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A003051
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Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.
(Formerly M0420)
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24
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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
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OFFSET
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1,3
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COMMENTS
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The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
If only primitive sublattices are considered, we get A003050.
Here only rotations and reflections preserving the parent hexagonal lattice are allowed. If reflections are not allowed, we get A145394. If any rotations and reflections are allowed, we get A300651.
In other words, the parent lattice of the sublattices under consideration has Patterson symmetry group p6mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6).
Rutherford says at p. 161 that his sequence for p6mm differs from this sequence, but it seems that with the current definition and terms of this sequence, this actually is his p6mm sequence, and the sequence he thought to be this one is actually A300651. Also, he says that a(n) != A300651(n) only when A002324(n) > 2 (first time happens at n = 49), but actually these two sequences differ at other terms, too, for example, at n = 42 (see illustration). (End)
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = Sum_{ m^2 | n } A003050(n/m^2).
a(n) = Sum_{ d|n } floor(d/6) + 1 - 1*[d == 2 or 6 (mod 12)] + 1*[d == 4 (mod 12)]. [Kurth] - Brahadeesh Sankarnarayanan, Feb 24 2023
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MATHEMATICA
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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}] (* Jean-François Alcover, Oct 11 2011, after given formula *)
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CROSSREFS
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Cf. A003050, A054384, A001615, A006984, A054345, A054346, A000203, A069734, A145391, A145392, A145393, A145394, A112689, A159842.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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