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Fixed point of the morphism 0->001, 1->010.
27

%I #22 Sep 09 2022 05:27:12

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

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

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

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

%C A189628 is one of many 01-sequences fixed by morphisms. An extension of the list begun at A189576 is continued here with sequences of type 3,3.

%C Each row shows a morphism, followed by four sequences:

%C (1) the fixed sequence a [starting from a(0)=0],

%C (2) positions of 0 in a,

%C (3) positions of 1 in a,

%C (4) partial sums of a.

%C Some lower-numbered entries are conjectural.

%C 0->001, 1->010..A189628..A189629..A189630..A189631

%C 0->001, 1->011..A116178..A189636..A189637..A189638

%C 0->001, 1->100..A189632..A189633..A189634..A189635

%C 0->001, 1->101..A189640..A026138..A026323..A189641

%C 0->001, 1->110..A064990..A189658..A189659..A189660

%C 0->010, 1->001..A189664..A189665..A189666..A189667

%C 0->010, 1->011..A080846..A026225..A026179..A189672

%C 0->010, 1->100..A189668..A189669..A189670..A189671

%C 0->010, 1->101..A000035..A005408..A005843..A004526

%C 0->010, 1->110..A189673..A026227..A026138..A189674

%C 0->011, 1->001..A189706..A189707..A189708..A189709

%C 0->011, 1->010..A156595..A189715..A189716..A189717

%C 0->011, 1->100..A189718..A189719..A189720..A189721

%C 0->011, 1->101..A189723..A189724..A189725..A189726

%C 0->011, 1->110..A189727..A189728..A189729..A189730

%C Each of the 15 sequences in column 3 (i.e., A189628 to A189727) is generated by a 3-part recurrence, as in the Formula section. Two other sequences generated by such a recurrence are A189816 and A189820.

%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

%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)=a(k), a(3k)=1-a(k) for k>=1. - corrected by _Michel Dekking_, Sep 09 2022

%e 0->001->001001010->->

%t t = Nest[Flatten[# /. {0->{0,0,1}, 1->{0,1,0}}] &, {0}, 5] (*A189628*)

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

%t Flatten[Position[t, 0]] (*A189629*)

%t Flatten[Position[t, 1]] (*A189630*)

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

%t Table[s[n], {n, 1, 120}] (*A189631*)

%Y Cf. A189629, A189630, A189631, A189576.

%K nonn

%O 1

%A _Clark Kimberling_, Apr 24 2011