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

1,4

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.]

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)

Prog

(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

Crossrefs

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

Sequence in context: A108103 A111376 A241664 * A156249 A187808 A317673

Adjacent sequences: A157223 A157224 A157225 * A157227 A157228 A157229

Keyword

nonn

Author

N. J. A. Sloane, Feb 25 2009

Extensions

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

Status

approved