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A035263 Trajectory of 1 under the morphism 1 -> 10, 0 -> 11. 52
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

First Feigenbaum symbolic (or period-doubling) sequence, corresponding to the accumulation point of the 2^{k} cycles through successive bifurcations.

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, Dec 17 2002

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, Jan 08 2003 [Note that if we fill in the holes with the terms of S itself, we get A141260. - N. J. A. Sloane, Jan 14 2009]

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___...

If we fill the holes with S we get A141260:

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,

........1.........0.........1.........1.........0.......1.........1.........0...

- the result is

1..0..1.1.1..0..1.0.1..0..1.1.1..0..1.1.1..0..1.0.1.... = A141260.

But instead, if we define T recursively by filling the holes in S with the terms of T itself, we get 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,

........1.........0.........1.........1.........1.......0.........1.........0...

- the result is

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.

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, Jan 15 2003

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, Sep 21 2003

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, Sep 21 2003

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), ... - Philippe Deléham, Jan 02 2004

Parity of A007913, A065882 and A065883. - Philippe Deléham, Mar 28 2004

The length of n-th run of 1's in this sequence is A080426(n). - Philippe Deléham, Apr 19 2004

Also parity of A005043, A005773, A026378, A104455, A117641. - Philippe Deléham, Apr 28 2007

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 W. Adamson, May 11 2007

A conjectured identity relating to the partition sequence, A000041: p(x) = A(x) * A(x^2) when A(x) = the Euler transform of A035263 = polcoeff A174065: (1 + x + x^2 + 2x^3 + 3x^4 + 4x^5 + ...). - Gary W. Adamson, Mar 21 2010

a(n) = A010060(n) XOR A010060(n+1); a(A079523(n)) = 0; a(A121539(n)) = 1. - Reinhard Zumkeller, Mar 01 2012

a(n) is 1 if the number of trailing zeroes in the binary representation of n is even. - Ralf Stephan, Aug 22 2013

REFERENCES

B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the real axis and representation of number, Ann. Inst. Henri Poincaré, Section A: Physique Théorique, Vol. XXIX no. 3, 305-356 (1978).

Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 892, column 2, Note on p. 84, part (a).

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461.

J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, _Simon Plouffe_ and B. E. Sagan, A relative of the Thue-Morse sequence

J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.. In F. Axel and D. Gratias, editors, Beyond Quasicrystals, pages 293-367. Les Editions de Physique/Springer, 1995.

J.-P. Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.

Joerg Arndt, Matters Computational (The Fxtbook), pp.9-10, pp.734-738

Benoit Cloitre, Fractal walk on the 130000 first terms (step of unit length moving with angle Pi/3 if 0 and -Pi/3 if 1 starting at (0,0)

A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website

K. Karamanos, Symbolic Dynamics to a Digital Approach, Int. J. of Bifurcation and Chaos, 11(6), 1683-1694 (2001).

K. Karamanos and G. Nicolis, Symbolic Dynamics and Entropy Analysis of Feigenbaum Limit Sets<:a>, Chaos, Solitons and Fractals 10(7), 1135 - 1150 (1999).

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

N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, A 15 (1973), 25-44.

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Eric Weisstein's World of Mathematics, Binary carry sequence.

Eric Weisstein's World of Mathematics, Double-free set.

Index entries for characteristic functions

Index entries for sequences related to binary expansion of n

FORMULA

Absolute values of first differences (A029883) of Thue-Morse sequence (A001285 or A010060). Self-similar under 10->1 and 11->0.

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

a(n) = sum(k>=0, (-1)^k*(floor((n+1)/2^k)-floor(n/2^k))) - Benoit Cloitre, Jun 03 2003

Another g.f.: sum(k>=0, x^(2^k)/(1+(-1)^k*x^(2^k))). - Ralf Stephan, Jun 13 2003

a(2*n) = 1-a(n), a(2*n+1) = 1. - Ralf Stephan, Jun 13 2003

a(n) = parity of A033485(n). - Philippe Deléham, Aug 13 2003

Equals A088172 mod 2, where A088172 = 1, 2, 3, 7, 13, 26, 53, 106, 211, 422, 845, ... (first differences of A019300). - Gary W. Adamson, Sep 21 2003

a(n) = a(n-1) - (-1)^n*a(floor(n/2)). - Benoit Cloitre, Dec 02 2003

a(1) = 1 and a(n) = abs(a(n-1)-a(floor(n/2))). - Benoit Cloitre, Dec 02 2003

a(n) = 1 - A096268(n+1); A050292 gives partial sums. - Reinhard Zumkeller, Aug 16 2006

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, May 18 2005.

a(-n) = a(n). a(0)=0. - Michael Somos, Sep 04 2006

Dirichlet g.f.: zeta(s)*2^s/(2^s+1). - Ralf Stephan, Jun 17 2007

a(n+1) = a(n) XOR a(ceiling(n/2)), a(1) = 1. - Reinhard Zumkeller, Jun 11 2009

Let D(x) be the generating function, then D(x) + D(x^2) == x/(1-x). - Joerg Arndt, May 11 2010

a((2*n-1)*2^p) = (p+1) mod 2, p  >= 0 and n >= 1. - Johannes W. Meijer, Feb 07 2013

a(n) = A000035(A001511(n)). - Omar E. Pol, Oct 29 2013

MAPLE

nmax:=105: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+1) mod 2 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 07 2013

MATHEMATICA

a[n_] := a[n] = If[ EvenQ[n], 1 - a[n/2], 1]; Table[ a[n], {n, 1, 105}] (* Or *)

Rest[ CoefficientList[ Series[ Sum[ x^(2^k)/(1 + (-1)^k*x^(2^k)), {k, 0, 20}], {x, 0, 105}], x]]

f[1] := True; f[x_] := Xor[f[x - 1], f[Floor[x/2]]]; a[x_] := Boole[f[x]] (* Ben Branman, Oct 04 2010 *)

a[n_] := If[n == 0, 0, 1 - Mod[ IntegerExponent[n, 2], 2]]; (* Jean-François Alcover, Jul 19 2013, after Michael Somos *)

Nest[ Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {0}, 7] (* Robert G. Wilson v, Jul 23 2014 *)

PROG

(PARI) {a(n) = if( n==0, 0, 1 - valuation(n, 2)%2)}; /* Michael Somos, Sep 04 2006 */

(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 */

(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 */

(Haskell)

import Data.Bits (xor)

a035263 n = a035263_list !! (n-1)

a035263_list = zipWith xor a010060_list $ tail a010060_list

-- Reinhard Zumkeller, Mar 01 2012

CROSSREFS

Parity of A001511. Anti-parity of A007814.

Absolute values of first differences of A010060. Apart from signs, same as A029883.

Swapping 0 and 1 gives A096268.

Cf. A033485, A050292, A088172, A019300, A010060, A039982, A073675, A121701, A141260, A000041, A174065, A220466.

Sequence in context: A104106 A141260 A029883 * A089045 A070749 A059778

Adjacent sequences:  A035260 A035261 A035262 * A035264 A035265 A035266

KEYWORD

nonn,easy,nice,mult,changed

AUTHOR

Karamanos Konstantinos, N. J. A. Sloane

STATUS

approved

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Last modified October 22 20:39 EDT 2014. Contains 248401 sequences.