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A189687 Fixed point of the morphism 0->011, 1->01. 6

%I

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

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

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

%N Fixed point of the morphism 0->011, 1->01.

%C From _Danny Rorabaugh_, Mar 14 2015: (Start)

%C Let x(i) and y(i) be the number of 0s and 1s, respectively, after the i-th stage of generating this word, so x(0) = 1, y(0) = 0, x(i+1) = x(i) + y(i), and y(i+1) = 2x(i) + y(i). Equivalently: x(0) = 1, x(1) = 1, x(i+2) = 2x(i+1) + x(i), y(0) = 0, y(1) = 2, and y(i+2) = 2y(i+1) + y(i).

%C The number of 0s after the i-th stage is x(i) = A001333(i).

%C The number of 1s after the i-th stage is y(i) = 2*A000129(i) = A163271(i+1) = A001333(i+1) - A001333(i).

%C Let S(n) = Sum_{j<=n} a(j) be the partial sums of this sequence, so S(x(i)+y(i)) = y(i). Consequently, if the Cesàro sum of a(n) exists, then it is lim_{n->infinity} S(n)/n = lim_{i->infinity} A163271(i+1)/A001333(i+1) = 2 - sqrt(2).

%C (End)

%e 0->011->0110101->01101010110101101->

%t t = Nest[Flatten[# /. {0->{0,1,1}, 1->{0,1}}] &, {0}, 5] (*A189687*)

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

%t Flatten[Position[t, 0]] (* A086377 conjectured *)

%t Flatten[Position[t, 1]] (* A081477 conjectured *)

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

%t Table[s[n], {n, 1, 120}] (*A189688*)

%Y Cf. A189688, A086377, A189688.

%Y Fixed points of similar morphisms: A004641, A005614, A080764, A159684, A171588, A189572.

%K nonn

%O 1

%A _Clark Kimberling_, Apr 25 2011

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Last modified December 9 02:36 EST 2016. Contains 278959 sequences.