This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A088705 #49 Sep 18 2024 10:23:27
%S A088705 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 A088705 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 A088705 1,0,1,-5,1,0,1,-1,1,0,1,-2,1,0,1,-1,1,0,1,-3,1,0,1
%N A088705 First differences of A000120. One minus exponent of 2 in n.
%C A088705 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 A088705 Reinhard Zumkeller, <a href="/A088705/b088705.txt">Table of n, a(n) for n = 0..10000</a>
%H A088705 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 A088705 Kevin Ryde, <a href="http://user42.tuxfamily.org/c-curve/index.html">Iterations of the Lévy C Curve</a>, see index "turn".
%H A088705 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences with (relatively) simple ordinary generating functions</a>, 2004.
%H A088705 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>.
%H A088705 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%F A088705 For n > 0: a(n) = A000120(n) - A000120(n-1) = 1 - A007814(n).
%F A088705 Multiplicative with a(2^e) = 1-e, a(p^e) = 1 otherwise. - _David W. Wilson_, Jun 12 2005
%F A088705 G.f.: Sum{k>=0} t/(1+t), t=x^2^k.
%F A088705 a(0) = 0, a(2*n) = a(n) - 1, a(2*n+1) = 1.
%F A088705 Let T(x) be the g.f., then T(x)-T(x^2)=x/(1+x). - _Joerg Arndt_, May 11 2010
%F A088705 Dirichlet g.f.: zeta(s) * (2-2^s)/(1-2^s). - _Amiram Eldar_, Sep 18 2023
%p A088705 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 A088705 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 A088705 (PARI) a(n)=if(n<1,0,if(n%2==0,a(n/2)-1,1))
%o A088705 (PARI) a(n)=if(n<1,0,1-valuation(n,2))
%o A088705 (Haskell)
%o A088705 a088705 n = a088705_list !! n
%o A088705 a088705_list = 0 : zipWith (-) (tail a000120_list) a000120_list
%o A088705 -- _Reinhard Zumkeller_, Dec 11 2011
%o A088705 (Python)
%o A088705 def A088705(n): return 1-(~n & n-1).bit_length() # _Chai Wah Wu_, Sep 18 2024
%Y A088705 Cf. A000120, A007814, A079559.
%K A088705 sign,easy,mult
%O A088705 0,9
%A A088705 _Ralf Stephan_, Oct 10 2003