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A159842 Number of symmetrically-distinct supercells (sublattices) of the fcc and bcc lattices (n is the "volume factor" of the supercell). 5
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 (list; graph; refs; listen; history; text; internal format)
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 ? [From R. J. Mathar, Apr 28 2009]

This sequence also gives number of sublattices of index n for the diamond lattice - see Hanany, Orlando & Reffert, sec. 6.3 (they call it the tetrahedral lattice). Indeed: the point group of the diamond with respect to the node is the same as the quotient of the automorphism group of the fcc lattice by the space inversion, but space inversion adds nothing to the equivalence of sublattices 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, arXiv:1002.3609 [hep-th]

Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th]

Gus L. W. Hart and Rodney W. Forcade, Algorithm for generating derivative superstructures, Phys. Rev. B 77, 224115 (2008), DOI: 10.1103/PhysRevB.77.224115

Materials Simulation Group

Materials Simulation Group, Derivative structure enumeration library

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

MAPLE

See published paper by Hart & Forcade

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

More terms from Andrey Zabolotskiy, Mar 18 2018

STATUS

approved

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Last modified June 20 05:01 EDT 2019. Contains 324229 sequences. (Running on oeis4.)