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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 11 13:41 EDT 2021. Contains 343791 sequences. (Running on oeis4.)