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A088705
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First differences of A000120. One minus exponent of 2 in n.
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11
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0, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 1, -3, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 1, -3, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 1, -5, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 1, -3, 1, 0, 1
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OFFSET
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0,9
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COMMENTS
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The number of 1's in the binary expansion of n+1 minus the number of 1's in the binary expansion of n.
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LINKS
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FORMULA
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Multiplicative with a(2^e) = 1-e, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
G.f.: Sum{k>=0} t/(1+t), t=x^2^k.
a(0) = 0, a(2*n) = a(n) - 1, a(2*n+1) = 1.
Let T(x) be the g.f., then T(x)-T(x^2)=x/(1+x). - Joerg Arndt, May 11 2010
Dirichlet g.f.: zeta(s) * (2-2^s)/(1-2^s). - Amiram Eldar, Sep 18 2023
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MAPLE
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add(x^(2^k)/(1+x^(2^k)), k=0..20); series(%, x, 1001); seriestolist(%); # To get up to a million terms, from N. J. A. Sloane, Aug 31 2014
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MATHEMATICA
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a[n_] := If[n<1, 0, If[Mod[n, 2] == 0, a[n/2] - 1, 1]]; Array[a, 60, 0] (* Amiram Eldar, Nov 26 2018 *)
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PROG
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(PARI) a(n)=if(n<1, 0, if(n%2==0, a(n/2)-1, 1))
(PARI) a(n)=if(n<1, 0, 1-valuation(n, 2))
(Haskell)
a088705 n = a088705_list !! n
a088705_list = 0 : zipWith (-) (tail a000120_list) a000120_list
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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