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Search: id:A004012
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| A004012 |
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Theta series of hexagonal close-packing.
(Formerly M4817)
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+0
1
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| 1, 0, 0, 12, 0, 0, 6, 0, 2, 18, 0, 12, 6, 0, 0, 12, 0, 12, 6, 6, 12, 24, 6, 0, 0, 12, 0, 12, 0, 24, 12, 12, 2, 12, 6, 24, 6, 12, 0, 24, 0, 12, 0, 6, 24, 12, 12, 24, 6, 12, 0, 24, 0, 24, 18, 12, 12, 24, 0, 12, 0, 12, 0, 36, 0, 24, 12, 18, 12, 24, 12, 48, 2, 0, 0, 36, 0, 0, 24, 12, 12
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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The theta series of even layers is a(q^3)*theta_3(q^8) and odd layers is c(q^3)*theta_2(q^8). - Michael Somos Aug 15 2006
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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. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143, esp. p. 143.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..5000
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FORMULA
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{t3(8z/3)-t2(8z/3)/2}*{t3(z)t3(3z)+t2(z)t2(3z)}+(1/2)*t2(8z/3)*{t3(z/3)t3(z)+t2(z/3)t2(z)}, where t3=theta_3, t2=theta_2.
Expansion of a(q^3)theta_3(q^8) +c(q^3)theta_2(q^8) in powers of q where a(q), c(q) are cubic AGM analog functions (see A004016, A005882).
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PROGRAM
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(PARI) {a(n)=local(A, A0, A1); if(n<0, 0, A=x*O(x^n); A1=x^3*eta(x^9+A)^3*eta(x^32+A)^2/eta(x^3+A)/eta(x^16+A); A0=sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1+A)* sum(k=1, sqrtint(n\8), 2*x^(8*k^2), 1+A)* sum(k=1, sqrtint(n\9), 2*x^(9*k^2), 1+A); polcoeff(2*A0+6*A1-subst(A0, x, -x), n)) } /* Michael Somos Aug 03 2006 */
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CROSSREFS
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Sequence in context: A063863 A101364 A104203 this_sequence A072837 A023917 A064141
Adjacent sequences: A004009 A004010 A004011 this_sequence A004013 A004014 A004015
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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