OFFSET
1
COMMENTS
Old name was: 1-limiting word of reversed iterates of the mapping 0->01, 1->0, starting with 0.
The first seven iterates of the mapping 0->01, 1->0, starting with 0, are:
0
01
010
01001
01001010
0100101001001
010010100100101001010
with limit the infinite Fibonacci word A003849.
The corresponding reversed iterates are as follows:
0
10
010
10010
01010010
1001001010010
010100101001001010010
The 1-limiting word is the limit of the n-th iterates for n == 1 mod 2. (The 0-limiting word is A189661.)
There is a simple way to construct this. Let sigma be the time-reversed Fibonacci morphism, i.e., the morphism sigma: 0->10, 1->0. Then sigma^2 is given by 0->010, 1->10, and sigma^2 has two fixed points. The fixed point starting with 1 is the present sequence. - Michel Dekking, Oct 18 2018
Sturmian word with slope (3 - sqrt(5))/2 and intercept sqrt(5) - 2; see Formula section. - Jeffrey Shallit, Sep 05 2024
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = floor(alpha + (n+1)*beta) - floor(alpha + n*beta), where alpha = sqrt(5) - 2 and beta = (3 - sqrt(5))/2. - Jeffrey Shallit, Sep 05 2024
EXAMPLE
The first four n-th reversed iterates for n == 1 mod 2 are these:
10
10010
1001001010010
1001001010010010100101001001010010
MATHEMATICA
z = 13; (*number of iterates*)
s = {0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"0" -> "01", "1" -> "0"}];
r[n_] := StringReverse[w[n]];
TableForm[Table[r[n], {n, 0, 8}]]
rw = ToCharacterCode[r[z]] - 48 (* A289034 *)
p0 = Flatten[Position[rw, 0]]; (* A099267 *)
p1 = Flatten[Position[rw, 1]]; (* A007066 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 10 2017
EXTENSIONS
Name changed by Michel Dekking, Oct 20 2018
STATUS
approved