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A033687
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Theta series of hexagonal lattice A_2 with respect to deep hole.
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29
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1, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 2, 0, 2, 0, 3, 2, 0, 0, 2, 1, 2, 0, 2, 2, 2, 0, 0, 0, 4, 0, 2, 1, 2, 0, 2, 2, 0, 0, 1, 2, 2, 0, 4, 0, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 3, 2, 2, 0, 2, 0, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 2, 4, 0, 0, 1, 4, 0, 0, 2, 2, 0, 2, 0, 2, 0, 1, 2, 0, 0, 4, 2, 2, 0, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Denoted by g_1(q) in Cynk and Hulek in Remark 3.4 on page 12
a(n)=0 if and only if A000731(n)=0 (see the Han-Ono paper). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 16 2008
Number of 3-core partitions of n (denoted c_3(n) in Granville and Ono, p. 340). - Brian Hopkins (bhopkins(AT)spc.edu), May 13 2008
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REFERENCES
| 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 697.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.35) and (32.351).
G.-N. Han and Ken Ono, Hook lengths and 3-cores (available at http://www-irma.u-strasbg.fr/~guoniu/hook/hh3core).
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
J. Lovejoy and O. Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., 6 (2008), 23-36. [From Jeremy Lovejoy (lovejoy(AT)liafa.jussieu.fr), Jun 12 2009]
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
| Euler transform of period 3 sequence [1, 1, -2, ...].
Expansion of eta(q^3)^3/(eta(q)q^(1/3)) in powers of q.
a(4n+1)=a(n). - Michael Somos Dec 06 2004
a(n)=b(3n+1) where b(n) is multiplicative and b(p^e) = 0 if p = 3, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6). - Michael Somos May 20 2005
Given g.f. A(x), B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u^2*w-2u*w^2-v^3. - Michael Somos Dec 06 2004
Given g.f. A(x), B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1*u3^2+u1*u6^2-u1*u3*u6-u2^2*u3. - Michael Somos May 20 2005
Given g.f. A(x), B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u2*u3^2+2*u2*u3*u6+4*u2*u6^2-u1^2*u6. - Michael Somos May 20 2005
G.f.: Product_{k>0} (1-q^(3k))^3/(1-q^k).
G.f.: Sum_{k} x^k/(1-x^(3k+1)) = Sum_{k} x^k/(1-x^(6k+2)) . - Michael Somos Nov 03 2005
Expansion of q^(-1) * c(q^3) / 3 = q^(-1) * (a(q) - b(q)) / 9 in powers of q^3 where a(), b(), c() are cubic AGM analog functions. - Michael Somos Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 3^(1/2) (t/i) g(t) where t = exp(2 pi i t) and g(t) is g.f. for A005928.
a(n) = Sum_{d|3n+1} LengendreSymbol{d,3} - Brian Hopkins (bhopkins(AT)spc.edu), May 13 2008
q-series for a(n): Sum_{n >= 0} q^(n^2+n)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))). [From Jeremy Lovejoy (lovejoy(AT)liafa.jussieu.fr), Jun 12 2009]
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EXAMPLE
| eta(q^9)^3/eta(q^3) = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 +...
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PROG
| (PARI) a(n)=if(n<0, 0, sumdiv(3*n+1, d, kronecker(-3, d))) /* Michael Somos Nov 03 2005 */
(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x^3+A)^3/eta(x+A), n))
(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p!=3, if(p%6==1, e+1, !(e%2))))))} /* Michael Somos May 20 2005 */
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CROSSREFS
| A002324(3n+1)=a(n). A005882(n)=3a(n). A033685(3n+1)=3a(n). - Michael Somos, Apr 04 2003
Cf. A000731.
Cf. A045831, A053723, A081622.
Sequence in context: A166348 A127543 A068907 * A133457 A068067 A046926
Adjacent sequences: A033684 A033685 A033686 * A033688 A033689 A033690
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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