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A250005 a(n) = 1, unless forbidden by the "iterated cubefree rule", in which case a(n) = 2. 2


%S 1,1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,

%T 1,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,2,1,2,1,1,2,1,1,

%U 2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1

%N a(n) = 1, unless forbidden by the "iterated cubefree rule", in which case a(n) = 2.

%C Inspired by the Kolakoski sequence A000002.

%C The "iterated cubefree rule" defined here means that the sequence a does not contain words repeated three times, and neither does any of the sequences RL^n(a), n >= 0, where RL is the runlength transform (in other words, the sequence a is smooth, that is differentiable arbitrarily many times - see for example the link by Fedou and Fici for a definition of differentiable sequences). The OK sequence A000002 obviously satisfies this rule because RL(OK)=OK.

%C The patterns appearing in this sequence look very similar to those in A000002. Indeed, the frequency of terms which value is not constrained by the iterated cubefree rule diminishes as n increases in both sequences and seems to tend to zero (A250006 gives the ranks of these terms for this sequence).

%C Thus all binary sequence satisfying the iterated cubefree rule might share similar limiting properties, in particular: does the limiting frequency of 1's exist for such a sequence? If yes, is it equal to 1/2, despite the priority given to 1's in this sequence?

%H Jean-Christophe Hervé, <a href="/A250005/b250005.txt">Table of n, a(n) for n = 1..10000</a>

%H F. M. Dekking, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002544490">On the structure of self-generating sequences</a>, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.

%H J. M. Fedou and G. Fici, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL13/Fici/fici.html">Some remarks on differentiable sequences and recursivity</a>, Journal of Integer Sequences 13(3): Article 10.3.2 (2010).

%H C. Kimberling, Integer Sequences and Arrays, <a href="http://faculty.evansville.edu/ck6/integer/index.html">Illustration of the Kolakoski sequence</a>

%H W. Kolakoski and N. Ucoluk, <a href="http://www.jstor.org/stable/2314839">Problem 5304: Self Generating Runs</a>, Amer. Math. Monthly, 72 (1965), 674; 73 (1966), 681-682.

%e a(1)=a(2)=1 because it is not forbidden by the cubefree rule. But a(3) cannot be equal to 1, thus a(3)=2; Then a(4)=a(5)= 1, a(6)=2, a(7) = 1 but a(8)=2, because otherwise we would have the cube 112112112.

%o (R)

%o runlen<- function(seq) {

%o n<- length(seq)

%o if(n==1) return(c())

%o else {rl<-c()

%o i<-1

%o while(i < n) { k<-1

%o while(i+k<=n) {

%o if(seq[i+k]!=seq[i]) break

%o else k<-k+1}

%o if (i+k!=2) rl<-c(rl, k)

%o i<-i+k }

%o return(rl)}}

%o # following recursive function not optimized

%o isOK<-function(x,seq) {

%o n<-length(seq)

%o if(n<=1) return(1)

%o if(seq[n]==seq[n-1]&seq[n-1]==x) return(0)

%o rl<-runlen(c(seq,x))

%o lrl<-length(rl)

%o return(isOK(rl[lrl],rl[1:lrl-1]))}

%o sequence<-function(n) {

%o if(n<=0) return(c())

%o seq<-c(1)

%o for(i in 2:n) {

%o if(isOK(1,seq)) seq<-c(seq, 1)

%o else seq<-c(seq, 2)}

%o return(cbind(c(1:n),seq))}

%Y Cf. A000002, A250006.

%K nonn

%O 1,3

%A _Jean-Christophe Hervé_, Nov 10 2014

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Last modified April 23 09:49 EDT 2021. Contains 343204 sequences. (Running on oeis4.)