

A277735


Unique fixed point of the morphism 0 > 01, 1 > 20, 2 > 0.


5



0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1
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OFFSET

1,3


COMMENTS

From Clark Kimberling, May 21 2017: (Start)
Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 1.8392867552141611325518525646532866...,
V = U^2 = 3.3829757679062374941227085364...,
W = U^3 = 6.2222625231203986266745611011....
If n >=2, then u(n)  u(n1) is in {1,2,3}, v(n)  v(n1) is in {2,4,5}, and w(n)  w(n1) is in {4,7,9}. (u = A277736, v = A277737, w = A277738). (End)
Although I believe the assertions in Kimberling's comment above to be correct, these results are quite tricky to prove, and unless a formal proof is supplied at present these assertions must be regarded as conjectures.  N. J. A. Sloane, Aug 20 2018
From Michel Dekking, Oct 03 2019: (Start)
Here is a proof of Clark Kimberling's conjectures (and more).
The incidence matrix of the defining morphism
sigma: 0 > 01, 1 > 20, 2 > 0
is the same as the incidence matrix of the tribonacci morphism
0 > 01, 1 > 02, 2 > 0
(see A080843 and/or A092782).
This implies that the frequencies f0, f1 and f2 of the letters 0,1, and 2 in (a(n)) are the same as the corresponding frequencies in the tribonacci word, which are 1/t, 1/t^2 and 1/t^3 (see, e.g., A092782).
Since U = 1/f0, V = 1/f1, and W = 1/f2, we conclude that
U = t = A058265, V = t^2 = A276800 and W = t^3 = A276801.
The statements on the difference sequences u, v, and w of the positions of 0,1, and 2 are easily verified by applying sigma to the return words of these three letters.
Here the return words of an arbitrary word w in a sequence x are all the words occurring in x with prefix w that do not have other occurrences of w in them.
The return words of 0 are 0, 01, and 012, which indeed have length 1, 2
and 3. Since
sigma(0) = 01, sigma(1) = 0120, and sigma(012) = 01200,
one sees that u is the unique fixed point of the morphism
1 > 2, 2> 31, 3 >311.
With a little more work, passing to sigma^2, and rotating, one can show that v is the unique fixed point of the morphism
2>52, 4>5224, 5>52244 .
Similarly, w is the unique fixed point of the morphism
4>94, 7>9447, 9>94477.
Interestingly, the three morphisms having u,v, and w as fixed point are essentially the same morphism (were we replaced sigma by sigma^2) with standard form
1>12, 2>1223, 3>12233.
(End)
The kind of phenomenon observed at the end of the previous comment holds in a very strong way for the tribonacci word. See Theorem 5.1. in the paper by Huang and Wen.  Michel Dekking, Oct 04 2019


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Y.K. Huang, Z.Y. Wen, Kernel words and gap sequence of the Tribonacci sequence, Acta Mathematica Scientia (Series B). 36.1 (2016) 173194.
Victor F. Sirvent, Semigroups and the selfsimilar structure of the flipped tribonacci substitution, Applied Math. Letters, 12 (1999), 2529.
Victor F. Sirvent, The common dynamics of the Tribonacci substitutions, Bulletin of the Belgian Mathematical SocietySimon Stevin 7.4 (2000): 571582.
Index entries for sequences that are fixed points of mappings


MAPLE

with(ListTools);
T:=proc(S) Flatten(subs( {0=[0, 1], 1=[2, 0], 2=[0]}, S)); end;
S:=[0];
for n from 1 to 10 do S:=T(S); od:
S;


MATHEMATICA

s = Nest[Flatten[# /. {0 > {0, 1}, 1 > {2, 0}, 2 > 0}] &, {0}, 10] (* A277735 *)
Flatten[Position[s, 0]] (* A277736 *)
Flatten[Position[s, 1]] (* A277737 *)
Flatten[Position[s, 2]] (* A277738 *)
(* Clark Kimberling, May 21 2017 *)


CROSSREFS

Cf. A277736, A277737, A277738.
Equals A100619(n)1.
Sequence in context: A321013 A284258 A322389 * A248911 A116681 A131371
Adjacent sequences: A277732 A277733 A277734 * A277736 A277737 A277738


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Nov 07 2016


EXTENSIONS

Name clarified by Michel Dekking, Oct 03 2019


STATUS

approved



