OFFSET
1,2
COMMENTS
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
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
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
EXAMPLE
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 + ...
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 *)
PROG
(PARI) {a(n) = if( n<1, 0, numdiv(n) + if( n%2, 0, numdiv( n / 2)))} /* Michael Somos, Mar 24 2012 */
CROSSREFS
KEYWORD
mult,easy,nonn
AUTHOR
Valery A. Liskovets, Apr 07 2002
EXTENSIONS
Corrected by T. D. Noe, Nov 13 2006
STATUS
approved