%I #6 Jan 16 2013 18:51:55

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

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

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

%N Kolakoski fan based on A000034 with initial row 1.

%C Conjecture (following Benoit Cloitre's conjecture at A111090): if L(n) is the number (assumed finite) of terms in row n of K, then L(n)*(2/3)^n approaches a constant. (L= A143590.)

%F Introduced here is an array K called the "Kolakoski fan based on a sequence s with initial row w": suppose that s=(s(1),s(2),...) is a sequence of 1's and 2's and that w=(w(1),w(2),...) is a finite or infinite sequence of 1's and 2's. Assume that s(1)=w(1) and that if w(1)=1 then s contains at least one 2. Row 1 of the array K is w. Subsequent rows are defined inductively: the first term of row n is s(n) and the remaining terms are defined by Kolakoski substitution; viz., each number in row n-1 tells the string-length (1 or 2) of the next string in row n, each term being either 1 or 2.

%e s=(1,2,1,2,1,2,1,2,...) and w=1, so the first 7 rows are

%e 1

%e 2

%e 1 1

%e 2 1

%e 1 1 2

%e 2 1 2 2

%e 1 1 2 1 1 2 2

%Y Cf. A000002, A143477, A143490.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Aug 25 2008