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.

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

Author

N. J. A. Sloane, Feb 25 2009

Keywords

• nonn

Comments

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

Links

• 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.]

Crossrefs

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

Programs

PARI

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

Formula

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)

Extensions

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