|
|
A159842
|
|
Number of symmetrically-distinct supercells (sublattices) of the fcc and bcc lattices (n is the "volume factor" of the supercell).
|
|
7
|
|
|
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.
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
|
|
|
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)])
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Gus Hart (gus_hart(AT)byu.edu), Apr 23 2009
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|