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

0,9

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.

Index entries for sequences related to binary expansion of n.

Formula

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

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

Crossrefs

Cf. A000120, A007814, A079559.

Sequence in context: A016398 A024359 A354512 * A065712 A153172 A242498

Adjacent sequences: A088702 A088703 A088704 * A088706 A088707 A088708

Keyword

sign,easy,mult

Author

Ralf Stephan, Oct 10 2003

Status

approved