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

1,2

Links

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.]

Formula

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

Example

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.

Mathematica

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 *)

Crossrefs

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.

Sequence in context: A256918 A099871 A157220 * A256284 A322979 A106288

Adjacent sequences: A304179 A304180 A304181 * A304183 A304184 A304185

Keyword

nonn,mult

Author

Andrey Zabolotskiy, May 07 2018

Status

approved