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 A099054 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. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. a(n) = A219762(n+1) + 1. - Reinhard Zumkeller, Aug 08 2014 REFERENCES S. E. Arshon, A proof of the existence of infinite asymmetric sequences on n symbols. Matematicheskoe Prosveshchenie (Mathematical Education), 2 (1935) 24-33 (in Russian); http://ilib.mccme.ru/djvu/mp1/mp1-2.htm S. E. Arshon, A proof of the existence of infinite asymmetric sequences on n symbols. Mat. Sb., 2 (1937) 769-779 (in Russian, with French abstract). Zheng-Pan Wang, Some combinatorial properties of Arshon sequences of arbitrary orders, J. Algebra and Applications, 12 (2013), article #1250210. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 James D. Currie: No iterated morphism generates any Arshon Sequence of Odd Order, Discrete Math. 259 (2002), no. 1-3, 277-283. 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] MATHEMATICA 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 *) PROG (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 -- Reinhard Zumkeller, Aug 08 2014 CROSSREFS Cf. A100336, A100337, A003270. Cf. A219762, A241418 (first differences). Sequence in context: A301630 A063047 A003270 * A071282 A107796 A194308 Adjacent sequences:  A099051 A099052 A099053 * A099055 A099056 A099057 KEYWORD nonn,nice,easy AUTHOR Sergey Kitaev, Nov 14 2004 EXTENSIONS More terms from Robert G. Wilson v and John W. Layman, Nov 15 2004 STATUS approved

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Last modified January 19 19:49 EST 2019. Contains 319309 sequences. (Running on oeis4.)