

A287104


Start with 0 and repeatedly substitute 0>10, 1>12, 2>0.


5



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

1,2


COMMENTS

The fixed point of the morphism 0>10, 1>12, 2>0. 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. It appears that 1/U + 1/V + 1/W = 1, where
U = 3.079595623491438786010417...,
V = 2.324717957244746025960908...,
W = U + 1 = 4.079595623491438786010417....
From Michel Dekking, Sep 15 2019: (Start)
The incidence matrix of the morphism sigma: 0>10, 1>12, 2>0 has characteristic polynomial chi(u) = u^32u^2+u1. The real root of chi is lambda := Q/6 + 2/3*1/Q + 2/3, where
Q = ( 100 + 12*sqrt(69) )^1/3.
An eigenvector of lambda is (1, lambda^2lambda, lambda1).
The PerronFrobenius Theorem gives that the asymptotic frequencies f0, f1 and f2 of the letters 0, 1, and 2 are
f0 = 1/lambda^2,
f1 = (lambda^2  lambda +1)/lambda^3,
f2 = (lambda  1)/lambda^2.
Algebraic expressions for the constants U,V and W are then given by
U = 1/f0, V = 1/f1, W = 1/f2.
In particular, this shows that W = U + 1.
(End)
Conjecture: if n >=2, then u(n)  u(n1) is in {2,3,4}, v(n)  v(n1) is in {2,3}, and w(n)  w(n1) is in {3,4,5}.
See A287105, A287106, and A287107 for proofs of these conjectures, with explicit expressions for u, v, and w.  Michel Dekking, Sep 15 2019


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000
Index entries for sequences that are fixed points of mappings


MATHEMATICA

s = Nest[Flatten[# /. {0 > {1, 0}, 1 > {1, 2}, 2 > 0}] &, {0}, 10] (* A287104 *)
Flatten[Position[s, 0]] (* A287105 *)
Flatten[Position[s, 1]] (* A287106 *)
Flatten[Position[s, 2]] (* A287107 *)


CROSSREFS

Cf. A287105, A287106, A287107.
Sequence in context: A025884 A257024 A124433 * A190483 A090239 A165276
Adjacent sequences: A287101 A287102 A287103 * A287105 A287106 A287107


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 21 2017


STATUS

approved



