OFFSET
1
COMMENTS
Iterates of the mapping, starting with 00:
00
1000
001000
1000001000
0010001000001000
10000010000010001000001000
001000100000100010000010000010001000001000
The 1-limiting word is the limit of the n-th iterates for n == 1 mod 2.
The number of letters (0's and 1's) in the n-th iterate is given by 2*F(n+2) for n >= 0, where F = A000045 (Fibonacci numbers), as follows from the observation that this sequence is the {0->00, 1->10}-transform of the mapping 0->10, 1->0; see the Example section.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
EXAMPLE
The first three n-th iterates for n == 1 (mod 3) are
1000
1000001000
10000010000010001000001000
Regarding the connection to Fibonacci numbers mentioned in Comments, iterates of the morphism 0->10, 1->0, starting with 0th iterate 0, are
0
10
010
10010
01010010
1001001010010
where the length of the n-th iterate is F(n+2). The {0->00, 1->10}-transform of the iterates is then 00, 1000, 001000, ..., as indicated in Comments.
MATHEMATICA
s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "00"}]
Table[w[n], {n, 0, 8}]
st = ToCharacterCode[w[9]] - 48 (* A288711 *)
Flatten[Position[st, 0]] (* A288712 *)
Flatten[Position[st, 1]] (* A288713 *)
Table[StringLength[w[n]], {n, 0, 20}] (* 2*A000045 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 16 2017
STATUS
approved