%I A035263
%S A035263 1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,
%T A035263 1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,
%U A035263 1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1
%N A035263 Trajectory of 1 under the morphism 1 -> 10, 0 -> 11.
%C A035263 First Feigenbaum symbolic (or period-doubling) sequence, corresponding 
               to the accumulation point of the 2^{k} cycles through successive 
               bifurcations.
%C A035263 To construct the sequence: start with 1 and concatenate: 1,1, then change 
               the last term (1->0; 0->1) gives: 1,0. Concatenate those 2 terms: 
               1,0,1,0, change the last term: 1,0,1,1. Concatenate those 4 terms: 
               1,0,1,1,1,0,1,1 change the last term: 1,0,1,1,1,0,1,0 etc. - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Dec 17 2002
%C A035263 Let T denote the present sequence. Here is another way to construct T. 
               Start with the sequence S = 1,0,1,_,1,0,1,_,1,0,1,_,1,0,1,_,... and 
               fill in the successive holes with the successive terms of the sequence 
               T (from paper by Allouche et al). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jan 08 2003. Note that if we fill in the holes with the terms of 
               S itself, we get A141260. - N. J. A. Sloane (njas(AT)research.att.com), 
               Jan 14 2009.
%C A035263 In more detail: define S to be 1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 
               0, 1___1,0,1___...
%C A035263 If we fill the holes with S we get A141260:
%C A035263 1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1,0,1___1, 0, 1___1, 
               0, 1___1, 0,
%C A035263 ........1.........0.........1.........1.........0.......1.........1.........0...
%C A035263 - the result is
%C A035263 1..0..1.1.1..0..1.0.1..0..1.1.1..0..1.1.1..0..1.0.1.... = A141260
%C A035263 But instead, if we define T recursively by filling the holes in S with 
               the
%C A035263 terms of T itself, we get A035263:
%C A035263 1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1,0,1___1, 0, 1___1, 
               0, 1___1, 0,
%C A035263 ........1.........0.........1.........1.........1.......0.........1.........0...
%C A035263 - the result is
%C A035263 1..0..1.1.1..0..1.0.1..0..1.1.1..0..1.1.1..0..1.1.1.0.1.0.1..0..1.1.1..0..1.0.1.. 
               = A035263
%C A035263 Characteristic function of A003159, i.e. A035263(n)=1 if n is in A003159 
               and A035263(n)=0 otherwise (from paper by Allouche et al.). - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2003
%C A035263 This is the sequence of R (=1), L (=0) moves in the Tower of Hanoi game: 
               R, L, R, R, R, L, R, L, R, L, R, R, R... - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 21 2003
%C A035263 Manfred Schroeder, p. 279 states, "... the kneading sequences for unimodal 
               maps in the binary notation, 0, 1, 0, 1, 1, 1, 0, 1..., are obtained 
               from the Morse-Thue sequence by taking sums mod 2 of adjacent elements." 
               On p. 278, in the chapter "Self-Similarity in the Logistic Parabola", 
               he writes, "Is there a closer connection between the Morse-Thue sequence 
               and the symbolic dynamics of the superstable orbits? There is indeed. 
               To see this, let us replace R by 1 and C and L by 0." - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Sep 21 2003
%C A035263 Partial sums modulo 2 of the sequence 1, a(1), a(1), a(2), a(2), a(3), 
               a(3), a(4), a(4), a(5), a(5), a(6), a(6), ... - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), Jan 02 2004
%C A035263 Parity of A007913, A065882 and A065883. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 28 2004
%C A035263 The length of n-th run of 1's in this sequence is A080426(n). - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Apr 19 2004
%C A035263 Also parity of A005043, A005773, A026378, A104455, A117641 . - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Apr 28 2007
%C A035263 Equals parity of the Tower of Hanoi, or ruler sequence (A001511), where 
               the Tower of Hanoi sequence (1, 2, 1, 3, 1, 2, 1, 4,...) denotes 
               the disc moved, labeled (1, 2, 3,...) starting from the top; and 
               the parity of (1, 2, 1, 3,...) denotes the direction of the move, 
               CW or CCW. The frequency of CW moves converges to 2/3. - Gary Adamson, 
               May 11 2007
%D A035263 B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the 
               real axis and representation of number, Ann. Inst. Henri Poincar\'e, 
               Section A: Physique Th\'eorique, Vol. XXIX no. 3, 305-356 (1978).
%D A035263 K. Karamanos, From Symbolic Dynamics to a Digital Approach, Int. J. of 
               Bifurcation and Chaos, 11(6), 1683-1694 (2001).
%D A035263 K. Karamanos and G. Nicolis, Symbolic Dynamics and Entropy Analysis of 
               Feigenbaum Limit Sets, Chaos, Solitons and Fractals 10(7), 1135 - 
               1150 (1999).
%D A035263 N. Metropolis, M. L. Stein and P. R. Stein, On Finite Limit Sets for 
               Transformations on the Unit Interval, J. Combinat. Theory, Vol. A 
               15, 25-44 (1973).
%D A035263 Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991
%D A035263 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 892, column 
               2, Note on p. 84, part (a).
%H A035263 T. D. Noe, <a href="b035263.txt">Table of n, a(n) for n=1..10000</a>
%H A035263 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A035263 J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, 
               Simon Plouffe and Bruce E. Sagan, <a href="http://www-igm.univ-mlv.fr/
               ~berstel/Articles/Relative.ps">A sequence related to that of Thue-Morse</
               a>, Discrete Math., 139 (1995), 455-461.
