

A288711


1limiting 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
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OFFSET

1


COMMENTS

Iterates of the mapping, starting with 00:
00
1000
001000
1000001000
0010001000001000
10000010000010001000001000
001000100000100010000010000010001000001000
The 1limiting word is the limit of the nth iterates for n == 1 mod 2.
The number of letters (0s and 1s) in the nth 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 nth 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 nth 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



