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Search: id:A113062
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| A113062 |
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Expansion of theta series of hexagonal net with respect to a node. |
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4
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| 1, 3, 0, 6, 3, 0, 0, 6, 0, 6, 0, 0, 6, 6, 0, 0, 3, 0, 0, 6, 0, 12, 0, 0, 0, 3, 0, 6, 6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, 12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, 6, 0, 0, 0, 0, 12, 0, 0, 0, 6, 0, 12, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 6, 6, 0, 0, 6, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 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 net is the familiar 2-dimensional honeycomb (not a lattice) in which each node has 3 neighbors.
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REFERENCES
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N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
A. F. Wells, Structural Inorganic Chemistry, Oxford, 5th ed., 1984; see Fig. 3.9(a.1).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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Moebius transform is period 9 sequence [3, -3, 3, 3, -3, -3, 3, -3, 0, ...].
a(3n+2)=0. a(3n+1)=A033685(3n+1), a(3n)=A004016(n).
Expansion of a(q^3)+c(q^3) in powers of q where a(q),c(q) are cubic AGM analog functions. - Michael Somos Aug 15 2006
a(3n+2)=0. a(3n+1)=A005882(n)=A033685(3n+1), a(3n)=A004016(n).
a(n)=3*b(n) where b(n) is multiplicative and b(p^e) = 2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, 3*sumdiv(n, d, [0, 1, -1, 1, 1, -1, -1, 1, -1][d%9+1]))
(PARI) {a(n)=local(A, p, e); if(n<1, n==0, A=factor(n); 3*prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 2, if(p%6==1, e+1, !(e%2))))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); abs(polcoeff( eta(x+A)^3/eta(x^3+A), n)))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)^3+6*x*eta(x^9+A)^3)/eta(x^3+A), n))} /* Michael Somos Aug 15 2006 */
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CROSSREFS
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A005928(3n)=a(3n), A005928(3n+1)=-a(3n+1). A113063(n)=a(n)/3 if n>0.
Sequence in context: A077187 A011079 A005928 this_sequence A005889 A095712 A085753
Adjacent sequences: A113059 A113060 A113061 this_sequence A113063 A113064 A113065
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Oct 13 2005; definition corrected Oct 17 2005
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