A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.

1, 3, 2, 5, 2, 6, 2, 7, 3, 6, 2, 10, 2, 6, 4, 9, 2, 9, 2, 10, 4, 6, 2, 14, 3, 6, 4, 10, 2, 12, 2, 11, 4, 6, 4, 15, 2, 6, 4, 14, 2, 12, 2, 10, 6, 6, 2, 18, 3, 9, 4, 10, 2, 12, 4, 14, 4, 6, 2, 20, 2, 6, 6, 13, 4, 12, 2, 10, 4, 12, 2, 21, 2, 6, 6, 10, 4, 12, 2, 18, 5, 6, 2, 20, 4, 6, 4, 14, 2, 18

Offset: 1

Author

Valery A. Liskovets, Apr 07 2002

Keywords

• mult
• easy
• nonn

Comments

Dirichlet convolution of A000012 by A040001. - R. J. Mathar, Mar 30 2011

a(n) is the number of full-dimensional lattices with volume n in Z^2 which are symmetric about a coordinate axis (equivalently, about both). - Álvar Ibeas, Mar 19 2021

Examples

			x + 3*x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 3*x^9 + 6*x^10 + ...
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
• Valery A. Liskovets and Alexander Mednykh, Number of non-orientable coverings of the Klein bottle, 2002.
• 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

Crossrefs

Equals row sums of triangle A143110. - Gary W. Adamson, Jul 25 2008

Cf. A000005, A000012, A001227, A001620, A040001, A059841, A069734, A099774, A183063, A304182, A320111.

Programs

Maple

read("transforms") : nmax := 100 :
L := [1,1,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017
with(NumberTheory): seq(tau(n) + `if`(n::odd, 0, tau(n/2)), n=1..100); # Peter Luschny, Mar 19 2021

Mathematica

d[n_] := DivisorSigma[0, n];
a[n_] := If[EvenQ[n], d[n] + d[n/2], d[n]];
Array[a, 100] (* Jean-François Alcover, Aug 27 2019 *)

PARI

{a(n) = if( n<1, 0, numdiv(n) + if( n%2, 0, numdiv( n / 2)))} /* Michael Somos, Mar 24 2012 */

Formula

Multiplicative with a(2^e)=2e+1 and a(p^e)=e+1 for e>0 and an odd prime p.

a(n) = d(n)+d(n/2) for even n and a(n) = d(n) otherwise where d(n) is the number of divisors of n (A000005).

G.f.: Sum_{k>0} x^k*(1+2*x^k)/(1-x^(2*k)). - Vladeta Jovovic, Dec 16 2002

Dirichlet g.f.: (1+2^(-s))*zeta^2(s) [ Rutherford]. - N. J. A. Sloane, Feb 23 2009

Moebius transform is period 2 sequence [ 1, 2, ...]. - Michael Somos, Mar 24 2012

a(2*n - 1) = A099774(n).

a(n) = Sum_{ m: m^2|n } A304182(n/m^2). - Andrey Zabolotskiy, May 07 2018

Sum_{k=1..n} a(k) ~ 3*n*log(n)/2 + (3*gamma - 3/2 - log(2)/2)*n, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 04 2019

a(n) = 3*tau(n) - tau(2*n). - Ridouane Oudra, Mar 15 2021

a(n) = A320111(n) + (A059841(n)*A000005(n)), i.e. a(n) = A320111(n) if n is odd, and a(n) = A320111(n) + A000005(n) if n is even. - Antti Karttunen, Mar 17 2021

a(n) = A000005(n) + A183063(n) = 2*A000005(n) - A001227(n). - Amiram Eldar, Dec 22 2023

Extensions

Corrected by T. D. Noe, Nov 13 2006