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A103440
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Sum[d|n, d==1 mod 3, d^2] - Sum[d|n, d==2 mod 3, d^2].
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2
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1, -3, 1, 13, -24, -3, 50, -51, 1, 72, -120, 13, 170, -150, -24, 205, -288, -3, 362, -312, 50, 360, -528, -51, 601, -510, 1, 650, -840, 72, 962, -819, -120, 864, -1200, 13, 1370, -1086, 170, 1224, -1680, -150, 1850, -1560, -24, 1584, -2208, 205, 2451, -1803, -288, 2210, -2808, -3, 2880, -2550
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OFFSET
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1,2
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REFERENCES
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G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).
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LINKS
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Table of n, a(n) for n=1..56.
J. Stienstra, Mahler measure, Eisenstein series and dimers
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FORMULA
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G.f.: F(q) = Sum[n>=1, A049347(n-1)*n^2*q^n/(1-q^n) ].
G.f.: F(q) = -qG'(q)/(9G(q)), with G(q) = Prod[n>=1, (1-q^n)^(9n*A049347(n-1)) ].
a(n) is multiplicative with a(3^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - z * a(p^(e-2)) where z = kronecker(-3, p) * p^2 and a(p) = z + 1.
a(3*n) = a(n).
G.f.: Sum_{k>0} x^k * (1 - x^k - 6*x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3. - Michael Somos Oct 21 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v + w + 3*v^2 - 8*w^2 + 6*v*w - 8*u*w + 6*u*v - 9*v^3 - 54*u*v*w + 72*u*w^2 - 9*u^2*w. - Michael Somos Dec 23 2007
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EXAMPLE
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q - 3*q^2 + q^3 + 13*q^4 - 24*q^5 - 3*q^6 + 50*q^7 - 51*q^8 + q^9 + ...
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( -3, d)))} /* Michael Somos Oct 21 20007 */
(PARI) {a(n) = local(A, p, e, a0, a1, x, y, z); if(n<1, 0, A=factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 1, z = kronecker( -3, p) * p^2 ; a0 = 1; a1 = y = z + 1; for(i=2, e, x = y * a1 - z * a0; a0 = a1; a1 = x); a1))))} /* Michael Somos Oct 21 20007 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^9 / eta(x^3 + A)^3) / 9, n))} /* Michael Somos Oct 21 20007 */
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CROSSREFS
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Equals A103637(n) - A103638(n). Cf. A002173.
A109041(n) = -9 * a(n) unless n=0. A014985(n) = a(2^n). -24 * A134340(n) = a(6*n+5).
Sequence in context: A184828 A053286 A008826 * A116483 A010290 A074960
Adjacent sequences: A103437 A103438 A103439 * A103441 A103442 A103443
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KEYWORD
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sign,mult
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AUTHOR
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Ralf Stephan, Feb 11 2005
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STATUS
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approved
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