

A003051


Number of inequivalent sublattices of index n in hexagonal lattice (two sublattices are equivalent if one can be rotated or reflected to give the other).
(Formerly M0420)


12



1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 18, 9, 17, 16, 13, 9, 28, 12, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 21, 27, 16, 26, 13, 24, 18, 26, 13, 40, 14
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OFFSET

1,3


COMMENTS

The hexagonal lattice is the familiar 2dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.


REFERENCES

A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201217.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201217. [Annotated and corrected scanned copy]
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 2939 (Abstract, pdf, ps).
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, 156163. [See Table 2].
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice


FORMULA

a(n) = Sum_{ m^2  n } A003050(n/m^2).
a(n) = (A000203 + 2*A002324 + 3*A145390)/6. [Rutherford]  N. J. A. Sloane, Mar 13 2009


MATHEMATICA

max = 73; A145390 = Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1  x^n), {n, 1, max}], {x, 0, max}], x], 1]; A002324[n_] := (dn = Divisors[n]; Count[dn, _?(Mod[#, 3] == 1 & )]  Count[dn, _?(Mod[#, 3] == 2 & )]); a[n_] := (DivisorSigma[1, n] + 2 A002324[n] + 3*A145390[[n]])/6; Table[a[n], {n, 1, max}] (* JeanFrançois Alcover, Oct 11 2011, after given formula *)


CROSSREFS

Cf. A003050, A054384, A001615, A006984, A054345.
Sequence in context: A030582 A036762 A032154 * A257396 A237582 A097352
Adjacent sequences: A003048 A003049 A003050 * A003052 A003053 A003054


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



