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A064990 If A_k denotes the first 3^k terms, then A_0 = 0, A_{k+1} = A_k A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's. 10


%S 0,0,1,0,0,1,1,1,0,0,0,1,0,0,1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0,0,1,1,

%T 1,0,0,0,1,0,0,1,1,1,0,1,1,0,1,1,0,0,0,1,1,1,0,1,1,0,0,0,1,1,1,0,1,1,

%U 0,0,0,1,0,0,1,0,0,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0,1,0,0,1,1,1,0

%N If A_k denotes the first 3^k terms, then A_0 = 0, A_{k+1} = A_k A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's.

%C Called the Mephisto Waltz sequence (or the Mephisto Waltz infinite word).

%C May also be obtained by starting with 0 and iterating the morphism 0 -> 001, 1 -> 110.

%C The sequence is fourth-power free.

%C The sequence gives A_oo. For the concatenation A_0, A_1, A_2, ... see A134391.

%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 25.

%D Konrad Jacobs, Invitation to Mathematics, Princeton, 1992; pp. 105-106 and 215.

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 38.1.2, pp. 729-730

%H J. Endrullis, D. Hendriks and J. W. Klop, <a href="http://www.cs.vu.nl/~diem/publication/pdf/degrees.pdf">Degrees of streams</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MephistoWaltzSequence.html">Mephisto Waltz Sequence</a>

%F a(3k-2)=a(k), a(3k-1)=a(k), a(3k)=1-a(k) for k>=1, a(0)=0.

%e Here are A_0 through A_5:

%e 0

%e 001

%e 001001110

%e 001001110001001110110110001

%e 001001110001001110110110001001001110001001110110110001110110001110110001001001110

%e 001001110001001110110110001001001110001001110110110001110110001110110001001001110\

%e 00100111000100111011011000100100111000100111011011000111011000111011000100100111\

%e 0110110001110110001001001110110110001110110001001001110001001110001001110110110001

%p with(ListTools);

%p f2:=proc(S) map(x->x+1 mod 2, S); end;

%p f:=proc(S) global f2;

%p [op(S), op(S), op(f2(S))]; end;

%p S:=[0];

%p for n from 1 to 6 do S:=f(S): od:

%p S; # _N. J. A. Sloane_, Apr 30 2017

%t t = Nest[Flatten[# /. {0->{0,0,1}, 1->{1,1,0}}] &, {0}, 5] (*A064990*)

%t f[n_] := t[[n]]

%t Flatten[Position[t, 0]] (*A189658*)

%t Flatten[Position[t, 1]] (*A189659*)

%t s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;

%t Table[s[n], {n, 1, 120}] (*A189660*)

%t (* by Clark Kimberling, Apr 25 2011 *)

%t Nest[ Flatten[# /. # -> {#, #, Abs[# - 1]}] &, {0}, 5] (* _Robert G. Wilson v_, Sep 27 2011 *)

%Y Cf. Thue-Morse sequence A010060, A001285. Number of 0's in A_k gives A007051, number of 1's is A003462. See also A064991.

%Y Cf. A134391, A189628.

%Y A285196 is a similar sequence.

%K nonn,easy,nice

%O 0,1

%A Michael Gilleland (megilleland(AT)yahoo.com), Oct 31 2001

%E More terms from _Naohiro Nomoto_, Nov 29 2001

%E Corrected by _N. J. A. Sloane_, Jun 14 2010, at the suggestion of Chris Erickson

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Last modified December 6 21:48 EST 2019. Contains 329809 sequences. (Running on oeis4.)