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Search: id:A003050
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| A003050 |
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Number of primitive sublattices of index n in hexagonal lattice: triples x,y,z from Z/nZ with x+y+z=0, discarding any triple that can be obtained from another by multiplying by a unit and permuting.
(Formerly M0229)
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+0
7
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| 1, 1, 2, 2, 2, 3, 3, 4, 3, 4, 3, 6, 4, 5, 6, 6, 4, 7, 5, 8, 8, 7, 5, 12, 6, 8, 7, 10, 6, 14, 7, 10, 10, 10, 10, 14, 8, 11, 12, 16, 8, 18, 9, 14, 14, 13, 9, 20, 11, 16, 14, 16, 10, 19, 14, 20, 16, 16, 11, 28, 12, 17, 18, 18, 16, 26, 13, 20, 18, 26, 13, 28
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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.
Also the number of triangles with vertices on points of the hexagonal lattice that have area equal to n/2. - Amihay Hanany, Oct 15 2009
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REFERENCES
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A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.
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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T. D. Noe, Table of n, a(n) for n=1..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).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for sequences related to sublattices
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FORMULA
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Let n = Product_{i=1..w} p_i^e_i. Then a(n) = (1/6) * n prod_{i=1..w} (1 + 1/p_i) + (C_1)/3 + 2^(w-2+C_2), where
C_2 = 2 if n == 0 mod 8, 1 if n == 1, 3, 4, 5, 7 mod 8, 0 if n == 2, 6 mod 8.
C_1 = 0 if 2|n or 9|n, = prod_{i=1..w, p_i > 3} ( 1+ Legendre(p_i, 3)) otherwise and
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EXAMPLE
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For n = 6 the 3 primitive triples are 510, 411, 321.
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MATHEMATICA
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Join[{1}, Table[p=Transpose[FactorInteger[n]][[1]]; If[Mod[n, 2]==0 || Mod[n, 9]==0, c1=0, c1=Product[(1+JacobiSymbol[p[[i]], 3]), {i, Length[p]}]]; c2={2, 1, 0, 1, 1, 1, 0, 1}[[1+Mod[n, 8]]]; n*Product[(1+1/p[[i]]), {i, Length[p]}]/6+c1/3+2^(Length[p]-2+c2), {n, 2, 100}]] [From T. D. Noe (noe(AT)sspectra.com), Oct 18 2009]
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CROSSREFS
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Cf. A003051, A001615, A006984, A007997, A048259, A054345.
Adjacent sequences: A003047 A003048 A003049 this_sequence A003051 A003052 A003053
Sequence in context: A038809 A078342 A107325 this_sequence A070868 A155216 A064144
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KEYWORD
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nonn,nice,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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