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Search: id:A033685
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| A033685 |
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Theta series of hexagonal lattice A_2 with respect to deep hole. |
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+0
8
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| 0, 3, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 3, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 9, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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. 111.
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LINKS
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G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
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a(3n)=a(3n+2)=0.
G.f.: 3x Product_{k>0} (1-x^(9k))^3/(1-x^(3k)) = 3 Sum_{k>0} x^k(1-x^k)*(1-x^(2k))*(1-x^(4k))/(1-x^(9k)) . - Michael Somos Jul 15 2005
Expansion of 3*eta(q^9)^3/eta(q^3) in powers of q.
Expansion of c(q^3) in powers of q where c(q) is a cubic AGM analog function. - Michael Somos Oct 17 2006
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EXAMPLE
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3*q^(1/3)+3*q^(4/3)+6*q^(7/3)+6*q^(13/3)+3*q^(16/3)+O(q^(19/3))+...
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PROGRAM
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(PARI) a(n)=if((n<0)|(n%3!=1), 0, 3*sumdiv(n, d, kronecker(d, 3))) /* Michael Somos Jul 16 2005 */
(PARI) {a(n)=local(A); if((n<0)|(n%3!=1), 0, n=n\3; A=x*O(x^n); 3*polcoeff( eta(x^3+A)^3/eta(x+A), n))} /* Michael Somos Jul 16 2005 */
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CROSSREFS
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Essentially same as A005882 and A033687.
a(3n+1)=A005882(n)=3 A033687(n)=-A005928(3n+1)=A004016(3n+1)/2.
Sequence in context: A117138 A095104 A021337 this_sequence A063691 A075874 A111787
Adjacent sequences: A033682 A033683 A033684 this_sequence A033686 A033687 A033688
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KEYWORD
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nonn
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AUTHOR
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njas
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