OFFSET
1
COMMENTS
This coincides with the Pell word A171588, the fixed point of the morphism 0->001, 1->0. This was conjectured by R. J. Mathar, Jul 07 2017 and proved by Dekking and Keane in 2022. As observed by Michel Dekking, Mar 09 2022, this also proves the two conjectures about the positions of 0's and 1's stated in the MATHEMATICA section. - N. J. A. Sloane, Mar 09 2022
Conjecture: the number of letters (0's and 1's) in the n-th iterate of the mapping is given by A289004.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
Michel Dekking and Mike Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], 2022.
EXAMPLE
The first seven iterates of the mapping:
00
0010
0010010
00100100010
001001000100100010
0010010001001000100100100010010
0010010001001000100100100010010001001001000100100010
MATHEMATICA
z = 10; (* number of iterates *)
s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "001", "10" -> "010"}]
TableForm[Table[w[n], {n, 0, 10}]]
st = ToCharacterCode[w[z]] - 48 (* A289001 *)
Flatten[Position[st, 0]] (* A001951 conjectured *)
Flatten[Position[st, 1]] (* A001952 conjectured *)
Table[StringLength[w[n]], {n, 0, 20}] (* A289004 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 25 2017
STATUS
approved