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Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.
(Formerly M0420)
+10
24
1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 18, 9, 17, 16, 13, 9, 28, 12, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 21, 27, 16, 26, 13, 24, 18, 26, 13, 40, 14
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
If only primitive sublattices are considered, we get A003050. Here only rotations and reflections preserving the parent hexagonal lattice are allowed. If reflections are not allowed, we get A145394. If any rotations and reflections are allowed, we get A300651. In other words, the parent lattice of the sublattices under consideration has Patterson symmetry group p6mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6). Rutherford says at p. 161 that his sequence for p6mm differs from this sequence, but it seems that with the current definition and terms of this sequence, this actually is his p6mm sequence, and the sequence he thought to be this one is actually A300651. Also, he says that a(n) != A300651(n) only when A002324(n) > 2 (first time happens at n = 49), but actually these two sequences differ at other terms, too, for example, at n = 42 (see illustration). (End)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = Sum_{ m^2 | n } A003050(n/m^2). a(n) = Sum_{ d|n } floor(d/6) + 1 - 1*[d == 2 or 6 (mod 12)] + 1*[d == 4 (mod 12)]. [Kurth] - Brahadeesh Sankarnarayanan, Feb 24 2023
MATHEMATICA
max = 73; A145390 = Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1 - x^n), {n, 1, max}], {x, 0, max}], x], 1]; A002324[n_] := (dn = Divisors[n]; Count[dn, _?(Mod[#, 3] == 1 & )] - Count[dn, _?(Mod[#, 3] == 2 & )]); a[n_] := (DivisorSigma[1, n] + 2 A002324[n] + 3* A145390[[n]])/6; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Oct 11 2011, after given formula *)
CROSSREFS
Cf. A003050, A054384, A001615, A006984, A054345, A054346, A000203, A069734, A145391, A145392, A145393, A145394, A112689, A159842.
Number of inequivalent sublattices of index n in a square lattice, where two sublattices are considered equivalent if one can be rotated to give the other.
+10
8
1, 1, 2, 2, 4, 3, 6, 4, 8, 7, 8, 6, 14, 7, 12, 10, 16, 9, 20, 10, 18, 16, 18, 12, 30, 13, 20, 20, 28, 15, 30, 16, 32, 24, 26, 20, 46, 19, 30, 26, 38, 21, 48, 22, 42, 33, 36, 24, 62, 29, 38, 34, 46, 27, 60, 30, 60, 40, 44, 30, 70, 31, 48, 52, 64, 33, 72, 34, 60, 48, 60
COMMENTS
If reflections are allowed, we get A054346. If only rotations that preserve the parent square lattice are allowed, we get A145392. The analog for a hexagonal lattice is A054384.
EXAMPLE
For n = 1, 2, 3, 4 the sublattices are generated by the rows of:
[1 0] [2 0] [2 0] [3 0] [3 0] [4 0] [4 0] [2 0] [2 0]
[0 1] [0 1] [1 1] [0 1] [1 1] [0 1] [1 1] [0 2] [1 2].
PROG
(SageMath)
# see A159842 for the definitions of dc, fin, u, N def ff(m, k1, minus = True):
def f(n):
if n == 1: return 1
r = 1
for (p, k) in factor(n):
if p % 4 != m or k1 and k > 1: return 0
if minus: r *= (-1)**k
return r
return f
f1, f2, f3 = ff(1, True), ff(1, True, False), ff(3, False)
def a_SL(n):
return (dc(u, N, f1)(n) + dc(u, f3)(n)) / 2
Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other.
+10
7
1, 1, 2, 2, 4, 3, 5, 3, 7, 5, 7, 4, 11, 5, 8, 8, 12, 6, 13, 6, 15, 10, 11, 7, 21, 9, 13, 12, 18, 9, 21, 9, 21, 14, 16, 13, 29, 11, 17, 16, 28, 12, 28, 12, 25, 21, 20, 13, 39, 16, 24, 20, 29, 15, 34, 18, 36, 22, 25, 16, 47, 17, 26, 29, 38, 21, 40, 18, 36, 26, 36, 19, 58, 20
COMMENTS
If we count sublattices as equivalent only if they are related by a rotation, we get A054345 instead of this sequence. If we only allow rotations and reflections that preserve the parent (square) lattice, we get A145393; the first discrepancy is at n = 25 (see illustration), the second is at n = 30. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145392. The analog for the hexagonal lattice is A300651. - Andrey Zabolotskiy, Mar 12 2018
EXAMPLE
For n = 1, 2, 3, 4 the sublattices are generated by the rows of:
[1 0] [2 0] [2 0] [3 0] [3 0] [4 0] [4 0] [2 0] [2 0]
[0 1] [0 1] [1 1] [0 1] [1 1] [0 1] [1 1] [0 2] [1 2].
PROG
(SageMath)
def a_GL(n):
return (a_SL(n) + dc(fin(1, 0, 0, 1), u, u, f2)(n)) / 2
Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated to give the other.
