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A170823
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An infinite word on the alphabet 1, 2, 3 by Bollobas.
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3
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1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 1, 3, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1
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OFFSET
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0,2
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COMMENTS
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A concatenation of blocks u_k, k >= 0, where u_k has length 5^k. The sequence is defined recursively - see the Maple code.
Bollobás gives this sequence intending it to be a squarefree ternary word, where squarefree means nowhere a repeat w w for a block w of any length. However, squares do occur in it, for example a(18) onwards is 3212 3212, or a(19) onwards is 2123 2123.
In Bollobás' proof, the signs sequence is A337004. For blocks w of length l=4, the second signs subsequence presented (which should stop at length 7), does in fact occur, as does one other.
- - + + - - + \ two l=4 signs subsequences
- + + - - + + / in A337004 making squares here
All else in the argument holds, and in particular the "peaks" reduction means the only squares are lengths l = 4*5^k.
Zolotov shows this word is cubefree, and weakly squarefree (no x w w x where x is a single symbol and w is a block, possibly empty). However uniform cyclic squarefree must wait for Leech's order 13 morphism in A337005.
(End)
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REFERENCES
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B. Bollobas, The Art of Mathematics: Coffee Time in Memphis, Cambridge, 2006, pp. 226-228.
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LINKS
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MAPLE
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a:=[1, 2, 3, 2, 1]; b:=[2, 3, 1, 3, 2]; c:=[3, 1, 2, 1, 3]; S:=[1];
for m from 1 to 6 do S:=subs({1=a[], 2=b[], 3=c[]}, S); od: S;
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PROG
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(PARI) my(table=[0, 1, 2, 1, 0]); a(n) = my(v=digits(n, 5)); sum(i=1, #v, table[v[i]+1]) %3+1; \\ Kevin Ryde, Jul 31 2020
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CROSSREFS
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Cf. A337004 (first differences as +1,-1).
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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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