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1-limiting word of the morphism 0 -> 10, 1 -> 00
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%I #25 Nov 05 2019 08:39:35

%S 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,

%T 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,

%U 0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0

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

%C 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]

%C 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

%H Clark Kimberling, <a href="/A284948/b284948.txt">Table of n, a(n) for n = 1..10000</a>

%p f(0):= (0,0,1,0): f(1):= (1,0,1,0):

%p A:= [0]: # if start at 0 get A328979, if start at 1 get the present sequence

%p for i from 1 to 8 do A:= map(f, A) od:

%p A; # _N. J. A. Sloane_, Nov 05 2019

%t s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {0, 0}}] &, {0}, 7] (* A284948 *)

%t u = Flatten[Position[s, 0]] (* A171946 *)

%t v = Flatten[Position[s, 1]] (* A171947 *)

%Y Cf. A036554, A171946, A171947, A328979.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Apr 18 2017