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Search: id:A002324
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| A002324 |
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Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).
(Formerly M0016 N0002)
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+0
15
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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1,7
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COMMENT
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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -3.
(Number of points of norm n in hexagonal lattice) / 6, n>0.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 112, first display.
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. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
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]
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]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
G. E. Andrews, Three aspects of partitions
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
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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
Has a nice Dirichlet series expansion, see PARI line.
G.f.: Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 16 2002
a(3n+2)=0, a(3n)=a(n), a(3n+1)=A033687(n). - Michael Somos, Apr 04 2003
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
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
G.f.: Sum_{k>0} x^(3k-2)/(1-x^(3k-2)) -x^(3k-1)/(1-x^(3k-1)) . - Michael Somos Nov 02 2005
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]
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PROGRAM
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(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)
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, (d%3==1)-(d%3==2)))
(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 */
(PARI) a(n)=if(n<1, 0, qfrep([2, 1; 1, 2], n, 1)[n]/3) /* Michael Somos Jun 05 2005 */
(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 */
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CROSSREFS
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See A004016 (=6*A002324(n), n>0) for much more information, also A035019.
Cf. A145377.
Sequence in context: A145171 A117154 A074941 this_sequence A101671 A078979 A063974
Adjacent sequences: A002321 A002322 A002323 this_sequence A002325 A002326 A002327
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KEYWORD
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easy,nonn,nice,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David Radcliffe (radcl008(AT)umn.edu).
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