A088705 - OEIS

%I #49 Sep 18 2024 10:23:27

%S 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,

%T 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,

%U 1,0,1,-5,1,0,1,-1,1,0,1,-2,1,0,1,-1,1,0,1,-3,1,0,1

%N First differences of A000120. One minus exponent of 2 in n.

%C The number of 1's in the binary expansion of n+1 minus the number of 1's in the binary expansion of n.

%H Reinhard Zumkeller, <a href="/A088705/b088705.txt">Table of n, a(n) for n = 0..10000</a>

%H Yann Bugeaud and Guo-Niu Han, <a href="https://doi.org/10.37236/3831">A combinatorial proof of the non-vanishing of Hankel determinants of the Thue-Morse sequence</a>, Electronic Journal of Combinatorics 21(3) (2014), #P3.26. See F(z) in (1.1). - _N. J. A. Sloane_, Aug 31 2014

%H Kevin Ryde, <a href="http://user42.tuxfamily.org/c-curve/index.html">Iterations of the Lévy C Curve</a>, see index "turn".

%H Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences with (relatively) simple ordinary generating functions</a>, 2004.

%H Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%F For n > 0: a(n) = A000120(n) - A000120(n-1) = 1 - A007814(n).

%F Multiplicative with a(2^e) = 1-e, a(p^e) = 1 otherwise. - _David W. Wilson_, Jun 12 2005

%F G.f.: Sum{k>=0} t/(1+t), t=x^2^k.

%F a(0) = 0, a(2*n) = a(n) - 1, a(2*n+1) = 1.

%F Let T(x) be the g.f., then T(x)-T(x^2)=x/(1+x). - _Joerg Arndt_, May 11 2010

%F Dirichlet g.f.: zeta(s) * (2-2^s)/(1-2^s). - _Amiram Eldar_, Sep 18 2023

%p 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

%t 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 *)

%o (PARI) a(n)=if(n<1,0,if(n%2==0,a(n/2)-1,1))

%o (PARI) a(n)=if(n<1,0,1-valuation(n,2))

%o (Haskell)

%o a088705 n = a088705_list !! n

%o a088705_list = 0 : zipWith (-) (tail a000120_list) a000120_list

%o -- _Reinhard Zumkeller_, Dec 11 2011

%o (Python)

%o def A088705(n): return 1-(~n & n-1).bit_length() # _Chai Wah Wu_, Sep 18 2024

%Y Cf. A000120, A007814, A079559.

%K sign,easy,mult

%O 0,9

%A _Ralf Stephan_, Oct 10 2003