A157226 - OEIS

%I #20 Oct 01 2018 06:56:29

%S 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 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 2,6,1,4,1,3,2,4,2,6,1,4,1,3,1,8,2,3,2

%N 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 _Andrey Zabolotskiy_'s new formula confirms that a(n) indeed is a function of A305891(n). - _Antti Karttunen_, Oct 01 2018

%H Andrey Zabolotskiy, <a href="/A157226/b157226.txt">Table of n, a(n) for n = 1..5000</a>

%H 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 From _Andrey Zabolotskiy_, Sep 30 2018: (Start)

%F Let b(n) = A007875(n) for n>1, b(1) = 0. Then

%F a(n) = b(n) for odd n,

%F a(n) = b(n) + b(n/2) for even n.

%F Thus the sorted list of all terms (except for a(1)=0) is A029744. (End)

%o (PARI)

%o A007875(n) = eulerphi(2^omega(n));

%o A157226(n) = if(n<=2,n-1,(A007875(n) + if(!(n%2),A007875(n/2)))); \\ _Antti Karttunen_, Oct 01 2018

%Y Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157230, A157231, A304182, A007875, A029744.

%K nonn

%O 1,4

%A _N. J. A. Sloane_, Feb 25 2009

%E New name and more terms from _Andrey Zabolotskiy_, May 09 2018