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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
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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; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math, 31 (2001) 131-139. see p. 132 Theorem 1.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| 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); Dirichlet convolution of A010052 and A154955.
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).
|a(n)| = A093709(n).
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EXAMPLE
| 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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MATHEMATICA
| f[n_]:=Which[IntegerQ[Sqrt[n/2]], -1, IntegerQ[Sqrt[n]], 1, True, 0]; Array[f, 110] (* From Harvey P. Dale, Jul 07 2011 *)
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PROG
| (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
| Cf. A093709.
Sequence in context: A154269 A036987 A181101 * A113430 A113681 A179416
Adjacent sequences: A143256 A143257 A143258 * A143260 A143261 A143262
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Aug 02 2008
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