A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0

Offset

1,65

Comments

From Andrey Zabolotskiy, May 09 2018: (Start)

Also, the number of partitions of n into 2 distinct coprime squares.

All such sublattices (including non-primitive ones) are counted in A025441.

The primitive sublattices that have the same symmetries (including the orientation of the mirrors) as the parent lattice are not counted here; they are counted in A019590, and all such sublattices (including non-primitive ones) are counted in A053866.

For n > 2, equals A193138. (End)

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

a(n) = (A000089(n) - A019590(n)) / 2. - Andrey Zabolotskiy, May 09 2018

a(n) = 1 if n>2 is in A224450, a(n) = 2 if n is in A224770, a(n) is a higher power of 2 if n is in A281877 (first time reaches 8 at n = 32045). - Andrey Zabolotskiy, Sep 30 2018

a(n) = b(n) for odd n, a(n) = b(n/2) for even n, where b(n) = A024362(n). - Andrey Zabolotskiy, Jan 23 2022

Crossrefs

Cf. A193138, A145393 (all sublattices of the square lattice), A025441, A019590, A053866, A157226, A157230, A157231, A000089, A304182, A224450, A224770, A281877, A024362.

Sequence in context: A102354 A370222 A349615 * A193138 A255320 A256574

Adjacent sequences: A157225 A157226 A157227 * A157229 A157230 A157231

Keyword

nonn

Author

N. J. A. Sloane, Feb 25 2009

Extensions

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

Status

approved