A159842 Number of symmetrically-distinct supercells (sublattices) of the fcc and bcc lattices (n is the "volume factor" of the supercell).

1, 2, 3, 7, 5, 10, 7, 20, 14, 18, 11, 41, 15, 28, 31, 58, 21, 60, 25, 77, 49, 54, 33, 144, 50, 72, 75, 123, 49, 158, 55, 177, 97, 112, 99, 268, 75, 136, 129, 286, 89, 268, 97, 249, 218, 190, 113, 496, 146, 280, 203, 333, 141, 421, 207, 476, 247, 290, 171, 735

Offset: 1

Author

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

Keywords

• nonn

Comments

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.

Is this the same as A045790? - R. J. Mathar, Apr 28 2009

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

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
• J. Davey, A. Hanany and R. K. Seong, Counting Orbifolds, J. High Energ. Phys., 2010, 10; arXiv:1002.3609 [hep-th], 2010.
• Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, J. High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th], 2010.
• Gus L. W. Hart and Rodney W. Forcade, Algorithm for generating derivative structures, Phys. Rev. B 77, 224115 (2008), DOI: 10.1103/PhysRevB.77.224115.
• Materials Simulation Group at Brigham Young University, Derivative structure enumeration library.
• Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda and Isao Tanaka, Enumeration of nonequivalent substitutional structures using advanced data structure of binary decision diagram, J. Chem. Phys. 153, 104109 (2020); preprint: Derivative structure enumeration using binary decision diagram, arXiv:2002.12603 [physics.comp-ph], 2020.
• Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
• Index entries for sequences related to sublattices
• Index entries for sequences related to f.c.c. lattice
• Index entries for sequences related to b.c.c. lattice

Crossrefs

Cf. A045790.

Cf. A001001.

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

Cf. A173824, A173877, A173878.

Programs

Python

def dc(f, *r): # Dirichlet convolution of multiple sequences
    if not r:
        return f
    return lambda n: sum(f(d)*dc(*r)(n//d) for d in range(1, n+1) if n%d == 0)
def fin(*a): # finite sequence
    return lambda n: 0 if n > len(a) else a[n-1]
def per(*a): # periodic sequences
    return lambda n: a[n%len(a)]
u, N, N2 = lambda n: 1, lambda n: n, lambda n: n**2
def a(n): # Hanany, Orlando & Reffert, sec. 6.3
    return (dc(u, N, N2)(n) + 9*dc(fin(1, -1, 0, 4), u, u, N)(n)
            + 8*dc(fin(1, 0, -1, 0, 0, 0, 0, 0, 3), u, u, per(0, 1, -1))(n)
            + 6*dc(fin(1, -1, 0, 2), u, u, per(0, 1, 0, -1))(n))//24
print([a(n) for n in range(1, 300)])
# Andrey Zabolotskiy, Mar 18 2018

Extensions

Terms a(20) and beyond from Andrey Zabolotskiy, Mar 18 2018