+10
5
1, 1, 1, 2, 3, 2, 4, 3, 5, 5, 6, 4, 10, 5, 7, 8, 11, 6, 13, 7, 14, 10, 12, 8, 20, 11, 13, 14, 17, 10, 24, 11, 21, 16, 18, 14, 31, 13, 19, 18, 30, 14, 28, 15, 28, 26, 24, 16, 42, 17, 31, 24, 31, 18, 40, 24, 35, 26, 30, 20, 56, 21, 31, 31, 43, 26, 48, 23, 42, 32, 42, 24, 65
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
If reflections are allowed, we get A300651. If only rotations that preserve the parent hexagonal lattice are allowed, we get A145394. The analog for square lattice is A054345. - Andrey Zabolotskiy, Mar 10 2018
PROG
(SageMath)
# see A159842 for the definitions of dc, fin, u, N def gg(m, k1, minus = True):
def f(n):
if n == 1: return 1
r = 1
for (p, k) in factor(n):
if p % 3 != m or k1 and k > 1: return 0
if minus: r *= (-1)**k
return r
return f
g1, g2, g3 = gg(1, True), gg(1, True, False), gg(2, False)
def a_SL(n):
return (dc(u, N, g1)(n) + 2 * dc(u, g3)(n)) / 3
Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if they are related by any rotation or reflection.
+10
5
1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 17, 9, 17, 16, 13, 9, 28, 11, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 20, 27, 16, 26, 13, 24, 18, 24, 13, 40
COMMENTS
If we count sublattices as equivalent only if they are related by a rotation, we get A054384 instead of this sequence. If we only allow rotations and reflections that preserve the parent (hexagonal) lattice, we get A003051; the first discrepancy is at n = 42 (see illustration), the second is at n = 49. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145394. The analog for square lattice is A054346. Although A003051 has its counterpart A003050 which counts primitive sublattices only, this sequence has no such counterpart sequence because a primitive sublattice can turn to a non-primitive one via a non-parent-lattice-preserving rotation, so the straightforward definition of primitiveness does not work in this case.
PROG
(SageMath)
def a_GL(n):
return (a_SL(n) + dc(fin(1, -1, 0, 2), u, u, g2)(n)) / 2
Three-dimensional simplices of determinant n.
+10
4
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
COMMENTS
Two simplices are considered equal if some integer affine automorphism sends the first to the second.
AUTHOR
Jacques-Olivier Moussafir (msfr(AT)ceremade.dauphine.fr)
Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the simple cubic lattice of index n.
+10
4
1, 3, 3, 9, 5, 13, 7, 24, 14, 23, 11, 49, 15, 33, 31, 66, 21, 70, 25, 89, 49, 61, 33, 162, 50, 81, 75, 137, 49, 177, 55, 193, 97, 123, 99, 296, 75, 147, 129, 312, 89, 291, 97, 269, 218, 203, 113, 534, 146, 302, 203, 357, 141, 451, 207, 508, 247, 307, 171, 789
PROG
(Python)
# see A159842 for the definition of dc, fin, per, u, N, N2 def a(n): # from DeCross's slides
return (dc(u, N, N2)(n) + 6*dc(fin(1, -1, 0, 4), u, u, N)(n)
+ 3*dc(fin(1, 3), 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), u, u, per(0, 1, 0, -1))(n))//24
print([a(n) for n in range(1, 300)])
Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the 3D hexagonal lattice of index n.
+10
4
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
PROG
(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)])
Number of four-dimensional simplical toric diagrams with hypervolume n.
+10
3
1, 2, 4, 10, 8, 19, 13, 45, 33, 47, 30, 129, 43, 96, 108, 226, 78, 264, 102, 357, 226, 277, 163, 813, 260, 425, 436, 780, 297, 1092, 355, 1281, 678, 856, 712, 2215, 569, 1155, 1050, 2537, 752, 2544, 856, 2447, 2048, 1944, 1093, 5388, 1447, 3083, 2150, 3827
COMMENTS
Also gives the number of distinct abelian orbifolds of C^5/Gamma, Gamma in SU(5).
PROG
(Sage)
# see Python in A159842 for the definition of dc, fin, per, u, N, N2 def fin_d(d):
return fin(*(d.get(n+1, 0) for n in range(max(d))))
def a(n): # see Hanany & Seong 2011, Table 1 row D=5 and Table 9
return (dc(u, N, N2, lambda n: n**3)(n) +
10 * dc(u, u, N, N2, fin(1, -1, 0, 8))(n) +
15 * dc(u, u, N, N, fin_d({1: 1, 2: -3, 4: 14, 8: -12, 16: 16}))(n) +
20 * dc(u, u, N, per(0, 1, -1), fin(1, 0, -1, 0, 0, 0, 0, 0, 9))(n) +
20 * dc(u, u, u, per(0, 1, -1), fin(1, -1, 0, 2), fin(1, 0, -1, 0, 0, 0, 0, 0, 3))(n) +
30 * dc(u, u, u, per(0, 1, 0, -1), fin_d({1: 1, 2: -2, 4: 3, 16: 6, 32: -8, 64: 8}))(n) +
24 * dc(u, per(0, 1, -1, -1, 1), per(0, 1, I, -I, -1), per(0, 1, -I, I, -1))(n)) / 120
print([a(n) for n in range(1, 100)])
CROSSREFS
Cf. A003051 (No. of two-dimensional triangular toric diagrams of area n), A045790 (No. of three-dimensional tetrahedral toric diagrams of volume n), A173877, A173878.
AUTHOR
Rak-Kyeong Seong (rak-kyeong.seong(AT)imperial.ac.uk), Feb 25 2010
EXTENSIONS
a(16) corrected, terms a(31) and beyond added from Hanany & Seong 2011 by Andrey Zabolotskiy, Jun 30 2019
Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the tetragonal lattice of index n.
+10
3
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
PROG
(Python)
# 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)])
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