This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A170823 #28 Dec 19 2021 09:08:29 %S A170823 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, %T A170823 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, %U A170823 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 %N A170823 An infinite word on the alphabet 1, 2, 3 by Bollobas. %C A170823 A concatenation of blocks u_k, k >= 0, where u_k has length 5^k. The sequence is defined recursively - see the Maple code. %C A170823 From _Kevin Ryde_, Aug 11 2020: (Start) %C A170823 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. %C A170823 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. %C A170823 - - + + - - + \ two l=4 signs subsequences %C A170823 - + + - - + + / in A337004 making squares here %C A170823 All else in the argument holds, and in particular the "peaks" reduction means the only squares are lengths l = 4*5^k. %C A170823 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. %C A170823 (End) %D A170823 B. Bollobas, The Art of Mathematics: Coffee Time in Memphis, Cambridge, 2006, pp. 226-228. %H A170823 B. Bollobas, <a href="/A170823/a170823.pdf">The Art of Mathematics: Coffee Time in Memphis</a>, Cambridge 2006, scan of pages 226, 227 annotated by _N. J. A. Sloane_, Jul 31 2020. %H A170823 Boris Zolotov, <a href="https://arxiv.org/abs/1505.00019">Another Solution to the Thue Problem of Non-Repeating Words</a>, arXiv:1505.00019 [math.CO], 2015. (Section 5 morphism 1, then section 6.) %H A170823 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a> %p A170823 a:=[1,2,3,2,1]; b:=[2,3,1,3,2]; c:=[3,1,2,1,3]; S:=[1]; %p A170823 for m from 1 to 6 do S:=subs({1=a[],2=b[],3=c[]},S); od: S; %o A170823 (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 %Y A170823 Cf. A010060, A005678, A005679, A005680, A005681, A006156, A007413. %Y A170823 Cf. A337004 (first differences as +1,-1). %K A170823 nonn %O A170823 0,2 %A A170823 _N. J. A. Sloane_, Dec 25 2009