cp's OEIS Frontend

A300782 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the simple cubic lattice of index n.

%I A300782 #20 May 09 2023 10:26:46
%S A300782 1,3,3,9,5,13,7,24,14,23,11,49,15,33,31,66,21,70,25,89,49,61,33,162,
%T A300782 50,81,75,137,49,177,55,193,97,123,99,296,75,147,129,312,89,291,97,
%U A300782 269,218,203,113,534,146,302,203,357,141,451,207,508,247,307,171,789
%N A300782 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the simple cubic lattice of index n.
%H A300782 Andrey Zabolotskiy, <a href="/A300782/b300782.txt">Table of n, a(n) for n = 1..1000</a>
%H A300782 Matt DeCross, <a href="https://mdecross.github.io/LatticePolytopesandOrbifolds.pdf">Lattice Polytopes and Orbifolds</a>, 2015.
%H A300782 Matt DeCross, <a href="https://mdecross.github.io/OrbifoldsTalk.pdf">Lattice Polytopes and Orbifolds in Quiver Gauge Theories</a>, 2015. See slides 18-21.
%H A300782 Gus L. W. Hart and Rodney W. Forcade, <a href="https://scholarsarchive.byu.edu/facpub/180/">Algorithm for generating derivative structures</a>, Phys. Rev. B 77, 224115 (2008), <a href="https://doi.org/10.1103/PhysRevB.77.224115">DOI: 10.1103/PhysRevB.77.224115</a> [see Table IV].
%H A300782 Materials Simulation Group, <a href="https://github.com/msg-byu/enumlib">Derivative structure enumeration library</a>
%H A300782 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%H A300782 <a href="/index/Cu#cubic_lattice">Index entries for sequences related to cubic lattice</a>
%o A300782 (Python)
%o A300782 # see A159842 for the definition of dc, fin, per, u, N, N2
%o A300782 def a(n): # from DeCross's slides
%o A300782     return (dc(u, N, N2)(n) + 6*dc(fin(1, -1, 0, 4), u, u, N)(n)
%o A300782       + 3*dc(fin(1, 3), u, u, N)(n)
%o A300782       + 8*dc(fin(1, 0, -1, 0, 0, 0, 0, 0, 3), u, u, per(0, 1, -1))(n)
%o A300782       + 6*dc(fin(1, 1), u, u, per(0, 1, 0, -1))(n))//24
%o A300782 print([a(n) for n in range(1, 300)])
%o A300782 # _Andrey Zabolotskiy_, Sep 02 2019
%Y A300782 Cf. A159842, A300783, A300784, A003051, A145393, A001001, A128119, A160870, A145396, A145398.
%K A300782 nonn
%O A300782 1,2
%A A300782 _Andrey Zabolotskiy_, Mar 12 2018
%E A300782 Terms a(11) and beyond from _Andrey Zabolotskiy_, Sep 02 2019