A288707 0-limiting word of the mapping 00->1000, 10->00, starting with 00.

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, 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 0-limiting word is the limit of the n-th iterates for n == 0 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.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

The first four n-th iterates for n == 0 mod 3 are
00
001000
0010001000001000
001000100000100010000010000010001000001000

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[10]] - 48   (* A288707 *)
Flatten[Position[st, 0]]  (* A288708 *)
Flatten[Position[st, 1]]  (* A288709 *)
Table[StringLength[w[n]], {n, 0, 20}] (* 2*A000045 *)

Crossrefs

Cf. A000045, A288708, A288709, A288711 (1-limiting word).

Sequence in context: A286044 A129272 A059648 * A079261 A354028 A359152

Adjacent sequences: A288704 A288705 A288706 * A288708 A288709 A288710

Keyword

nonn,easy

Author

Clark Kimberling, Jun 16 2017

Status

approved