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A069735
Number of regular orientable coverings of the Klein bottle with 2n lists.
9
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
1,2
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
LINKS
Valery A. Liskovets and Alexander Mednykh, Number of non-orientable coverings of the Klein bottle, 2002.
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
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
Equals row sums of triangle A143110. - Gary W. Adamson, Jul 25 2008
Sequence in context: A057034 A075410 A023513 * A358536 A274457 A328579
KEYWORD
mult,easy,nonn
AUTHOR
Valery A. Liskovets, Apr 07 2002
EXTENSIONS
Corrected by T. D. Noe, Nov 13 2006
STATUS
approved