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A048298
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a(n) = n if n=2^i, i=0,1,2,3,...; else = 0.
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13
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0, 1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,3
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COMMENTS
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Expand x/(x-1) = Sum_{n >= 0} 1/x^n as Sum a(n) / (1+x^n).
Nim-binomial transform of the natural numbers. If {t(n)} is the Nim-binomial transform of {a(n)}, then t(n)=(S^n)a(0), where Sf(n) denotes the Nim-sum of f(n) and f(n+1); and S^n=S(S^(n-1)). - John W. Layman, Mar 06 2001
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REFERENCES
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J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
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LINKS
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Table of n, a(n) for n=0..102.
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
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FORMULA
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Multiplicative with a(2^e)=2^e and a(p^e)=0 for p > 2. - Vladeta Jovovic, Jan 27 2002
Inverse mod 2 binomial transform of n. a(n)=sum{k=0..n, (-1)^A010060(n-k)*mod(C(n, k), 2)*k} - Paul Barry, Jan 03 2005
Dirichlet g.f.: 2^s/(2^s-2). - Ralf Stephan, Jun 17 2007
For n>=1, we have a recursion Sum{d|n}(-1)^(1+(n/d))a(d)=1. [From Vladimir Shevelev, Jun 09 2009]
for n>=1 there is the recurrence n=sum(k=1,n,a(k)*g(n/k)) where g(x)=floor(x)-2*floor(x/2) [From Benoit Cloitre, Nov 11 2010]
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EXAMPLE
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If n=1 we have a(n)=1; if n=p is prime, then (-1)^(p+1)+a(p)=1, thus a(2)=2, and a(p)=0,if p>2. [From Vladimir Shevelev, Jun 09 2009]
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CROSSREFS
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A kind of inverse to A048272. Cf. A060147.
This is Guy Steele's sequence GS(5, 1) (see A135416).
Sequence in context: A104774 A087263 A099894 * A123565 A081120 A200038
Adjacent sequences: A048295 A048296 A048297 * A048299 A048300 A048301
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KEYWORD
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easy,nonn,mult,changed
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AUTHOR
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Adam Kertesz (adamkertesz(AT)worldnet.att.net)
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EXTENSIONS
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More terms from Keiko L. Noble (s1180624(AT)cedarville.edu).
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STATUS
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approved
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