A112689 -id:A112689 - cp's OEIS frontend

A003051 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.

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

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
From Andrey Zabolotskiy, Mar 10 2018: (Start)
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)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217. [Annotated and corrected scanned copy]
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see Table 3).
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024.
W. Kurth, Enumeration of Platonic maps on the torus, Discrete Math. 61 (1986), 71-83.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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 2].
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003050, A054384, A001615, A006984, A054345, A054346, A000203, A069734, A145391, A145392, A145393, A145394, A112689, A159842.

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 *)

a(n) = Sum_{ m^2 | n } A003050(n/m^2).
a(n) = (A000203(n) + 2*A002324(n) + 3*A145390(n))/6. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ d|n } A112689(d+1). - Andrey Zabolotskiy, Aug 29 2019
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

A008679 Expansion of 1/((1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 8, 7

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 3 and 4. - Reinhard Zumkeller, Feb 09 2009
Convolution of A112689 (shifted left once) by A033999. - R. J. Mathar, Feb 13 2009
With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose largest three parts are equal. - Wesley Ivan Hurt, Jan 06 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 216
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,-1).

Cf. A005044, A033999, A112689.

a:=[1,0,0,1,1,0,1,1];; for n in [8..90] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # G. C. Greubel, Sep 09 2019

R:=PowerSeriesRing(Integers(), 90); Coefficients(R!( 1/((1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 09 2019

seq(coeff(series(1/((1-x^3)*(1-x^4)), x, n+1), x, n), n = 0..90); # G. C. Greubel, Sep 09 2019

LinearRecurrence[{0,0,1,1,0,0,-1}, {1,0,0,1,1,0,1}, 90] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
CoefficientList[Series[1/((1-x)^2(1+x)(1+x+x^2)(1+x^2)), {x,0,90}], x] (* Vincenzo Librandi, Jun 11 2013 *)

my(x='x+O('x^90)); Vec(1/((1-x^3)*(1-x^4))) \\ G. C. Greubel, Sep 09 2019

def A008679_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^3)*(1-x^4))).list()
A008679_list(90) # G. C. Greubel, Sep 09 2019

a(n+12) = a(n) + 1. - Reinhard Zumkeller, Feb 09 2009
G.f.: 1/((1-x)^2*(1+x)*(1+x+x^2)*(1+x^2)). - R. J. Mathar, Feb 13 2009
a(n) = 1 + floor(n/3) + floor(-n/4). - Tani Akinari, Sep 02 2013
E.g.f.: (1/72)*(9*exp(-x)+21*exp(x)+6*exp(x)*x+18*cos(x)+24*exp(-x/2)*cos(sqrt(3)*x/2)-18*sin(x)+8*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 09 2019
a(n) = A005044(n+3) - A005044(n+1). - Yuchun Ji, Oct 10 2020
From Wesley Ivan Hurt, Jan 17 2021: (Start)
a(n) = a(n-3) + a(n-4) - a(n-7).
a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i = n+4-i-k-j], where [ ] is the Iverson bracket. (End)

A291983 Expansion of 1/((1+x)(1+x^2)(1+x^3)).

Original entry on oeis.org

1, -1, 0, -1, 2, -1, 1, -2, 2, -2, 2, -2, 3, -3, 2, -3, 4, -3, 3, -4, 4, -4, 4, -4, 5, -5, 4, -5, 6, -5, 5, -6, 6, -6, 6, -6, 7, -7, 6, -7, 8, -7, 7, -8, 8, -8, 8, -8, 9, -9, 8, -9, 10, -9, 9, -10, 10, -10, 10, -10, 11, -11, 10, -11, 12, -11, 11, -12, 12, -12, 12

Offset: 0

Seiichi Manyama, Sep 07 2017

sign

Index entries for linear recurrences with constant coefficients, signature (-1, -1, -2, -1, -1, -1).

Cf. A112689, A291954, A291984, A291985.

CoefficientList[Series[1/((1+x)(1+x^2)(1+x^3)),{x,0,70}],x] (* or *) LinearRecurrence[ {-1,-1,-2,-1,-1,-1},{1,-1,0,-1,2,-1},80] (* Harvey P. Dale, Jun 12 2022 *)

Vec(1/((1+x)*(1+x^2)*(1+x^3)) + O(x^100))

A112690 Expansion of 1/(1 + x^2 - x^3 - x^5).

Original entry on oeis.org

0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1

Offset: 0

Paul Barry, Sep 15 2005

Partial sums are A112689.

Index entries for linear recurrences with constant coefficients, signature (0,-1,1,0,1).

Cf. A000100, A057077, A061347, A112689.

LinearRecurrence[{0, -1, 1, 0, 1}, {0, 1, 0, -1, 1}, 100] (* Vincenzo Librandi, Jul 07 2016 *)
Join[{0},CoefficientList[Series[1/(1+x^2-x^3-x^5),{x,0,100}],x]] (* Harvey P. Dale, Jul 30 2024 *)

concat(0, Vec(1/(1+x^2-x^3-x^5) + O(x^80))) \\ Michel Marcus, Jul 07 2016

a(n) = round(real((exp(-2/3*I*n*Pi)*(-4+(3+3*I)*exp((I*n*Pi)/6) + 2*exp((2*I*n*Pi)/3) + (3-3*I)*exp((7*I*n*Pi)/6) - 4*exp((4*I*n*Pi)/3)))/12)) \\ Colin Barker, Jul 07 2016

G.f.: 1/((1+x^2)*(1-x^3)).
a(n) = Sum_{k=0..n} Sum_{j=0..floor((k+1)/2)} (-1)^(k-j)*C(k-j+1, j-1).
a(n+12) = a(n) = 1/6 + A057077(n+1)/2 + A061347(n+1)/3. - R. J. Mathar, Feb 23 2009
a(n+10) = (A000100(n) mod 2)*(-1)^(1 + floor(n/2)). - John M. Campbell, Jul 07 2016
From Ilya Gutkovskiy, Jul 07 2016: (Start)
E.g.f.: (3*sin(x) + 3*cos(x) + exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/6.
a(n) = (3*sin(Pi*n/2) + 3*cos(Pi*n/2) - 4*cos(2*Pi*n/3) + 1)/6. (End)
a(n) = 2*floor(n/4) + floor((n+2)/3) - floor(n/3) - floor(n/2). - Ridouane Oudra, Mar 11 2023

cp's OEIS Frontend

A003051 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.

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A008679 Expansion of 1/((1-x^3)*(1-x^4)).

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A291983 Expansion of 1/((1+x)*(1+x^2)*(1+x^3)).

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A112690 Expansion of 1/(1 + x^2 - x^3 - x^5).

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A291983 Expansion of 1/((1+x)(1+x^2)(1+x^3)).