A159842 - OEIS

%I #41 Apr 06 2024 20:30:41

%S 1,2,3,7,5,10,7,20,14,18,11,41,15,28,31,58,21,60,25,77,49,54,33,144,

%T 50,72,75,123,49,158,55,177,97,112,99,268,75,136,129,286,89,268,97,

%U 249,218,190,113,496,146,280,203,333,141,421,207,476,247,290,171,735

%N Number of symmetrically-distinct supercells (sublattices) of the fcc and bcc lattices (n is the "volume factor" of the supercell).

%C The number of fcc/bcc supercells (sublattices) as a function of n (volume factor) is equivalent to the sequence A001001. But many of these sublattices are symmetrically equivalent. The current sequence lists those that are symmetrically distinct.

%C Is this the same as A045790? - _R. J. Mathar_, Apr 28 2009

%C This sequence also gives number of sublattices of index n for the diamond structure - see Hanany, Orlando & Reffert, sec. 6.3 (they call it the tetrahedral lattice). Indeed: the diamond structure consists of two interpenetrating fcc lattices, and all sites of any sublattice should belong to the same fcc lattice because every sublattice is inversion-symmetric. - _Andrey Zabolotskiy_, Mar 18 2018

%H Andrey Zabolotskiy, <a href="/A159842/b159842.txt">Table of n, a(n) for n = 1..1000</a>

%H J. Davey, A. Hanany and R. K. Seong, <a href="https://doi.org/10.1007/JHEP06%282010%29010">Counting Orbifolds</a>, J. High Energ. Phys., 2010, 10; arXiv:<a href="https://arxiv.org/abs/1002.3609">1002.3609</a> [hep-th], 2010.

%H Amihay Hanany, Domenico Orlando, and Susanne Reffert, <a href="https://doi.org/10.1007/JHEP06(2010)051">Sublattice counting and orbifolds</a>, J. High Energ. Phys., 2010 (2010), 51, <a href="https://arxiv.org/abs/1002.2981">arXiv.org:1002.2981 [hep-th]</a>, 2010.

%H Gus L. W. Hart and Rodney W. Forcade, <a href="https://hdl.lib.byu.edu/1877/2951">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>.

%H Materials Simulation Group at Brigham Young University, <a href="https://github.com/msg-byu/enumlib">Derivative structure enumeration library</a>.

%H Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda and Isao Tanaka, <a href="https://doi.org/10.1063/5.0021663">Enumeration of nonequivalent substitutional structures using advanced data structure of binary decision diagram</a>, J. Chem. Phys. 153, 104109 (2020); preprint: <a href="https://arxiv.org/abs/2002.12603">Derivative structure enumeration using binary decision diagram</a>, arXiv:2002.12603 [physics.comp-ph], 2020.

%H Andrey Zabolotskiy, <a href="https://arxiv.org/abs/2003.10251">Coweight lattice A^*_n and lattice simplices</a>, arXiv:2003.10251 [math.CO], 2020.

%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>

%H <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>

%H <a href="/index/Ba#bcc">Index entries for sequences related to b.c.c. lattice</a>

%o (Python)

%o def dc(f, *r): # Dirichlet convolution of multiple sequences

%o     if not r:

%o         return f

%o     return lambda n: sum(f(d)*dc(*r)(n//d) for d in range(1, n+1) if n%d == 0)

%o def fin(*a): # finite sequence

%o     return lambda n: 0 if n > len(a) else a[n-1]

%o def per(*a): # periodic sequences

%o     return lambda n: a[n%len(a)]

%o u, N, N2 = lambda n: 1, lambda n: n, lambda n: n**2

%o def a(n): # Hanany, Orlando & Reffert, sec. 6.3

%o     return (dc(u, N, N2)(n) + 9*dc(fin(1, -1, 0, 4), u, u, N)(n)

%o             + 8*dc(fin(1, 0, -1, 0, 0, 0, 0, 0, 3), u, u, per(0, 1, -1))(n)

%o             + 6*dc(fin(1, -1, 0, 2), u, u, per(0, 1, 0, -1))(n))//24

%o print([a(n) for n in range(1, 300)])

%o # _Andrey Zabolotskiy_, Mar 18 2018

%Y Cf. A045790.

%Y Cf. A001001.

%Y Cf. A003051, A145393, A145391, A145398, A300782, A300783, A300784.

%Y Cf. A173824, A173877, A173878.

%K nonn

%O 1,2

%A Gus Hart (gus_hart(AT)byu.edu), Apr 23 2009

%E Terms a(20) and beyond from _Andrey Zabolotskiy_, Mar 18 2018