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A189687 Fixed point of the morphism 0->011, 1->01. 7
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1

COMMENTS

From Danny Rorabaugh, Mar 14 2015: (Start)

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).

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

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

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).

(End)

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..120.

Wieb Bosma, Michel Dekking, Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), arXiv:1710.01498 [math.NT], 2017.

Wieb Bosma, Michel Dekking, Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A4.

EXAMPLE

0->011->0110101->01101010110101101->

MATHEMATICA

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

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

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

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

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

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

CROSSREFS

Cf. A189688, A086377, A189688.

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

Sequence in context: A192082 A296066 A073070 * A284653 A099104 A066829

Adjacent sequences:  A189684 A189685 A189686 * A189688 A189689 A189690

KEYWORD

nonn

AUTHOR

Clark Kimberling, Apr 25 2011

STATUS

approved

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Last modified March 23 12:43 EDT 2019. Contains 321430 sequences. (Running on oeis4.)