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A073057
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Define s(1)={1,2}, s(2)={1,3}, s(3)={4,2} and s(4)={4,3}. For a finite sequence A={a_1, ..., a_n}, with elements in {1,2,3,4}, define t(A) to be the concatenation of A, s(a_1), s(a_2), ... and s(a_n). Start with the sequence {1,2,3,4} and repeatedly apply t; limiting sequence is shown.
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1
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1, 2, 3, 4, 1, 2, 1, 3, 4, 2, 4, 3, 1, 2, 1, 3, 4, 2, 4, 3, 1, 2, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 4, 3, 4, 2, 1, 2, 1, 3, 4, 2, 4, 3, 1, 2, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 4, 3, 4, 2, 1, 2, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 4, 3, 4, 2, 1, 2, 1, 3, 1, 2, 4, 2, 1, 2, 1, 3, 4, 3, 1, 3, 4, 3, 4, 2, 1, 2, 4, 2, 4, 3, 4, 2, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A Rudin-Shapiro substitution sequence as a mapped nested list in Mathematica.
I was surprised to find that this kind of substitution gave different results from the classical ones quoted by Allouche. This sequence is related to paper folding sequences by a second substitution map of {a,b,c,d} ->{0,0,1,1}
Fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 42, 4 ->43, starting from a(1-4) = 1234. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 02 2009]
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REFERENCES
| J. P. Allouche, Schroedinger operators with Rudin-Shapiro potentials are not palindromic, J. Math. Phys. 38 (1997), no. 4, 1843-1848.
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EXAMPLE
| E.g. the first step takes {1,2,3,4} to {1,2,3,4, 1,2, 1,3, 4,2, 4,3}. The next takes this to {1,2,3,4,1,2,1,3,4,2,4,3, 1,2, 1,3, 4,2, 4,3, 1,2, 1,3, 1,2, 4,2, 4,3, 1,3, 4,3, 4,2}
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MATHEMATICA
| s[1]={1, 2}; s[2]={1, 3}; s[3]={4, 2}; s[4]={4, 3}; t[a_] := Join[a, Flatten[s/@a]]; t[t[t[t[{1, 2, 3, 4}]]]] (* Continue applying t for more terms *)
Nest[ Flatten[ Join[ #, # /. {1 -> {1, 2}, 2 -> {1, 3}, 3 -> {4, 2}, 4 -> {4, 3}}]] &, {1, 2, 3, 4}, 3] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 02 2009]
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CROSSREFS
| Cf. A020987, A073058.
Sequence in context: A104412 A195451 A118310 * A084310 A078978 A171171
Adjacent sequences: A073054 A073055 A073056 * A073058 A073059 A073060
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 16 2002
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