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Search: id:A143259
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| A143259 |
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a(n) = 1 if n is a nonzero square, -1 if n is twice a nonzero square, 0 otherwise. |
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+0
1
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| 1, -1, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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1,1
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FORMULA
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Expansion of (phi(q) - phi(q^2)) / 2 = q * psi(q^4) * f(-q, -q^7) / f(-q^3, -q^5) in powers of q where phi(), psi() and f() are Ramanujan theta functions.
Euler transform of period 8 sequence [ -1, 0, 1, 1, 1, 0, -1, -1, ...].
a(2*n) = -a(n).
a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 1 (mod 2).
Dirichlet g.f.: zeta(2*s) * (1 - 2^-s).
G.f. A(x) satisfies A(x) / A(x^2) = -1 + A111374(x).
G.f. A(x) satisfies A(x^2) = - (A(x) + A(-x)) / 2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u + v)^2 - v * (v + w) * (v + 2*w).
G.f.: (theta_3(q) - theta_3(q^2)) / 2 = Sum_{k>0} x^(k^2) - x^(2k^2).
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EXAMPLE
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q - q^2 + q^4 - q^8 + q^9 + q^16 - q^18 + q^25 - q^32 + q^36 + q^49 - q^50 + ...
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PROGRAM
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(PARI) {a(n) = issquare(n) - issquare(2*n)}
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod(k=1, n, (1 - x^k)^([1, 1, 0, -1, -1, -1, 0, 1][k%8 + 1]), 1 + x * O(x^n)), n))}
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CROSSREFS
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A093709(n) = |a(n)|.
Sequence in context: A058840 A154269 A036987 this_sequence A113430 A113681 A155972
Adjacent sequences: A143256 A143257 A143258 this_sequence A143260 A143261 A143262
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Aug 02 2008
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