Search: id:A002324
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%I A002324 M0016 N0002
%S A002324 1,0,1,1,0,0,2,0,1,0,0,1,2,0,0,1,0,0,2,0,2,0,0,0,1,0,1,2,0,0,2,0,0,0,0,
               1,2,0,
%T A002324 2,0,0,0,2,0,0,0,0,1,3,0,0,2,0,0,0,0,2,0,0,0,2,0,2,1,0,0,2,0,0,0,0,0,2,
               0,1,2,
%U A002324 0,0,2,0,1,0,0,2,0,0,0,0,0,0,4,0,2,0,0,0,2,0,0,1,0,0,2,0,0,0,0,1,2,0,2,
               2,0,0
%N A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 
               2 (mod 3).
%C A002324 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,
               p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -3.
%C A002324 (Number of points of norm n in hexagonal lattice) / 6, n>0.
%C A002324 The hexagonal lattice is the familiar 2-dimensional lattice in which 
               each point has 6 neighbors. This is sometimes called the triangular 
               lattice.
%D A002324 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 112, first display.
%D A002324 J. W. L. Glaisher, Table of the excess of the number of (3k+1)-divisors 
               of a number over the number of (3k+2)-divisors, Messenger Math., 
               31 (1901), 64-72.
%D A002324 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 
               105, National Research Council, Washington, DC, 1941, pp. 7-10.
%D A002324 J. S. Rutherford, Generating functions for the cage isomers of the C_{20n} 
               isohedral fullerenes, J. Mathematical Chem., 14 (1993), 385-390. 
               [From N. J. A. Sloane, Mar 12 2009]
%D A002324 J. S. Rutherford, Sublattice enumeration. IV. Equivalence classes of 
               plane sublattices by parent Patterson symmetry and colour lattice 
               group type, Acta Cryst. (2009). A65, 156163. [See Table 1]. [From 
               N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]
%D A002324 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002324 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002324 T. D. Noe, <a href="b002324.txt">Table of n, a(n) for n=1..10000</a>
%H A002324 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">
               Three aspects of partitions</a>
%H A002324 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/A2.html">Home page for hexagonal (or triangular) lattice 
               A2</a>
%F A002324 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2-3v^2+4w^2-2uw+w-v. 
               - Michael Somos, Jul 20 2004
%F A002324 Has a nice Dirichlet series expansion, see PARI line.
%F A002324 G.f.: Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Dec 16 2002
%F A002324 a(3n+2)=0, a(3n)=a(n), a(3n+1)=A033687(n). - Michael Somos, Apr 04 2003
%F A002324 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, 
               u3, u6)=(u1-u3)(u3-u6)-(u2-u6)^2 . - Michael Somos May 20 2005
%F A002324 Multiplicative with a(3^e)=1, a(p^e)=e+1 if p=1 (mod 3), a(p^e)=(1+(-1)^e)/
               2 if p=2 (mod 3) . - Michael Somos May 20 2005
%F A002324 G.f.: Sum_{k>0} x^(3k-2)/(1-x^(3k-2)) -x^(3k-1)/(1-x^(3k-1)) . - Michael 
               Somos Nov 02 2005
%F A002324 q-series for a(n): Sum_{n >= 1} q^(n^2)(1-q)(1-q^2)...(1-q^(n-1))/(((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). 
               [From Jeremy Lovejoy (lovejoy(AT)liafa.jussieu.fr), Jun 12 2009]
%o A002324 (PARI) a(n)=if(n<0,0,polcoeff(sum(k=1,n,x^k/(1+x^k+x^(2*k)),x*O(x^n)),
               n)) (from Michael Somos)
%o A002324 (PARI) a(n)=if(n<1,0,sumdiv(n,d,(d%3==1)-(d%3==2)))
%o A002324 (PARI) {a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], 
               if(p=A[k,1], e=A[k,2]; if(p==3, 1, if(p%3==1, e+1, !(e%2))))))} /
               * Michael Somos May 20 2005 */
%o A002324 (PARI) a(n)=if(n<1, 0, qfrep([2,1;1,2],n,1)[n]/3) /* Michael Somos Jun 
               05 2005 */
%o A002324 (PARI) a(n)=if(n<1, 0, direuler(p=2,n,1/(1-X)/(1-kronecker(-3,p)*X))[n]) 
               /* Michael Somos Jun 05 2005 */
%Y A002324 See A004016 (=6*A002324(n), n>0) for much more information, also A035019.
%Y A002324 Cf. A145377.
%Y A002324 Sequence in context: A145171 A117154 A074941 this_sequence A101671 A078979 
               A063974
%Y A002324 Adjacent sequences: A002321 A002322 A002323 this_sequence A002325 A002326 
               A002327
%K A002324 easy,nonn,nice,mult
%O A002324 1,7
%A A002324 N. J. A. Sloane (njas(AT)research.att.com).
%E A002324 More terms from David Radcliffe (radcl008(AT)umn.edu).

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