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A255648
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Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.
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2
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1, 1, 2, 1, 0, 2, 2, 1, 2, 0, 0, 2, 2, 2, 0, 1, 0, 2, 2, 0, 4, 0, 0, 2, 1, 2, 2, 2, 0, 0, 2, 1, 0, 0, 0, 2, 2, 2, 4, 0, 0, 4, 2, 0, 0, 0, 0, 2, 3, 1, 0, 2, 0, 2, 0, 2, 4, 0, 0, 0, 2, 2, 4, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 4, 2, 0, 2, 0, 0, 4, 0, 2, 0
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of (b(q^2)^2 / b(q) + b(q^6)^2 / b(q^3) - 2) / 3 in powers of q where b() is a cubic AGM theta function.
Expansion of (psi(q)^3 / psi(q^3) + psi(q^3)^3 / psi(q^9) - 2) / 3 in powers of q where psi() is a Ramanujan theta function.
Moebius transform is period 18 sequence [ 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2 if e>1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k))^2 + (x^(3*k) + x^(9*k)) / (1 + x^(6*k))^2.
a(2*n) = a(n). a(3*n) = 2 * A035178(n). a(3*n + 1) = A033687(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 22 2023
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EXAMPLE
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G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^6 + 2*q^7 + q^8 + 2*q^9 + 2*q^12 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, Sum[ { 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0}[[Mod[ d, 18, 1]]], { d, Divisors[ n]}]];
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1][d%18 + 1]))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 2, p%6==1, e+1, 1-e%2)))};
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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