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A035263
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Trajectory of 1 under the morphism 1 -> 10, 0 -> 11.
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52
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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
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OFFSET
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1,1
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COMMENTS
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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 DELEHAM, 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 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
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REFERENCES
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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).
K. Karamanos, From 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, Chaos, Solitons and Fractals 10(7), 1135 - 1150 (1999).
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).
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).
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LINKS
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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, Fxtbook, pp.9-10, pp.734-738
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).
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences
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
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FORMULA
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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
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MAPLE
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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]
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MATHEMATICA
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Nest[ Function[l, {Flatten[(l /. {0 -> {1, 1}, 1 -> {1, 0}})]}], {1}, 7] (* Or *)
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]
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PROG
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(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
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CROSSREFS
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Parity of A001511. Anti-parity of A007814.
Absolute values of first differences of A010060. Apart from signs, same as A029883.
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
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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Karamanos Konstantinos, N. J. A. Sloane.
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STATUS
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approved
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