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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). 6
 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 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, arXiv:1002.3609 [hep-th], 2010. Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, 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 superstructures, Phys. Rev. B 77, 224115 (2008), DOI: 10.1103/PhysRevB.77.224115. Materials Simulation Group, Derivative structure enumeration library Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda, Isao Tanaka, 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. 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 July 28 17:15 EDT 2021. Contains 346335 sequences. (Running on oeis4.)