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
Ulrich Reitebuch, Henriette-Sophie Lipschütz, and Konrad Polthier, Visualizing the Kolakoski Sequence, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 481-484.
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
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