A350871 - OEIS

%I #16 Jan 24 2022 06:41:52

%S 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,

%T 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,

%U 0,2,0,1,2,2,4,0,2,0,0,6,1,2,0,2,4,0,0

%N Number of well-rounded sublattices of index n in square lattice.

%C A sublattice is well-rounded if the linear span of its vectors of minimal length is the whole space.

%C 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.

%C In this sequence, any two sublattices differing by any isometry are counted as distinct.

%H Andrey Zabolotskiy, <a href="/A350871/b350871.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Baake and Peter Zeiner, <a href="https://arxiv.org/abs/1709.07317">Geometric enumeration problems for lattices and embedded Z-modules</a>, arXiv:1709.07317 [math.MG], 2017; in: <a href="http://www.aperiodicorder.org">Aperiodic Order</a>, 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.

%H Peter Zeiner, <a href="https://core.ac.uk/display/211846842">Coincidence Site Lattices and Coincidence Site Modules</a>, Thesis, Universität Bielefeld, 2015.

%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>

%H <a href="/index/Sq#sqlatt">Index entries for sequences related to square lattice</a>

%F See Zeiner (2015), Theorem 7.3.1. [Note that the formula from Baake & Zeiner (2017) contains an error.]

%e a(4) = 1 well-rounded index-4 sublattice has basis (2, 0), (0, 2).

%e a(5) = 2 w.-r. index-5 sublattices have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).

%e 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).

%t fa[s_] := Count[Divisors[s], _?(#^2 < (s/#)^2 < 3 #^2 &)];

%t f0[s_] := If[OddQ[s], 0, 2 fa[s/2]];

%t f1[s_] := With[{e2 = IntegerExponent[s, 2]}, 2 (-1)^e2 fa[s/2^e2]];

%t pr[s_] := Count[Range[s], _?(Mod[#^2 + 1, s] == 0 &)]; (*A000089*)

%t sq[n_] := Sum[pr[n/d^2], {d, Select[Range[n], Mod[n, #^2] == 0 &]}]; (*A002654*)

%t a[n_] := sq[n] + Sum[pr[n/d] (f0[d] + f1[d]), {d, Divisors[n]}];

%t Array[a, 87]

%Y 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).

%Y Cf. A097584.

%K nonn

%O 1,5

%A _Andrey Zabolotskiy_, Jan 20 2022