OFFSET
0,9
COMMENTS
The number of 1's in the binary expansion of n+1 minus the number of 1's in the binary expansion of n.
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.
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
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
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 *)
PROG
(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
-- Reinhard Zumkeller, Dec 11 2011
(Python)
def A088705(n): return 1-(~n & n-1).bit_length() # Chai Wah Wu, Sep 18 2024
CROSSREFS
KEYWORD
sign,easy,mult
AUTHOR
Ralf Stephan, Oct 10 2003
STATUS
approved