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A145390 Number of sublattices of index n of a centered rectangular lattice fixed by a reflection. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois Alcover, Sep 20 2011, after formula *)

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: A286529 A306225 A077199 * A270026 A128049 A104543

Adjacent sequences:  A145387 A145388 A145389 * A145391 A145392 A145393

KEYWORD

nonn,mult

AUTHOR

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

EXTENSIONS

New name from Andrey Zabolotskiy, Mar 10 2018

STATUS

approved

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Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)