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%I #12 Mar 30 2012 18:57:24
%S 0,1,1,1,0,1,1,0,1,1,0,0,1,1,1,0,1,1,0,0,1,1,1,0,1,1,0,0,1,0,1,1,1,0,
%T 1,1,0,1,1,0,0,1,1,1,0,1,1,0,0,1,0,1,1,1,0,1,1,0,1,1,0,0,1,1,1,0,1,1,
%U 0,0,1,0,1,1,1,0,0,1,1,1,0,1,1,0,1,1,0,0,1,1,1,0,1,1,0,0,1,1,1,0,1,1,0,0,1,0,1,1,1,0,1,1,0,1,1,0,0,1,1,1
%N Fixed point of the morphism 0->01, 1->110.
%C A189576 is one of many 01-sequences fixed by morphisms. It is helpful to classify a few such sequences:
%C Type 2,2: morphism: 0->01, 1->10, A010060 (Thue-Morse)
%C ..
%C Type 2,3: 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->01, 1->001..A189572..A189573..A080652..A088462
%C 0->01, 1->010..A159684..A003152..A003151..A097508
%C 0->01, 1->011..A096270..A026352..A004956..A005206
%C 0->01, 1->100..A189476..A189477..A189478..A189575
%C 0->01, 1->101..A189479..A007066..A099267..A189480
%C 0->01, 1->110..A189576..A189577..A189578..A189579
%C ..
%C Type 3,2: (rows as for type 2,3)
%C 0->001, 1->01..A188432..A026351..A026352..A060144
%C 0->001, 1->10..A189624..A189625..A189626..A189627
%C 0->010, 1->01..A003849..A000201..A001950..A060144
%C 0->010, 1->10..A189661..A189662..A026356..A189663
%C 0->011, 1->01..A189687..A086377..A081477..A189688
%C 0->011, 1->10..A189702..A189703..A189704..A189705
%C ..
%C Type 3,3: (See A189628 for a list and discussion.)
%e 0->01->01110->0111011011001->
%t t = Nest[Flatten[# /. {0->{0,1}, 1->{1,1,0}}] &, {0}, 6] (*A189576*)
%t f[n_] := t[[n]]
%t Flatten[Position[t, 0]] (*A189577*)
%t Flatten[Position[t, 1]] (*A189578*)
%t s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
%t Table[s[n], {n, 1, 120}] (*A189579*)
%Y Cf. A189577, A189578, A189579, A189628.
%K nonn
%O 1
%A _Clark Kimberling_, Apr 23 2011
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