0,1

a(n+1) = a(floor(n/2)) * (n mod 2); a(0)=1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 02 2002

Sum {0..infinity} 1/10^(2^n) = 0.110100010000000100000000000000010...

Binary representation of Kempner-Mahler number sum(k>=0,1/2^(2^k)).

Also a(n) == mod(A(n), 2) where A is any of A001700, A005573, A007854, A026641, A049027, A064063, A064088, A064090, A064092, A064 325, A064327, A064329, A064331, A064613, A076026, A105523, A123273, A126694, A126930, A126931, A126982, A126983, A126987, A127016, A127053, A127358, A127360, A127361, A127363 - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 26 2007

a(n) = (product of digits of n; n in binary notation) mod 2. This sequence is a transformation of the Thue-Morse sequence (A010060), since there exists a function f such that f(sum of digits of n) = (product of digits of n). - Ctibor O. ZIZKA (ctibor.zizka(AT)seznam.cz), Feb 12 2008

Moebius transform of A001511. [From Mats Granvik (mats.granvik(AT)abo.fi), Jan 03 2009]

Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Jan 14 2009: (Start)

It appears that the result of ln(n)mod(ln(2)) is 0 if n is a power of 2 and not

more than ln(2)-ln(1)=ln(2) in other cases.

(End)

a(n-1), n>=1, the characteristic sequence for powers of 2, A000079, is the unique solution of the following formal product and formal power series identity: product(1+a(j-1)*x^j,j=1..infinity) = 1+sum(x^k,k=1..infinity) = 1/(1-x). The product is therefore product(1+x^(2^l),l=1..infinity). Proof. Compare coefficients of x^n and use the binary representation of n. Uniqueness follows from the recurrence relation given for the general case under A147542. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 05 2009]

a(n) is also the number of orbits of length n for the map x->1-cx^2 on [ -1,1] at the Feigenbaum critical value c=1.401155... [From Thomas Ward (t.ward(AT)uea.ac.uk), Apr 08 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2009: (Start)

A054525 (Mobius transform) * A001511 = A036987 = A047999^(-1) * A001511 = the

inverse of Sierpinski's gasket * the ruler sequence. (End)

H. Niederreiter and M. Vielhaber, Tree complexity and a doubly ..., J. Complexity, 12 (1996), 187-198.

Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4. [From Thomas Ward (t.ward(AT)uea.ac.uk), Apr 08 2009]

D. Bailey et al., On the binary expansions of algebraic numbers

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.

D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions

E. Sheppard, net.math post (1985)

Stephen Wolfram, [Page 1092] A New Kind of Science | Online.

Index entries for characteristic functions

1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.

Right-shifted sequence is multiplicative with a(2^e) = 1, a(p^e) = 0 otherwise. - Mitch Harris, Apr 19 2005.

1 if n=0, [log2(n+1)]-[log2(n)] else. G.f.: (1/x) * Sum(k>=0, x^(2^k)) = Sum(k>=0, x^(2^k-1)). - Ralf Stephan, Apr 28 2003

a(n)=-sumdiv(n+1, d, mu(2*d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 24 2003

Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).

a(n)=mod(A000108(n), 2)=mod(A001405(n), 2) - Paul Barry (pbarry(AT)wit.ie), Nov 22 2004

a(n)=sum{k=0..n, (-1)^(n-k)*C(n,k)*sum{j=0..k, C(k,2^j-1)}}; - Paul Barry (pbarry(AT)wit.ie), Jun 01 2006

A036987 := n -> `if`(((2^floor_log_2(n+1)) = (n+1)), 1, 0);

floor_log_2 := proc(n) local nn, i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;

RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]

(PARI) a(n)=if(n<0, 0, n++; n==2^valuation(n, 2))

Cf. A007404, A078885, A078585, A078886, A078887, A078888, A078889, A078890.

The first row of A073346. Occurs for first time in A073202 as the row 6 (and 8).

Congruent to any of the sequences A000108, A007460, A007461, A007463, A007464, A061922, A068068 reduced modulo 2. Characteristic function of A000225.

If interpreted with offset=1 instead of 0 (i.e. a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n (A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359.

A043545(n)=1-a(n).

This is Guy Steele's sequence GS(1, 3), also GS(3, 1) (see A135416).

Cf. A054525, A047999 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2009]

Sequence in context: A039963 A058840 A154269 this_sequence A143259 A113430 A113681

Adjacent sequences: A036984 A036985 A036986 this_sequence A036988 A036989 A036990

nonn,mult,new

N. J. A. Sloane (njas(AT)research.att.com).