A088705 First differences of A000120. One minus exponent of 2 in n.
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
Offset: 0
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
- Yann Bugeaud and Guo-Niu Han, A combinatorial proof of the non-vanishing of Hankel determinants of the Thue-Morse sequence, Electronic Journal of Combinatorics 21(3) (2014), #P3.26. See F(z) in (1.1). - _N. J. A. Sloane_, Aug 31 2014
- Kevin Ryde, Iterations of the Lévy C Curve, see index "turn".
- Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
- Ralf Stephan, Table of generating functions.
- Index entries for sequences related to binary expansion of n.
Programs
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Haskell
a088705 n = a088705_list !! n a088705_list = 0 : zipWith (-) (tail a000120_list) a000120_list -- Reinhard Zumkeller, Dec 11 2011
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Maple
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
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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PARI
a(n)=if(n<1,0,if(n%2==0,a(n/2)-1,1))
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PARI
a(n)=if(n<1,0,1-valuation(n,2))
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Python
def A088705(n): return 1-(~n & n-1).bit_length() # Chai Wah Wu, Sep 18 2024
Formula
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
Comments