%H A035263 J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, S. Plouffe 
               and B. E. Sagan, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/
               Relative.ps">A relative of the Thue-Morse sequence</a>
%H A035263 J.-P. Allouche and M. Mendes France, <a href="http://www.lri.fr/~allouche/
               ">Automata and Automatic Sequences.</a>. In F. Axel and D. Gratias, 
               editors, Beyond Quasicrystals, pages 293-367. Les \'Editions de Physique/
               Springer, 1995.
%H A035263 J.-P. Allouche and J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/
               ~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</
               a>, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences 
               and Their Applications: Proceedings of SETA '98, Springer-Verlag, 
               1999, pp. 1-16.
%H A035263 K. Karamanos, <a href="http://www.worldscinet.com/ijbc/11/1106/S0218127401002924.html">
               Symbolic Dynamics to a Digital Approach</a>, Int. J. of Bifurcation 
               and Chaos, 11(6), 1683-1694 (2001).
%H A035263 D. Kohel, S. Ling and C. Xing, <a href="http://www.maths.usyd.edu.au/
               u/kohel/doc/perfect.ps">Explicit Sequence Expansions</a>
%H A035263 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer 
               generating functions. I. Elementary sequences</a>
%H A035263 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A035263 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A035263 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinaryCarrySequence.html">Link to a section of The World of Mathematics.</
               a>
%H A035263 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Double-FreeSet.html">Link to a section of The World of Mathematics.</
               a>
%H A035263 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%F A035263 Absolute values of first differences (A029883) of Thue-Morse sequence 
               (A001285 or A010060). Self-similar under 10->1 and 11->0.
%F A035263 Series expansion: (1/x) * Sum(i=0, infinity, (-1)^(i+1)*x^(2^i)/(x^(2^i)-1) 
               ). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 
               2003
%F A035263 a(n)=sum(k>=0, (-1)^k*(floor((n+1)/2^k)-floor(n/2^k))) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jun 03 2003
%F A035263 Another g.f.: sum(k>=0, x^(2^k)/(1+(-1)^k*x^(2^k))). - Ralf Stephan, 
               Jun 13 2003
%F A035263 a(2n) = 1-a(n), a(2n+1) = 1. - Ralf Stephan, Jun 13 2003
%F A035263 a(n) = parity of A033485(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Aug 13 2003
%F A035263 Equals A088172 mod 2, where A088172 = 1, 2, 3, 7, 13, 26, 53, 106, 211, 
               422, 845...(first differences of A019300). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 21 2003
%F A035263 a(n)=a(n-1)-(-1)^n*a(floor(n/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Dec 02 2003
%F A035263 a(1)=1 and a(n)=abs(a(n-1)-a(floor(n/2))) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Dec 02 2003
%F A035263 a(n) = 1 - A096268(n+1); A050292 gives partial sums. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Aug 16 2006
%F A035263 Multiplicative with a(2^k) = 1 - (k mod 2), a(p^k) = 1, p>2. Dirichlet 
               g.f.: prod{n = 4 or an odd prime}(1/(1-1/n^s)). Christian G. Bower 
               (bowerc(AT)usa.net) May 18, 2005.
%F A035263 a(-n)=a(n). a(0)=0. - Michael Somos Sep 04 2006
%F A035263 Dirichlet g.f.: zeta(s)*2^s/(2^s+1). - Ralf Stephan, Jun 17 2007
%F A035263 a(n+1) = a(n) XOR a(ceiling(n/2)), a(1) = 1. [From Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Jun 11 2009]
%t A035263 Nest[ Function[l, {Flatten[(l /. {0 -> {1, 1}, 1 -> {1, 0}})]}], {1}, 
               7] (* Or *)
%t A035263 a[n_] := a[n] = If[ EvenQ[n], 1 - a[n/2], 1]; Table[ a[n], {n, 1, 105}] 
               (* Or *)
%t A035263 Rest[ CoefficientList[ Series[ Sum[ x^(2^k)/(1 + (-1)^k*x^(2^k)), {k, 
               0, 20}], {x, 0, 105}], x]]
%o A035263 (PARI) {a(n)=if(n==0, 0, 1-valuation(n, 2)%2)} /* Michael Somos Sep 04 
               2006 */
%o A035263 (PARI) {a(n)=if(n==0, 0, n=abs(n);subst(Pol(binary(n))-Pol(binary(n-1)), 
               x, 1)%2)} /* Michael Somos Sep 04 2006 */
%o A035263 (PARI) {a(n)=if(n==0, 0, n=abs(n); direuler(p=2, n, 1/(1-X^((p<3)+1)))[n])} 
               /* Michael Somos Sep 04 2006 */
%Y A035263 Parity of A001511. Anti-parity of A007814.
%Y A035263 Absolute values of first differences of A010060. Apart from signs, same 
               as A029883.
%Y A035263 Cf. A033485, A050292, A088172, A019300, A010060, A039982, A073675, A121701, 
               A141260.
%Y A035263 Sequence in context: A104106 A141260 A029883 this_sequence A089045 A070749 
               A059778
%Y A035263 Adjacent sequences: A035260 A035261 A035262 this_sequence A035264 A035265 
               A035266
%K A035263 nonn,easy,nice,mult
%O A035263 1,1
%A A035263 Karamanos Konstantinos (kkaraman(AT)ulb.ac.be), N. J. A. Sloane (njas(AT)research.att.com).