A145390 - OEIS

%I #33 Aug 27 2023 04:22:52

%S 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,

%T 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,

%U 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

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

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

%H Andrey Zabolotskiy, <a href="/A145390/b145390.txt">Table of n, a(n) for n = 1..10000</a>

%H Amihay Hanany, Domenico Orlando, and Susanne Reffert, <a href="https://doi.org/10.1007/JHEP06(2010)051">Sublattice counting and orbifolds</a>, High Energ. Phys., 2010 (2010), 51, <a href="https://arxiv.org/abs/1002.2981">arXiv.org:1002.2981 [hep-th]</a> (see Table 3).

%H John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009

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

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

%F 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

%F a(n) = Sum_{ m: m^2|n } A060594(n/m^2). - _Andrey Zabolotskiy_, May 07 2018

%F 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

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

%p nmax := 100 :

%p L := [1,-1,0,2,seq(0,i=1..nmax)] :

%p MOBIUSi(%) :

%p MOBIUSi(%) ; # _R. J. Mathar_, Sep 25 2017

%t 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 *)

%t 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 *)

%o (PARI) t1=direuler(p=2,200,1/(1-X)^2)

%o t2=direuler(p=2,2,1-X+2*X^2,200)

%o t3=dirmul(t1,t2)

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

%K nonn,easy,mult

%O 1,3

%A _N. J. A. Sloane_, Feb 23 2009, Mar 13 2009

%E New name from _Andrey Zabolotskiy_, Mar 10 2018