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 A189687 Fixed point of the morphism 0->011, 1->01. 7

%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)

%C The Cesàro sum of (a(n)) DOES exist. It is well known that the frequencies of letters exist in sequences generated by (primitive) morphisms. The frequencies are given by the normalized right eigenvector (belonging to the Perron-Frobenius eigenvalue) of the incidence matrix of the morphism. - _Michel Dekking_, Feb 02 2017

%D Martine Queffélec, Substitution dynamical systems—spectral analysis, 2nd ed., Lecture Notes in Mathematics, vol. 1294, Springer-Verlag, Berlin, 2010.

%H Wieb Bosma, Michel Dekking, Wolfgang Steiner, <a href="https://arxiv.org/abs/1710.01498">A remarkable sequence related to Pi and sqrt(2)</a>, arXiv:1710.01498 [math.NT], 2017.

%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,changed

%O 1

%A _Clark Kimberling_, Apr 25 2011

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