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A112658
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Dean's Word: Omega 2,1: the trajectory of 0 -> 01, 1 -> 21, 2 -> 03, 3 -> 23.
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5
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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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.)
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> {0, 3}, 3 -> {2, 3}}] &, {0}, 7] (* Robert G. Wilson v, Dec 27 2005 *)
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PROG
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(PARI) a(n) = 2*bittest(n, valuation(n, 2)+1) + !(n%2); \\ Kevin Ryde, Sep 09 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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