A003051 -id:A003051 - cp's OEIS frontend

A173878 Number of six-dimensional simplical toric diagrams with hypervolume n.

1, 3, 7, 23, 19, 65, 46, 202, 156, 281, 183, 972, 333, 903, 1029, 2507, 912

Offset: 1

Rak-Kyeong Seong (rak-kyeong.seong(AT)imperial.ac.uk), Mar 01 2010

Also gives the number of distinct abelian orbifolds of C^7/Gamma, Gamma in SU(7).

Gabriele Balletti, Dataset of "small" lattice polytopes
J. Davey, A. Hanany and R. K. Seong, Counting Orbifolds, J. High Energ. Phys. (2010) 2010: 10; arXiv:1002.3609 [hep-th], 2010.
A. Hanany and R. K. Seong, Symmetries of abelian orbifolds, J. High Energ. Phys. (2011) 2011: 27; arXiv:1009.3017 [hep-th], 2010-2011.
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.

Cf. A003051 (No. of two-dimensional triangular toric diagrams of area n), A045790 (No. of three-dimensional tetrahedral toric diagrams of volume n), A173824 (No. of four-dimensional simplical toric diagrams of hypervolume n), A173877 (No. of five-dimensional simplical toric diagrams of hypervolume n).

a(8)-a(16) from Balletti's data and a(17) from Table 15 of Hanany & Seong 2011 added by Andrey Zabolotskiy, Mar 13 2020

A300784 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the tetragonal lattice of index n.

Original entry on oeis.org

1, 5, 5, 17, 9, 29, 13, 51, 28, 53, 25, 115, 33, 81, 73, 153, 51, 176, 61, 219, 121, 161, 85, 403, 126, 213, 188, 353, 129, 473, 145, 487, 257, 335, 261, 776, 201, 405, 345, 815, 243, 801, 265, 731, 584, 569, 313, 1407, 398, 838, 559, 975, 393, 1256, 573, 1375

Offset: 1

Andrey Zabolotskiy, Mar 12 2018

nonn

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
Index entries for sequences related to sublattices

Cf. A159842, A300782, A300783, A003051, A145393.

# see A159842 for the definition of dc, fin, per, u, N, N2
def a(n):
    return (dc(u, N, N2)(n) + 2*dc(fin(1, -1, 0, 4), u, u, N)(n)
      + 3*dc(fin(1, 3), u, u, N)(n)
      + 2*dc(fin(1, 1), u, u, per(0, 1, 0, -1))(n)) // 8
print([a(n) for n in range(1, 300)])
# Andrey Zabolotskiy, Jan 31 2020

Terms a(11) and beyond from Andrey Zabolotskiy, Jan 31 2020

A157235 Number of primitive inequivalent oblique sublattices of hexagonal (triangular) lattice of index n (equivalence and symmetry of sublattices are determined using only parent lattice symmetries).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 2, 1, 3, 2, 2, 2, 5, 2, 4, 3, 5, 3, 4, 4, 6, 5, 6, 4, 10, 4, 6, 6, 8, 6, 10, 5, 9, 7, 8, 6, 14, 6, 10, 10, 11, 7, 12, 8, 14, 10, 12, 8, 17, 10, 12, 11, 14, 9, 20, 9, 15, 14, 14, 12, 22, 10, 16, 14, 22, 11, 20, 11, 18, 18, 18

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 5.]

Cf. A003051 (all sublattices), A003050 (all primitive sublattices), A154272 (primitive sublattices fully inheriting the parent lattice symmetry, inlcuding the orientation of the mirrors), A000086 (primitive rotation-symmetric sublattices, counting mirror images as distinct), A060594 (primitive mirror-symmetric sublattices), A145377 (all sublattices inheriting the parent lattice symmetry), A304182.

a(n) = A003050(n) - (A000086(n)-A154272(n))/2 - A060594(n). - Andrey Zabolotskiy, Mar 19 2021

New name and a(1)=0 prepended by Andrey Zabolotskiy, May 09 2018

Terms a(31) and beyond from Andrey Zabolotskiy, Mar 19 2021

cp's OEIS Frontend

A173878 Number of six-dimensional simplical toric diagrams with hypervolume n.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A300784 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the tetragonal lattice of index n.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Extensions

A157235 Number of primitive inequivalent oblique sublattices of hexagonal (triangular) lattice of index n (equivalence and symmetry of sublattices are determined using only parent lattice symmetries).

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions