A284948 1-limiting word of the morphism 0 -> 10, 1 -> 00

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0

Offset

1

Comments

Consider iterations of the morphism defined by 0 -> 10, 1 -> 00: 0 -> 10 -> 0010 -> 10100010 -> 0010001010100010 -> ... There are two limiting words, one of which has initial term 1 and the other, 0. These are fixed points of the morphism squared: 0-> 0010, 1->1010. [Corrected by Michel Dekking, Jan 06 2019]

The 0-limiting word is 0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,... (A328979). It is the characteristic sequence of those natural numbers whose binary representation ends in an odd numbers of zeros, sequence A036554, but with offset 0 (easy to see from the fact that if the binary representation of N is equal to w, then the binary representations of 4N, 4N+1, 4N+2 and 4N+3 are w00, w01, w10 and w11). - Michel Dekking, Jan 06 2019

Links

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

Maple

f(0):= (0, 0, 1, 0): f(1):= (1, 0, 1, 0):
A:= [0]: # if start at 0 get A328979, if start at 1 get the present sequence
for i from 1 to 8 do A:= map(f, A) od:
A; # N. J. A. Sloane, Nov 05 2019

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {0, 0}}] &, {0}, 7] (* A284948 *)
u = Flatten[Position[s, 0]]  (* A171946  *)
v = Flatten[Position[s, 1]]  (* A171947 *)

Crossrefs

Cf. A036554, A171946, A171947, A328979.

Sequence in context: A267525 A014429 A354036 * A011637 A016229 A015869

Adjacent sequences: A284945 A284946 A284947 * A284949 A284950 A284951

Keyword

nonn,easy

Author

Clark Kimberling, Apr 18 2017

Status

approved