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A112658 Dean's Word: Omega 2,1: the trajectory of 0 -> 01, 1 -> 21, 2 -> 03, 3 -> 23.

%I A112658 #25 May 02 2021 11:36:00
%S A112658 0,1,2,1,0,3,2,1,0,1,2,3,0,3,2,1,0,1,2,1,0,3,2,3,0,1,2,3,0,3,2,1,0,1,
%T A112658 2,1,0,3,2,1,0,1,2,3,0,3,2,3,0,1,2,1,0,3,2,3,0,1,2,3,0,3,2,1,0,1,2,1,
%U A112658 0,3,2,1,0,1,2,3,0,3,2,1,0,1,2,1,0,3,2,3,0,1,2,3,0,3,2,3,0,1,2,1,0,3,2,1,0
%N A112658 Dean's Word: Omega 2,1: the trajectory of 0 -> 01, 1 -> 21, 2 -> 03, 3 -> 23.
%C A112658 Even-indexed terms of this sequence are the sequence A099545. - _Alexandre Wajnberg_, Jan 02 2006
%C A112658 Fractal sequence: odd terms are 0, 2, 0, 2,...; the subsets formed with the terms of index (2^i)n, with i>0, are identical: a(2n)=a(4n)=a(8n)=a(16n)=... - _Alexandre Wajnberg_, Jan 02 2006
%H A112658 J.-P. Allouche and M. Mendes France, <a href="https://webusers.imj-prg.fr/~jean-paul.allouche/allmendeshouches.pdf">Automata and Automatic Sequences</a>, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. See page 6.
%H A112658 J.-P. Allouche and M. Mendes France, <a href="/A003842/a003842.pdf">Automata and Automatic Sequences</a>, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. See page 6. [Local copy]
%H A112658 Kirby A. Baker, George F. McNulty, Walter Taylor, <a href="https://doi.org/10.1016/0304-3975(89)90071-6">Growth Problems For Avoidable Words</a>, Theoretical Computer Science, volume 69, number 3, 1989, pages 319-345.  (See morphism start of section 3, page 325.)
%H A112658 Richard A. Dean, <a href="http://www.jstor.org/stable/2313498">A sequence without repeats on x, ...</a>, Amer. Math. Monthly 72, 1965. pp. 383-385. MR 31 #350.
%H A112658 George F. McNulty, <a href="http://people.math.sc.edu/mcnulty/talks/victoria.pdf">Avoidable Words</a>, conference slides, 2003, slides 38-39.  (Also <a href="http://web.archive.org/web/20150911215528id_/http://at.yorku.ca/cgi-bin/amca/cala-20">conference abstract</a>.)
%H A112658 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%H A112658 <a href="/index/Sq#square_free">Index entries for sequences related to squarefree words</a>
%F A112658 It should be easy to prove that a(4n) = 0, a(4n+2) = 2, a(8n+1) = 1, a(8n+5) = 3, a(4n+3) = a(2n+1). This would imply that a(2n) = 2(n mod 2), a(2n+1) = 1 + 2*A014707(n), with A014707(n) the classical paperfolding curve. - _Ralf Stephan_, Dec 28 2005
%e A112658 The first few iterations of the morphism, starting with 0:
%e A112658 Start: 0
%e A112658 Rules:
%e A112658   0 --> 01
%e A112658   1 --> 21
%e A112658   2 --> 03
%e A112658   3 --> 23
%e A112658 -------------
%e A112658 0:   (#=1)
%e A112658   0
%e A112658 1:   (#=2)
%e A112658   01
%e A112658 2:   (#=4)
%e A112658   0121
%e A112658 3:   (#=8)
%e A112658   01210321
%e A112658 4:   (#=16)
%e A112658   0121032101230321
%e A112658 5:   (#=32)
%e A112658   01210321012303210121032301230321
%e A112658 6:   (#=64)
%e A112658   0121032101230321012103230123032101210321012303230121032301230321
%e A112658 /* _Joerg Arndt_, Jul 18 2012 */
%t A112658 Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> {0, 3}, 3 -> {2, 3}}] &, {0}, 7] (* _Robert G. Wilson v_, Dec 27 2005 *)
%o A112658 (PARI) a(n) = 2*bittest(n,valuation(n,2)+1) + !(n%2); \\ _Kevin Ryde_, Sep 09 2020
%Y A112658 Essentially the same: A343180, also A122002 (map 0123 -> 1537), A125047 (map 0123 -> 2134).
%Y A112658 Cf. A003324.
%K A112658 nonn
%O A112658 1,3
%A A112658 _Jeremy Gardiner_, Dec 27 2005