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A162397
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a(n) = n * Kronecker(-3, n).
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1
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1, -2, 0, 4, -5, 0, 7, -8, 0, 10, -11, 0, 13, -14, 0, 16, -17, 0, 19, -20, 0, 22, -23, 0, 25, -26, 0, 28, -29, 0, 31, -32, 0, 34, -35, 0, 37, -38, 0, 40, -41, 0, 43, -44, 0, 46, -47, 0, 49, -50, 0, 52, -53, 0, 55, -56, 0, 58, -59, 0, 61, -62, 0, 64, -65, 0
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OFFSET
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1,2
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COMMENTS
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In Gil and Robins 2003 on page 33 the g.f. is denoted by f_{4, 2}(x). - Michael Somos, Sep 04 2015
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REFERENCES
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George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 319, Equation (14.3.6).
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LINKS
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FORMULA
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Euler transform of length 3 sequence [ -2, -1, 2].
a(n) is completely multiplicative with a(3^e) = 0^e, a(p^e) = p^e if p == 1 (mod 3), a(p^e) = (-p)^e if p == 2 (mod 3).
G.f.: (x - x^3) / (1 + x + x^2)^2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/3. - Amiram Eldar, Nov 23 2023
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EXAMPLE
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G.f. = x - 2*x^2 + 4*x^4 - 5*x^5 + 7*x^7 - 8*x^8 + 10*x^10 - 11*x^11 + ...
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MATHEMATICA
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Table[n*KroneckerSymbol[-3, n], {n, 80}] (* Harvey P. Dale, Mar 14 2015 *)
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PROG
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(PARI) {a(n) = n * kronecker(-3, n)};
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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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STATUS
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approved
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