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A350871
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Number of well-rounded sublattices of index n in square lattice.
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2
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1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 2, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 4, 3, 2, 0, 0, 2, 2, 0, 1, 0, 2, 2, 1, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 8, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 0, 1, 2, 2, 4, 0, 2, 0, 0, 6, 1, 2, 0, 2, 4, 0, 0
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OFFSET
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1,5
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COMMENTS
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A sublattice is well-rounded if the linear span of its vectors of minimal length is the whole space.
A sublattice of the square lattice is well-rounded when it is square or centered rectangular (rhombic) with not too oblong unit cell: the angles of the rhombus should be at least Pi/3.
In this sequence, any two sublattices differing by any isometry are counted as distinct.
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LINKS
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FORMULA
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See Zeiner (2015), Theorem 7.3.1. [Note that the formula from Baake & Zeiner (2017) contains an error.]
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EXAMPLE
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a(4) = 1 well-rounded index-4 sublattice has basis (2, 0), (0, 2).
a(5) = 2 w.-r. index-5 sublattices have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).
At index 12, for the first time centered rectangular sublattices occur, there are a(12) = 2 of them with the bases (3, 2), (3, -2) and (2, 3), (-2, 3).
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MATHEMATICA
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fa[s_] := Count[Divisors[s], _?(#^2 < (s/#)^2 < 3 #^2 &)];
f0[s_] := If[OddQ[s], 0, 2 fa[s/2]];
f1[s_] := With[{e2 = IntegerExponent[s, 2]}, 2 (-1)^e2 fa[s/2^e2]];
pr[s_] := Count[Range[s], _?(Mod[#^2 + 1, s] == 0 &)]; (*A000089*)
sq[n_] := Sum[pr[n/d^2], {d, Select[Range[n], Mod[n, #^2] == 0 &]}]; (*A002654*)
a[n_] := sq[n] + Sum[pr[n/d] (f0[d] + f1[d]), {d, Divisors[n]}];
Array[a, 87]
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CROSSREFS
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Cf. enumeration of other classes of sublattices of Z^2: A000203 (all sublattices), A002654 (square sublattices), A000089 (primitive square sublattices), A350872 (coincidence sublattices), A145393 (all sublattices up to isometries of the parent lattice).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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