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

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, 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, 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

Offset: 1

Author

Jeremy Gardiner, Dec 27 2005

Keywords

• nonn

Comments

Even-indexed terms of this sequence are the sequence A099545. - Alexandre Wajnberg, Jan 02 2006

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

Examples

			The first few iterations of the morphism, starting with 0:
Start: 0
Rules:
  0 --> 01
  1 --> 21
  2 --> 03
  3 --> 23
-------------
0:   (#=1)
  0
1:   (#=2)
  01
2:   (#=4)
  0121
3:   (#=8)
  01210321
4:   (#=16)
  0121032101230321
5:   (#=32)
  01210321012303210121032301230321
6:   (#=64)
  0121032101230321012103230123032101210321012303230121032301230321
/* _Joerg Arndt_, Jul 18 2012 */
		

Links

• J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, 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.
• J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, 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]
• Kirby A. Baker, George F. McNulty, Walter Taylor, Growth Problems For Avoidable Words, Theoretical Computer Science, volume 69, number 3, 1989, pages 319-345. (See morphism start of section 3, page 325.)
• Richard A. Dean, A sequence without repeats on x, ..., Amer. Math. Monthly 72, 1965. pp. 383-385. MR 31 #350.
• George F. McNulty, Avoidable Words, conference slides, 2003, slides 38-39. (Also conference abstract.)
• Index entries for sequences that are fixed points of mappings
• Index entries for sequences related to squarefree words

Crossrefs

Essentially the same: A343180, also A122002 (map 0123 -> 1537), A125047 (map 0123 -> 2134).

Cf. A003324.

Programs

Mathematica

Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> {0, 3}, 3 -> {2, 3}}] &, {0}, 7] (* Robert G. Wilson v, Dec 27 2005 *)

PARI

a(n) = 2*bittest(n,valuation(n,2)+1) + !(n%2); \\ Kevin Ryde, Sep 09 2020

Formula

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