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A101455
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a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0...
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19
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1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Called X(n) (i.e. Chi(n)) in Hardy and Wright (p. 241), who show that X(n*m) = X(n)*X(m) for all n and m (i.e. X(n) is completely multiplicative) since (n*m - 1)/2 - (n - 1)/2 - (m - 1)/2 = (n - 1)*(m - 1)/2 = 0 (mod 2) when n and m are odd. Same as A056594 but with offset 1.
Multiplicative with a(2^e) = 0, a(p^e) = (-1)^((p^e-1)/2) otherwise. Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) May 17, 2005.
Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 15 2010: (Start)
The sequence is the non-principal Dirichlet character mod 4 (The principal character is A000035.)
Associated Dirichlet L-functions are for example L(1,chi)= sum_{n>=1} a(n)/n = A003881, or L(2,chi)= sum_{n>=1} a(n)/n^2 = A006752, or L(3,chi)= sum_{n>=1} a(n)/n^3 = A153071. (End)
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REFERENCES
| G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1979, p. 241.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (0,-1).
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FORMULA
| Euler transform of length 4 sequence [0, -1, 0, 1]. - Michael Somos Sep 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 * (1 + 2*v). - Michael Somos, Aug 04 2011
G.f.: (x - x^3) /(1 - x^4) = x/(1+x^2). - Michael Somos, Sep 02 2005
a(n + 2) = -a(n) = a(-n).
a(n)=sin(2*Pi*(n-1))/(4*cos(Pi/2*(n-1))) with n>=0 - Paolo P. Lava (paoloplava(AT)gmail.com), Jun 20 2006
a(n)=-(1/4)*{(n mod 4)-[(n+1) mod 4]-[(n+2) mod 4]+[(n+3) mod 4]}, with n>=1 [From Paolo P. Lava (paoloplava(AT)gmail.com), Aug 28 2009]
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PROG
| (PARI) {a(n) = if( n%2, ( -1)^(n\2))} /* Michael Somos, Sep 02 2005 */
sage: [lucas_number1(n, 0, 1) for n in xrange(1, 94)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 06 2008
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CROSSREFS
| Cf. A056594.
Sequence in context: A166698 A059841 A056594 * A091337 A179758 A174888
Adjacent sequences: A101452 A101453 A101454 * A101456 A101457 A101458
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KEYWORD
| easy,sign,mult
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AUTHOR
| Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jan 20 2005
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