A054384 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated to give the other.
1, 1, 1, 2, 3, 2, 4, 3, 5, 5, 6, 4, 10, 5, 7, 8, 11, 6, 13, 7, 14, 10, 12, 8, 20, 11, 13, 14, 17, 10, 24, 11, 21, 16, 18, 14, 31, 13, 19, 18, 30, 14, 28, 15, 28, 26, 24, 16, 42, 17, 31, 24, 31, 18, 40, 24, 35, 26, 30, 20, 56, 21, 31, 31, 43, 26, 48, 23, 42, 32, 42, 24, 65
Offset: 0
Links
- Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
- M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
- Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.1 for the Dirichlet g.f. zeta^SL_{x^2+xy+y^2}(s).
- G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
- 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 2]. - From _N. J. A. Sloane_, Feb 23 2009
- Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
- Index entries for sequences related to sublattices
- Index entries for sequences related to A2 = hexagonal = triangular lattice
Programs
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SageMath
# see A159842 for the definitions of dc, fin, u, N def gg(m, k1, minus = True): def f(n): if n == 1: return 1 r = 1 for (p, k) in factor(n): if p % 3 != m or k1 and k > 1: return 0 if minus: r *= (-1)**k return r return f g1, g2, g3 = gg(1, True), gg(1, True, False), gg(2, False) def a_SL(n): return (dc(u, N, g1)(n) + 2 * dc(u, g3)(n)) / 3 print([a_SL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024
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