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Search: id:A005882
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| A005882 |
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Theta series of planar hexagonal lattice (A2) with respect to deep hole.
(Formerly M2281)
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+0
9
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| 3, 3, 6, 0, 6, 3, 6, 0, 3, 6, 6, 0, 6, 0, 6, 0, 9, 6, 0, 0, 6, 3, 6, 0, 6, 6, 6, 0, 0, 0, 12, 0, 6, 3, 6, 0, 6, 6, 0, 0, 3, 6, 6, 0, 12, 0, 6, 0, 0, 6, 6, 0, 6, 0, 6, 0, 9, 6, 6, 0, 6, 0, 0, 0, 6, 9, 6, 0, 0, 6, 6, 0, 12, 0, 6, 0, 6, 0, 0, 0, 6, 6, 12, 0, 0, 3, 12, 0, 0, 6, 6, 0, 6, 0, 6, 0, 3, 6, 0, 0, 12
(list; graph; listen)
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OFFSET
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0,1
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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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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
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Expansion of 3eta(q^3)^3/(eta(q)q^(1/3)) in powers of q.
Expansion of q^(-1/3)c(q) in powers of q where c(q) is the third cubic AGM analog function.
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^3+2*u*w^2-u^2*w - Michael Somos Aug 15 2006
G.f.: 3 Product_{k>0} (1-q^(3k))^3/(1-q^k).
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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)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( 3*eta(x^3+A)^3/eta(x+A), n))} /* Michael Somos Aug 15 2006 */
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CROSSREFS
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Essentially same as A033685 and A033687.
a(n)=3 A033687(n). a(n)=A113062(3n+1)=A033685(3n+1).
Sequence in context: A110426 A093310 A132809 this_sequence A085572 A010609 A066519
Adjacent sequences: A005879 A005880 A005881 this_sequence A005883 A005884 A005885
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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