A288711 1-limiting word of the mapping 00->1000, 10->00, starting with 00.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0

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

Cf. A000045, A288708, A288709.

Sequence in context: A014189 A319691 A079979 * A347312 A089010 A162289

Adjacent sequences: A288708 A288709 A288710 * A288712 A288713 A288714

Keyword

nonn,easy

Author

Clark Kimberling, Jun 16 2017

Status

approved