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A244611
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Expansion of (phi(q) + phi(q^2) - phi(q^3) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.
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1
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1, 1, -1, 1, 0, -1, 0, 1, 1, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 1
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OFFSET
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1,1
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LINKS
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FORMULA
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Multiplicative with a(2^e) = 1, a(3^e) = (-1)^e, and a(p^e) = (1 + (-1)^e) / 2 if p>3.
G.f.: (theta_3(q) + theta_3(q^2) - theta_3(q^3) - theta_3(q^6)) / 2.
a(2*n) = a(n). a(2*n + 1) = A214505(n). a(3*n) = -a(n). a(3*n + 1) = A089801(n). a(6*n + 5) = 0.
Dirichlet g.f.: (1 + 1/2^s) * (1 - 1/3^s) * zeta(2*s).
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1 + 1/sqrt(2) - 1/sqrt(3) - 1/sqrt(6) = 0.721508... . (End)
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EXAMPLE
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G.f. = q + q^2 - q^3 + q^4 - q^6 + q^8 + q^9 - q^12 + q^16 + q^18 + ...
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PROG
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(PARI) {a(n) = issquare(n) + issquare(2*n) - issquare(3*n) - issquare(6*n)};
(PARI) {a(n) = if( n<1, 0, n/= 2^valuation(n, 2); issquare(n) - issquare(n*3))};
(PARI) {a(n) = local(A); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, (-1)^e, !(e%2))))))};
(Scheme)
;; Based on the given multiplicative formula, and using the memoization-macro definec:
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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