

A287112


1limiting word of the morphism 0>10, 1>20, 2>0.


5



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

1,3


COMMENTS

Starting with 0, the first 4 iterations of the morphism yield words shown here:
1st: 10
2nd: 2010
3rd: 0102010
4th: 1020100102010
The 1limiting word is the limit of the words for which the number of iterations is congruent to 1 mod 3.
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}, v(n)  v(n1) is in {2,3,4}, and w(n)  w(n1) is in {4,6,7}.
From Michel Dekking, Mar 29 2019: (Start)
This sequence is one of the three fixed points of the morphism alpha^3, where alpha is the defining morphism
0>10, 1>20, 2>0.
The other two fixed points are A286998 and A287174.
We have alpha = rho(tau), where tau is the Tribonacci morphism in A080843
0>01, 1>02, 2>0,
and rho is the rotation operator.
An eigenvector computation of the incidence matrix of the morphism gives that 0,1, and 2 have frequencies 1/t, 1/t^2 and 1/t^3, where t is the tribonacci constant A058265.
Apparently (u(n)) := A287113. Thus U, the limit of u(n)/n, equals 1/(1/t), the tribonacci constant t. Also, V = A276800, and W = A276801.
(End)


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000


EXAMPLE

1st iterate: 10
4th iterate: 1020100102010
7th iterate: 102010010201020100102010102010010201001020101020100102010201001020101020100102010.


MATHEMATICA

s = Nest[Flatten[# /. {0 > {1, 0}, 1 > {2, 0}, 2 > 0}] &, {0}, 10] (* A287112 *)
Flatten[Position[s, 0]] (* A287113 *)
Flatten[Position[s, 1]] (* A287114 *)
Flatten[Position[s, 2]] (* A287115 *)


CROSSREFS

Cf. A287113, A287114, A287115, A286998, A287174.
Sequence in context: A178781 A287174 A080843 * A296238 A221314 A087371
Adjacent sequences: A287109 A287110 A287111 * A287113 A287114 A287115


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 22 2017


STATUS

approved



