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A064990
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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.
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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, 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Called the Mephisto Waltz sequence (or the Mephisto Waltz infinite word).
May also be obtained by starting with 0 and iterating the morphism 0 -> 001, 1 -> 110.
The sequence is fourth-power free.
The sequence gives A_oo. For the concatenation A_0, A_1, A_2, ... see A134391.
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REFERENCES
| J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 25.
Konrad Jacobs, Invitation to Mathematics, Princeton, 1992; pp. 105-106 and 215.
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LINKS
| Joerg Arndt, Fxtbook, section 38.1.2, pp. 729-730
Eric Weisstein's World of Mathematics, Mephisto Waltz Sequence
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FORMULA
| a(3k-2)=a(k), a(3k-1)=a(k), a(3k)=1-a(k) for k>=1, a(0)=0.
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EXAMPLE
| Here are A_0 through A_5:
0
001
001001110
001001110001001110110110001
001001110001001110110110001001001110001001110110110001110110001110110001001001110
001001110001001110110110001001001110001001110110110001110110001110110001001001110\
00100111000100111011011000100100111000100111011011000111011000111011000100100111\
0110110001110110001001001110110110001110110001001001110001001110001001110110110001
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MATHEMATICA
| t = Nest[Flatten[# /. {0->{0, 0, 1}, 1->{1, 1, 0}}] &, {0}, 5] (*A064990*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (*A189658*)
Flatten[Position[t, 1]] (*A189659*)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189660*)
(* by Clark Kimberling, Apr 25 2011 *)
Nest[ Flatten[# /. # -> {#, #, Abs[# - 1]}] &, {0}, 5] (* Robert G. Wilson v, Sep 27 2011 *)
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CROSSREFS
| Cf. Thue-Morse sequence A010060, A001285. Number of 0's in A_k gives A007051, number of 1's is A003462. See also A064991.
Cf. A134391, A189628.
Sequence in context: A131378 A014707 A106138 * A059125 A111406 A156731
Adjacent sequences: A064987 A064988 A064989 * A064991 A064992 A064993
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Michael Gilleland (megilleland(AT)yahoo.com), Oct 31 2001
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EXTENSIONS
| More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Nov 29 2001
Corrected by N. J. A. Sloane, Jun 14 2010, at the suggestion of Chris Erickson.
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