cp's OEIS Frontend

A145392 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/2 to give the other.

%I A145392 #29 Apr 04 2021 08:36:36
%S A145392 1,2,2,4,4,6,4,8,7,10,6,14,8,12,12,16,10,20,10,22,16,18,12,30,17,22,
%T A145392 20,28,16,36,16,32,24,28,24,46,20,30,28,46,22,48,22,42,40,36,24,62,29,
%U A145392 48,36,50,28,60,36,60,40,46,30,84,32,48,52,64,44,72,34,64,48,72
%N A145392 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/2 to give the other.
%C A145392 From _Andrey Zabolotskiy_, Mar 12 2018: (Start)
%C A145392 The parent lattice of the sublattices under consideration has Patterson symmetry group p4, 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), A145393 (p4mm), A145394 (p6), A003051 (p6mm).
%C A145392 If we count sublattices related by parent-lattice-preserving reflection as equivalent, we get A145393 instead of this sequence. If we count sublattices related by rotation of the sublattice only (but not parent lattice; equivalently, sublattices related by rotation by angle which is not a multiple of Pi/2; see illustration in links) as equivalent, we get A054345. If we count sublattices related by any rotation or reflection as equivalent, we get A054346.
%C A145392 Rutherford says at p. 161 that a(n) != A054345(n) only when A002654(n) > 1, but actually these two sequences differ at other terms, too, for example, at n = 15 (see illustration). (End)
%H A145392 Antti Karttunen, <a href="/A145392/b145392.txt">Table of n, a(n) for n = 1..16384</a>
%H A145392 John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. [See Table 2; beware the typo in a(13).]
%H A145392 Andrey Zabolotskiy, <a href="/A145392/a145392.pdf">Sublattices of the square lattice</a> (illustrations for n = 1..6, 15, 25)
%H A145392 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%H A145392 <a href="/index/Sq#sqlatt">Index entries for sequences related to square lattice</a>
%F A145392 a(n) = (A000203(n) + A002654(n))/2. [Rutherford] - _N. J. A. Sloane_, Mar 13 2009
%F A145392 a(n) = Sum_{ m: m^2|n } A000089(n/m^2) + A157224(n/m^2) = A002654(n) + Sum_{ m: m^2|n } A157224(n/m^2). - _Andrey Zabolotskiy_, May 07 2018
%F A145392 a(n) = Sum_{ d|n } A004525(d). - _Andrey Zabolotskiy_, Aug 29 2019
%o A145392 (PARI)
%o A145392 A002654(n) = sumdiv(n, d, (d%4==1) - (d%4==3));
%o A145392 A145392(n) = ((sigma(n) + A002654(n))/2); \\ _Antti Karttunen_, Nov 23 2017
%Y A145392 Cf. A000203, A002654, A069734, A145391, A145393, A145394, A003051, A054345, A054346, A000089, A157224, A004525.
%K A145392 nonn
%O A145392 1,2
%A A145392 _N. J. A. Sloane_, Feb 23 2009
%E A145392 New name from _Andrey Zabolotskiy_, Mar 12 2018