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 A079729 Kolakoski-(1,2,3) sequence: a(n) is the length of the n-th run. 2
 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 2, 2, 3, 3, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 2, 2, 3, 3, 3, 1, 2, 3, 3, 1, 1, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Old name was: Kolakoski variation using (1,2,3) starting with 1,2. Partial sum sequence is expected to be asymptotic to 2*n. From Michel Dekking, Jan 31 2018: (Start) (a(n)) is the unique fixed point of the 3-block substitution beta given by     111 -> 123,            112 -> 1233,     122 -> 12233,          123 -> 122333,     222 -> 112233,         223 -> 1122333,     231 -> 112223,         233 -> 11222333,     311 -> 11123,          312 -> 111233,     331 -> 1112223,        333 -> 111222333. Here BL3 := {111, 112, 122, 123, 222, 223, 231, 233, 311, 312, 331, 333} is the set of all words of length 3 occurring at a position 1 mod 3 in (a(n)). This can be seen by splitting the  words beta(B) into words of length 3, and looking at the possible extensions of those words beta(B) that have a length which is not a multiple of 3. For example, beta(122) = 12233 can only be extended to 122331 or to 122333, and both words 331 and 333 are in BL3. Interestingly, BL3 is invariant for the permutation 1->3, 2->1, 3->2 (and its square).   Note: In general, a 3-block substitution beta maps a word w(1)...w(3n) to the word   beta(w(1)w(2)w(3))...beta(w(3n-2)w(3n-1)w(3n)).   If the length of a word w is 3n+r, with r=1 or r=2, then the last letter, respectively last 2 letters are ignored. (End) Conjecture: the frequencies of  1's, 2's and 3's in (a(n)) exist and are all equal to 1/3. This conjecture implies the conjecture of Benoit Cloitre on the partial sum sequence. - Michel Dekking, Jan 31 2018 LINKS Ivan Neretin, Table of n, a(n) for n = 1..10000 FORMULA Iterate beta: 122 -> 12233 ~ 122331 -> 122331112223 -> 12233111222312331122333, etc. Here a(6)=1 has been added to 12233 in step two to continue the iteration. - Michel Dekking, Jan 31 2018 EXAMPLE Sequence begins: 1,2,2,3,3,1,1,1,2,2,2,3,1,2,3,3,1,1,... read it as: (1),(2,2),(3,3),(1,1,1),(2,2,2),(3),(1),(2),(3,3),(1,1),... then count the terms in parentheses to get: 1,2,2,3,3,1,1,1,2,2,... which is the same sequence. MATHEMATICA seed = {1, 2, 3}; w = {}; i = 1; Do[w = Join[w, Array[seed[[Mod[i - 1, Length[seed]] + 1]] &, If[i > Length[w], seed, w][[i]]]]; i++, {n, 53}]; w (* Ivan Neretin, Apr 02 2015 *) PROG (PARI) a=[1, 2, 2]; for(n=3, 100, for(i=1, a[n], a=concat(a, 1+((n-1)%3)))); a; \\ Benoit Cloitre, Feb 13 2009 CROSSREFS Cf. A000002. Sequence in context: A300290 A068460 A143797 * A288723 A071859 A135695 Adjacent sequences:  A079726 A079727 A079728 * A079730 A079731 A079732 KEYWORD nonn AUTHOR Benoit Cloitre, Feb 17 2003 EXTENSIONS More terms from Philippe Deléham, Sep 24 2006 Name changed and text edited by Michel Dekking, Jan 31 2018 STATUS approved

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Last modified March 23 15:12 EDT 2018. Contains 301123 sequences. (Running on oeis4.)