A145390 Number of sublattices of index n of a centered rectangular lattice fixed by a reflection.

1, 1, 2, 3, 2, 2, 2, 5, 3, 2, 2, 6, 2, 2, 4, 7, 2, 3, 2, 6, 4, 2, 2, 10, 3, 2, 4, 6, 2, 4, 2, 9, 4, 2, 4, 9, 2, 2, 4, 10, 2, 4, 2, 6, 6, 2, 2, 14, 3, 3, 4, 6, 2, 4, 4, 10, 4, 2, 2, 12, 2, 2, 6, 11, 4, 4, 2, 6, 4, 4, 2, 15, 2, 2, 6, 6, 4, 4, 2, 14, 5, 2, 2, 12, 4, 2, 4, 10, 2, 6, 4, 6, 4, 2, 4, 18, 2, 3, 6, 9, 2

Offset

1,3

Comments

a(n) is the Dirichlet convolution of A000012 and A098178. - Domenico (domenicoo(AT)gmail.com), Oct 21 2009

Links

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000

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).

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 1]. - From N. J. A. Sloane, Feb 23 2009

Formula

Dirichlet g.f.: (1-2^(-s) + 2*4^(-s))*zeta^2(s).

G.f.: Sum_n (1 + cos(n*Pi/2)) x^n / (1 - x^n). - Domenico (domenicoo(AT)gmail.com), Oct 21 2009

If 4|n then a(n) = d(n) - d(n/2) + 2*d(n/4); else if 2|n then a(n) = d(n) - d(n/2); else a(n) = d(n); where d(n) is the number of divisors of n. [Rutherford] - Andrey Zabolotskiy, Mar 10 2018

a(n) = Sum_{ m: m^2|n } A060594(n/m^2). - Andrey Zabolotskiy, May 07 2018

Sum_{k=1..n} a(k) ~ n*(log(n) - 1 + 2*gamma - log(2)/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019

Multiplicative with a(2^e) = 2*e-1 and a(p^e) = e+1 for an odd prime p. - Amiram Eldar, Aug 27 2023

Maple

nmax := 100 :
L := [1, -1, 0, 2, seq(0, i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

Mathematica

m = 101; Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1 - x^n), {n, 1, m}], {x, 0, m}], x], 1] (* Jean-François Alcover, Sep 20 2011, after formula *)
f[p_, e_] := e+1; f[2, e_] := 2*e-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

Prog

(PARI) t1=direuler(p=2, 200, 1/(1-X)^2)
t2=direuler(p=2, 2, 1-X+2*X^2, 200)
t3=dirmul(t1, t2)

Crossrefs

Cf. A098178, A060594 (primitive sublattices only), A145391.

Sequence in context: A306225 A373249 A077199 * A270026 A340703 A128049

Adjacent sequences: A145387 A145388 A145389 * A145391 A145392 A145393

Keyword

nonn,easy,mult

Author

N. J. A. Sloane, Feb 23 2009, Mar 13 2009

Extensions

New name from Andrey Zabolotskiy, Mar 10 2018

Status

approved