cp's OEIS Frontend

A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.

%I A157228 #19 Jan 24 2022 04:43:34
%S A157228 0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,1,
%T A157228 0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,2,0,0,0,
%U A157228 0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,2,0,0
%N A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.
%C A157228 From _Andrey Zabolotskiy_, May 09 2018: (Start)
%C A157228 Also, the number of partitions of n into 2 distinct coprime squares.
%C A157228 All such sublattices (including non-primitive ones) are counted in A025441.
%C A157228 The primitive sublattices that have the same symmetries (including the orientation of the mirrors) as the parent lattice are not counted here; they are counted in A019590, and all such sublattices (including non-primitive ones) are counted in A053866.
%C A157228 For n > 2, equals A193138. (End)
%H A157228 Andrey Zabolotskiy, <a href="/A157228/b157228.txt">Table of n, a(n) for n = 1..5000</a>
%H A157228 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 5.]
%F A157228 a(n) = (A000089(n) - A019590(n)) / 2. - _Andrey Zabolotskiy_, May 09 2018
%F A157228 a(n) = 1 if n>2 is in A224450, a(n) = 2 if n is in A224770, a(n) is a higher power of 2 if n is in A281877 (first time reaches 8 at n = 32045). - _Andrey Zabolotskiy_, Sep 30 2018
%F A157228 a(n) = b(n) for odd n, a(n) = b(n/2) for even n, where b(n) = A024362(n). - _Andrey Zabolotskiy_, Jan 23 2022
%Y A157228 Cf. A193138, A145393 (all sublattices of the square lattice), A025441, A019590, A053866, A157226, A157230, A157231, A000089, A304182, A224450, A224770, A281877, A024362.
%K A157228 nonn
%O A157228 1,65
%A A157228 _N. J. A. Sloane_, Feb 25 2009
%E A157228 New name and more terms from _Andrey Zabolotskiy_, May 09 2018