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A170823 An infinite word on the alphabet 1, 2, 3 by Bollobas. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A concatenation of blocks u_k, k >= 0, where u_k has length 5^k. The sequence is defined recursively - see the Maple code.

From Kevin Ryde, Aug 11 2020: (Start)

Bollob├ís 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 Bollob├ís' 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)

REFERENCES

B. Bollobas, The Art of Mathematics: Coffee Time in Memphis, Cambridge, 2006, pp. 226-228.

LINKS

Table of n, a(n) for n=0..104.

B. Bollobas, The Art of Mathematics: Coffee Time in Memphis, Cambridge 2006, scan of pages 226,227 annotated by N. J. A. Sloane, Jul 31 2020.

Boris Zolotov, Another Solution to the Thue Problem of Non-Repeating Words, arXiv:1505.00019 [math.CO], 2015.  (Section 5 morphism 1, then section 6.)

Index entries for sequences that are fixed points of mappings

MAPLE

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;

PROG

(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

CROSSREFS

Cf. A010060, A005678, A005679, A005680, A005681, A006156, A007413.

Cf. A337004 (first differences as +1,-1).

Sequence in context: A260342 A281939 A002175 * A068073 A324504 A032452

Adjacent sequences:  A170820 A170821 A170822 * A170824 A170825 A170826

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 25 2009

STATUS

approved

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Last modified January 25 05:31 EST 2021. Contains 340416 sequences. (Running on oeis4.)