A281877 -id:A281877 - cp's OEIS frontend

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

N. J. A. Sloane, Feb 25 2009

nonn

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)

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.]

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

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

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

A037942 Numbers of the form x^2 + y^2 with x >= 0, y >= 0, gcd(x,y)=1, with multiplicity.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 25, 26, 29, 34, 37, 41, 50, 53, 58, 61, 65, 65, 73, 74, 82, 85, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 130, 137, 145, 145, 146, 149, 157, 169, 170, 170, 173, 178, 181, 185, 185, 193, 194

Offset: 1

Nissim Broudo (broudo(AT)brain.math.fau.edu)

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Eric Weisstein's World of Mathematics, Postage Stamp Problem.

If repetitions are deleted, same as A008784.

Cf. A224770 (multiplicity 2), A281877 (multiplicity >= 3).

s=14; v=[]; for(y=0,s,for(x=0,y,if(gcd(x,y)==1,r=x*x+y*y; if(r<=s*s,v=concat(v,r))))); vecsort(v) \\ Craig Clapp (craig_clapp(AT)earthlink.net), Apr 30 2010

Missing second instance of 85 (6^2+7^2 = 2^2+9^2 = 85) added by Craig Clapp (craig_clapp(AT)earthlink.net), Apr 30 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions