A300783 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the 3D hexagonal lattice of index n.

1, 3, 5, 11, 7, 19, 11, 34, 23, 33, 19, 77, 25, 53, 55, 104, 37, 115, 45, 143, 91, 105, 61, 272, 90, 139, 137, 235, 91, 309, 103, 331, 183, 219, 185, 516, 141, 267, 245, 544, 169, 529, 185, 485, 411, 375, 217, 952, 278, 550, 389, 647, 271, 829, 397, 922, 477

Offset: 1

Author

Andrey Zabolotskiy, Mar 12 2018

Keywords

• nonn

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
• 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 [see Table IV].
• Materials Simulation Group, 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.
• Index entries for sequences related to sublattices

Crossrefs

Cf. A159842, A300782, A300784, A003051, A145393.

Programs

Python

# see A159842 for the definitions of dc, fin, per, u, N, N2
def a(n):
    return (dc(u, N, N2)(n) + 6*dc(fin(1, -1, 0, 4), u, u, N)(n)
            + dc(fin(1, 3), u, u, N)(n)
            + 4*dc(fin(1, 0, 1), u, u, per(0, 1, -1))(n)) // 12
print([a(n) for n in range(1, 100)])
# Andrey Zabolotskiy, Feb 03 2020

Extensions

Terms a(11) and beyond from Andrey Zabolotskiy, Feb 03 2020