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A099054
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Arshon's sequence: start from 1 and replace the letters in odd positions using 1 -> 123, 2 -> 231, 3 -> 312 and the letters in even positions using 1 -> 321, 2-> 132, 3 -> 213.
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7
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1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 2, 1, 2, 3, 1, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1
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OFFSET
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0,2
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COMMENTS
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The first three iterations give 1; 123; 123132312; ... the limiting sequence is shown here. Properties: the sequence is squarefree and cannot be defined by iteration of a morphism.
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REFERENCES
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G. A. Gurevich, Nonrepeating sequences, pp. 61-66 of Kvant Selecta: Combinatorics I, ed. S. Tabachnikov, AMS, 2001.
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LINKS
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Sergey Kitaev, There are no iterated morphisms that define the Arshon sequence and the σ-sequence, Journal of Automata, Languages and Combinatorics 8 (2003) 1, 43-0-50. preprint, arXiv:math/0205216 [math.CO], 2002.
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MATHEMATICA
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f[n_List] := Block[{a = {}, l = Length[n], k = 1}, While[k < l + 1, If[ EvenQ[ k], Switch[ n[[k]], 1, AppendTo[a, 321], 2, AppendTo[a, 132], 3, AppendTo[a, 213]], Switch[ n[[k]], 1, AppendTo[a, 123], 2, AppendTo[a, 231], 3, AppendTo[a, 312]]]; k++ ]; Flatten[IntegerDigits /@ a]]; Take[ Nest[f, {1}, 5], 105] (* Robert G. Wilson v, Nov 15 2004 *)
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PROG
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(Haskell)
import Data.List (transpose, stripPrefix); import Data.Maybe (fromJust)
a099054 n = a099054_list !! n
a099054_list = 1 : concatMap fromJust (zipWith stripPrefix ass $ tail ass)
where ass = iterate f [1]
f xs = concat $ concat $ transpose [map g $ e xs, map h $ o xs]
g 1 = [1, 2, 3]; g 2 = [2, 3, 1]; g 3 = [3, 1, 2]
h 1 = [3, 2, 1]; h 2 = [1, 3, 2]; h 3 = [2, 1, 3]
e [] = []; e [x] = [x]; e (x:_:xs) = x : e xs
o [] = []; o [x] = []; o (_:x:xs) = x : o xs
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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