A099054 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

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.

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

G. A. Gurevich, Nonrepeating sequences, pp. 61-66 of Kvant Selecta: Combinatorics I, ed. S. Tabachnikov, AMS, 2001.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

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).

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).

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], 2002.

Zheng-Pan Wang, Some combinatorial properties of Arshon sequences of arbitrary orders, J. Algebra and Applications, 12 (2013), article #1250210.

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