

A159842


Number of symmetricallydistinct 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
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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 inversionsymmetric.  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 [hepth]
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hepth]
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[n1]
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



