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A003051
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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)
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11
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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REFERENCES
| A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.
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]. [From N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice
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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
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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}] (* From Jean-François Alcover, Oct 11 2011, after given formula *)
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CROSSREFS
| Cf. A003050, A054384, A001615, A006984, A054345.
Sequence in context: A030582 A036762 A032154 * A097352 A076050 A130799
Adjacent sequences: A003048 A003049 A003050 * A003052 A003053 A003054
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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