,0), (1,1)> and <(4,0), (3,1)>. The remaining 4 = a(4) lattices are fixed.

Álvar Ibeas, Table of n, a(n) for n = 1..10000
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 4. Contains errors for n = 24 and 28.]

Cf. A069735 (not only primitive sublattices), A304183 (primitive oblique sublattices), A069734 (all sublattices).
Cf. other columns of tables 4 and 5 from [Rutherford, 2009]: A001615, A060594, A157223, A000089, A157224, A000086, A157227, A019590, A157228, A157226, A157230, A157231, A154272, A157235.
Cf. A034444, A001221, A001620, A306016.

f[p_, e_] := If[p == 2, If[e == 1, 3, 4], 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

From Álvar Ibeas, Mar 18 2021: (Start)
For n odd, a(n) = A034444(n) = 2^(A001221(n)).
For n even, a(n) = A034444(n) + A034444(n/2). If 4|n, a(n) = 2^(A001221(n) + 1); otherwise, a(n) = 3 * 2^(A001221(n) - 1).
Multiplicative with a(2) = 3, a(2^e) = 4 (for e>1), and a(p^e) = 2 (for p>2).
Dirichlet g.f.: (1+2^(-s)) * zeta(s)^2 / zeta(2s).
(End)
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - log(2)/3 - 2*zeta'(2)/zeta(2) - 1)*9*n/Pi^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 31 2022

A003050 Number of primitive sublattices of index n in hexagonal lattice: triples x,y,z from Z/nZ with x+y+z = 0, discarding any triple that can be obtained from another by multiplying by a unit and permuting.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 3, 4, 3, 6, 4, 5, 6, 6, 4, 7, 5, 8, 8, 7, 5, 12, 6, 8, 7, 10, 6, 14, 7, 10, 10, 10, 10, 14, 8, 11, 12, 16, 8, 18, 9, 14, 14, 13, 9, 20, 11, 16, 14, 16, 10, 19, 14, 20, 16, 16, 11, 28, 12, 17, 18, 18, 16, 26, 13, 20, 18, 26, 13, 28

Offset: 1

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.
Also the number of triangles with vertices on points of the hexagonal lattice that have area equal to n/2. - Amihay Hanany, Oct 15 2009 [Here the area is measured in the units of the lattice unit cell area; since the number of the triangles of different shapes with the same half-integral area is infinite, the triangles are probably counted up to the equivalence relation defined in the Davey, Hanany and Rak-Kyeong Seong paper. Also, this comment probably belongs to A003051, not here. - Andrey Zabolotskiy, Mar 10 2018 and Jul 04 2019]
Also number of 2n-vertex connected cubic vertex-transitive graphs which are Cayley graphs for a dihedral group [Potočnik et al.]. - N. J. A. Sloane, Apr 19 2014

			For n = 6 the 3 primitive triples are 510, 411, 321.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217. [Annotated and corrected scanned copy]
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).
J. Davey, A. Hanany and Rak-Kyeong Seong, Counting Orbifolds, arXiv:1002.3609 [hep-th], 2010.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Primož Potočnik, Pablo Spiga and Gabriel Verret, A census of small connected cubic vertex-transitive graphs (See the sub-page Table.html, column headed "Dihedrants"). - _N. J. A. Sloane_, Apr 19 2014
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for sequences related to sublattices

Cf. A003051 (not only primitive sublattices), A001615, A006984, A007997, A048259, A054345, A154272, A157235.

Join[{1}, Table[p=Transpose[FactorInteger[n]][[1]]; If[Mod[n,2]==0 || Mod[n,9]==0, c1=0, c1=Product[(1+JacobiSymbol[p[[i]],3]), {i,Length[p]}]]; c2={2,1,0,1,1,1,0,1}[[1+Mod[n,8]]]; n*Product[(1+1/p[[i]]), {i, Length[p]}]/6+c1/3+2^(Length[p]-2+c2), {n,2,100}]] (* T. D. Noe, Oct 18 2009 *)

Let n = Product_{i=1..w} p_i^e_i. Then a(n) = (1/6)*n*Product_{i=1..w} (1 + 1/p_i) + (C_1)/3 + 2^(w-2+C_2),
where C_1 = 0 if 2|n or 9|n, = Product_{i=1..w, p_i > 3} (1 + Legendre(p_i, 3)) otherwise,
and C_2 = 2 if n == 0 (mod 8), 1 if n == 1, 3, 4, 5, 7 (mod 8), 0 if n == 2, 6 (mod 8).

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A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.

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