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Search: id:A101455
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| A101455 |
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For n > 0: 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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+0
15
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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; listen)
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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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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FORMULA
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Euler transform of length 4 sequence [0, -1, 0, 1]. - Michael Somos Sep 02 2005
G.f.: (x-x^3)/(1-x^4) . - Michael Somos Sep 02 2005
a(n)=sin(2*Pi*(n-1))/(4*cos(Pi/2*(n-1))) with n>=0 - Paolo P. Lava (ppl(AT)spl.at), 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 (ppl(AT)spl.at), Aug 28 2009]
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PROGRAM
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(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
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Cf. A056594.
Sequence in context: A016213 A015757 A166698 this_sequence A056594 A091337 A059841
Adjacent sequences: A101452 A101453 A101454 this_sequence A101456 A101457 A101458
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KEYWORD
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easy,sign,mult
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AUTHOR
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Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jan 20 2005
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