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Fixed point of the morphism 0->010, 1->001.
8

%I #53 Oct 16 2019 12:14:28

%S 0,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,0,0,1,0,

%T 1,0,0,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,0,1,

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

%N Fixed point of the morphism 0->010, 1->001.

%C a(n) is the parity of the number of ternary 1-digits below the lowest 0-digit in the ternary expansion of n-1. All fixed-width morphisms have a similar digital interpretation. - _Kevin Ryde_, Apr 26 2017

%H Joerg Arndt, <a href="/A189664/b189664.txt">Table of n, a(n) for n = 1..2187</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%F a(3k-2)=0, a(3k-1)=1-a(k), a(3k)=a(k) for k>=1, a(0)=0.

%F 2*a(n) = A284794(n) for n >= 1. - _Clark Kimberling_, Apr 14 2017

%F This formula cannot be correct, it belongs to A189665. - _Michel Dekking_, Oct 15 2019

%F a(n) = A284777(n) - 2n. - _Clark Kimberling_, Apr 15 2017

%e Iterating the morphism starting with 0:

%e 0: (#=1)

%e 0

%e 1: (#=3)

%e 010

%e 2: (#=9)

%e 010001010

%e 3: (#=27)

%e 010001010010010001010001010

%e 4: (#=81)

%e 010001010010010001010001010010001010010001010010010001010001010010010001010001010

%e etc.

%t t = Nest[Flatten[# /. {0->{0,1,0}, 1->{0,0,1}}] &, {0}, 5] (* A189664 *)

%t f[n_] := t[[n]]

%t Flatten[Position[t, 0]] (* A189665 *)

%t Flatten[Position[t, 1]] (* A189666 *)

%t s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;

%t Table[s[n], {n, 1, 120}] (* A189667 *)

%t Nest[Flatten[# /. a_Integer -> {0, Abs[a - 1], a}] &, {0}, 5] (* _Robert G. Wilson v_, Jul 16 2012 *)

%o (PARI) a(n) = n--; my(ret=0); while(n%3, if(n%3==1,ret=!ret); n\=3); ret; /* _Kevin Ryde_, Jul 23 2019 */

%Y Cf. A189628, A189665, A189666, A189667.

%K nonn

%O 1

%A _Clark Kimberling_, Apr 25 2011