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

1,2

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

Prog

(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

Crossrefs

Cf. A045790.

Cf. A001001.

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

Cf. A173824, A173877, A173878.

Sequence in context: A060203 A131880 A045790 * A085102 A087572 A085107

Adjacent sequences: A159839 A159840 A159841 * A159843 A159844 A159845

Keyword

nonn

Author

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

Extensions

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

Status

approved