OFFSET
0,3
COMMENTS
LINKS
Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.2 for the Dirichlet g.f. zeta^SL_{x^2+y^2}(s).
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. - From N. J. A. Sloane, Feb 23 2009
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25).
EXAMPLE
For n = 1, 2, 3, 4 the sublattices are generated by the rows of:
[1 0] [2 0] [2 0] [3 0] [3 0] [4 0] [4 0] [2 0] [2 0]
[0 1] [0 1] [1 1] [0 1] [1 1] [0 1] [1 1] [0 2] [1 2].
PROG
(SageMath)
# see A159842 for the definitions of dc, fin, u, N
def ff(m, k1, minus = True):
def f(n):
if n == 1: return 1
r = 1
for (p, k) in factor(n):
if p % 4 != m or k1 and k > 1: return 0
if minus: r *= (-1)**k
return r
return f
f1, f2, f3 = ff(1, True), ff(1, True, False), ff(3, False)
def a_SL(n):
return (dc(u, N, f1)(n) + dc(u, f3)(n)) / 2
print([a_SL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, May 06 2000
STATUS
approved