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A005928
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G.f.: s(1)^3/s(3), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
(Formerly M2202)
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8
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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
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OFFSET
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0,2
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COMMENTS
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Unsigned sequence is expansion of theta series of hexagonal net with respect to a node.
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REFERENCES
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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. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.34).
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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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a(n) is the coefficient of q^n in b(q)=eta(q)^3/eta(q^3) = (3/2)*a(q^3)-a(q)/2 where a(q)=theta(Hexagonal) - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002
Euler transform of period 3 sequence [ -3, -3, -2, ...]. - Michael Somos, May 20 2005
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). - Michael Somos, May 20 2005
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v^3-2*u*w^2+u^2*w. - Michael Somos, May 20 2005
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^2*u6 -2*u1*u2*u6 +4*u2^2*u6 -3*u2*u3^2. - Michael Somos, May 20 2005
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1*u2*u3 +u1^2*u3 -3*u1*u6^2 +u2^2*u3. - Michael Somos, May 20 2005
a(3n+2)=0. a(3n+1)=-A033685(3n+1), a(3n)=A004016(n). - Michael Somos, Jul 15 2005
Moebius transform is period 9 sequence [ -3, 3, 9, -3, 3, -9, -3, 3, 0, ...]. - Michael Somos, Dec 25 2007
Expansion of b(q) = a(q^3) - c(q^3) in powers of q 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/(3*t)) = 3^(3/2)*(t/i)*g(t) where t = exp(2*Pi*i*t) and g(t) is the g.f. for A033687.
G.f.: exp( sum(n>=1, (sigma(n)-sigma(3*n))*x^n/n ) ). [Joerg Arndt, Jul 30 2011]
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EXAMPLE
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1 - 3*q + 6*q^3 - 3*q^4 - 6*q^7 + 6*q^9 + 6*q^12 - 6*q^13 - 3*q^16 + ...
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PROG
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(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))))))} /* Michael Somos, May 20 2005 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x+A)^3/eta(x^3+A), n))} /* Michael Somos, May 20 2005 */
(PARI) {a(n) = if(n<1, n==0, sumdiv(n, d, [ 0, -3, 3, 9, -3, 3, -9, -3, 3][d%9+1]))} /* Michael Somos, Dec 25 2007 */
(PARI) N=66; x='x+O('x^N); /* that many terms */
gf=exp(sum(n=1, N, (sigma(n)-sigma(3*n))*x^n/n));
Vec(gf) /* show terms */ /* Joerg Arndt, Jul 30 2011 */
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CROSSREFS
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Cf. -3 * A123477(n) = a(n) unless n=0. A113062(n) = |a(n)|.
Sequence in context: A077187 A011079 * A113062 A005889 A095712 A085753
Adjacent sequences: A005925 A005926 A005927 * A005929 A005930 A005931
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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