OFFSET
1,5
COMMENTS
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.
LINKS
Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Michael Baake and Peter Zeiner, Geometric enumeration problems for lattices and embedded Z-modules, arXiv:1709.07317 [math.MG], 2017; in: Aperiodic Order, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172. See table "Some counts of the enumeration problems for Z^2"; beware of the typo in the 60th term.
Peter Zeiner, Coincidence Site Lattices and Coincidence Site Modules, Thesis, Universität Bielefeld, 2015.
FORMULA
See Zeiner (2015), Theorem 7.3.1. [Note that the formula from Baake & Zeiner (2017) contains an error.]
EXAMPLE
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).
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrey Zabolotskiy, Jan 20 2022
STATUS
approved