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A288711 1-limiting word of the mapping 00->1000, 10->00, starting with 00. 5
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 (list; graph; refs; listen; history; text; internal format)
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 (0s and 1s) 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 * A089010 A162289 A122276

Adjacent sequences:  A288708 A288709 A288710 * A288712 A288713 A288714

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 16 2017

STATUS

approved

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Last modified January 21 10:59 EST 2020. Contains 331105 sequences. (Running on oeis4.)