A125047 -id:A125047 - cp's OEIS frontend

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

Jeremy Gardiner, Dec 27 2005

nonn

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

			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 */

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

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

Cf. A003324.

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

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

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

A122002 a(0)=5; otherwise a(n) = (n mod 4) if n is odd, a(n) = h + 4, where h = (highest odd divisor of n) mod 4 if n is even.

Original entry on oeis.org

5, 1, 5, 3, 5, 1, 7, 3, 5, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 7, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 5, 1, 5, 3, 7, 1, 7, 3, 7, 1, 5, 3, 5, 1, 7, 3, 7, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 5, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 7, 1, 5, 3, 7, 1, 7, 3, 7, 1, 5, 3, 5

Offset: 0

N. J. A. Sloane, Aug 05 2008

nonn

a(n) in {1,3,5,7} for all n. a(4k+i) = i if i is odd.

There is a typo in Grytczuk's definition: he has "+ 5" instead of "+ 4".

A. Carpi, Multidimensional unrepetitive configurations, Theoret. Comput. Sci., 56 (1988), 233-241. a(n) = a_n of lemma 3.2 for the case p=2 and m=0 (which is corollary 3.3).
Jaroslaw Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429. [See Problem 15.]
Jui-Yi Kao, Narad Rampersad, Jeffrey Shallit, Manuel Silva, Words Avoiding Repetitions in Arithmetic Progressions, Theoretical Computer Science, volume 391, issues 1-2, February 2008, pages 126-137. And arXiv:math/0608607 [math.CO], 2006. (Extending to generalized paperfolding sequences.)
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to squarefree words

Essentially the same: A112658 (map 1357 -> 0213), A125047 (map 1357 -> 2314).

Cf. A003324.

a[0]=5;a[n_]:=If[OddQ[n],Mod[n,4],4+Mod[Select[Divisors[n],OddQ][[-1]],4]];Table[a[n],{n,0,100}] (* James C. McMahon, Oct 25 2024 *)

a(n) = 2*if(n,bittest(n,valuation(n,2)+1)) + if(n%2,1,5); \\ Kevin Ryde, Sep 09 2020

Morphism 1 -> 5,3; 3 -> 7,3; 5 -> 5,1; 7 -> 7,1 starting from 5 [Carpi, h in remark after lemma 3.2]. - Kevin Ryde, Sep 09 2020

Terms from a(26) on from John W. Layman, Aug 27 2008

cp's OEIS Frontend

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

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