cp's OEIS Frontend

A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

%I A157226 #20 Oct 01 2018 06:56:29
%S A157226 0,1,1,2,1,3,1,2,1,3,1,4,1,3,2,2,1,3,1,4,2,3,1,4,1,3,1,4,1,6,1,2,2,3,
%T A157226 2,4,1,3,2,4,1,6,1,4,2,3,1,4,1,3,2,4,1,3,2,4,2,3,1,8,1,3,2,2,2,6,1,4,
%U A157226 2,6,1,4,1,3,2,4,2,6,1,4,1,3,1,8,2,3,2
%N A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.
%C A157226 _Andrey Zabolotskiy_'s new formula confirms that a(n) indeed is a function of A305891(n). - _Antti Karttunen_, Oct 01 2018
%H A157226 Andrey Zabolotskiy, <a href="/A157226/b157226.txt">Table of n, a(n) for n = 1..5000</a>
%H A157226 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 A157226 From _Andrey Zabolotskiy_, Sep 30 2018: (Start)
%F A157226 Let b(n) = A007875(n) for n>1, b(1) = 0. Then
%F A157226 a(n) = b(n) for odd n,
%F A157226 a(n) = b(n) + b(n/2) for even n.
%F A157226 Thus the sorted list of all terms (except for a(1)=0) is A029744. (End)
%o A157226 (PARI)
%o A157226 A007875(n) = eulerphi(2^omega(n));
%o A157226 A157226(n) = if(n<=2,n-1,(A007875(n) + if(!(n%2),A007875(n/2)))); \\ _Antti Karttunen_, Oct 01 2018
%Y A157226 Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157230, A157231, A304182, A007875, A029744.
%K A157226 nonn
%O A157226 1,4
%A A157226 _N. J. A. Sloane_, Feb 25 2009
%E A157226 New name and more terms from _Andrey Zabolotskiy_, May 09 2018