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%I #18 Sep 20 2021 04:36:37
%S 0,1,2,2,1,0,0,2,1,0,2,1,2,1,0,0,1,2,0,1,2,0,2,1,2,1,0,0,1,2,0,2,1,2,
%T 1,0,0,2,1,2,1,0,0,1,2,0,1,2,2,1,0,0,2,1,0,1,2,2,1,0,0,2,1,0,1,2,0,2,
%U 1,2,1,0,0,2,1,2,1,0,0,1,2,0,1,2,2,1
%N Start with 0 and repeatedly substitute 0->012, 1->210, 2->021.
%C This is the fixed point of the morphism 0->012, 1->210, 2->021 starting with 0. Let u be the (nonperiodic) sequence of positions of 0, and likewise, v for 1 and w for 2; then u(n)/n -> 3, v(n)/n -> 3, w(n)/n -> 3.
%C In the following guide to related sequences, column 1 indexes fixed points on {1,2,3}, and columns 2,3,4 match the position sequences of 0, 1, 2. Those sequences therefore comprise a 3-way splitting of the positive integers.
%C Fixed point and morphism Position sequences
%C A287385: 0->012, 1->210, 2->021 A287386 A287387 A287388
%C A287397: 0->012, 1->210, 2->102 A287398 A287399 A287400
%C A287401: 0->012, 1->210, 2->120 A189728 A287403 A287404
%C A287407: 0->012, 1->210, 2->201 A287408 A287409 A287410
%C A287411: 0->012, 1->120, 2->021 A287412 A287413 A287414
%C A287418: 0->012, 1->120, 2->102 A287419 A287420 A287421
%C A053838: 0->012, 1->120, 2->201 A287435 A287436 A287437
%C A287438: 0->012, 1->120, 2->210 A189728 A189670 A287441
%C A287443: 0->012, 1->201, 2->021 A287444 A287445 A287446
%C A287447: 0->012, 1->201, 2->102 A189724 A287449 A287450
%C A287451: 0->012, 1->201, 2->120 A287452 A287453 A287454
%C A287455: 0->012, 1->201, 2->210 A287456 A189666 A287458
%C A287516: 0->012, 1->102, 2->021 A287517 A287518 A189630
%C A287520: 0->012, 1->102, 2->120 A287521 A287522 A189630
%C A287524: 0->012, 1->102, 2->201 A189724 A287526 A287527
%C A287528: 0->012, 1->102, 2->210 A287529 A189670 A189634
%H Clark Kimberling, <a href="/A287385/b287385.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%e First three iterations of the morphism: 012, 012210021, 012210021021210012012021210.
%t s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{2, 1, 0}, 2->{0, 2, 1}}] &, {0}, 9]; (*A287385*)
%t Flatten[Position[s, 0]]; (*A287386*)
%t Flatten[Position[s, 1]]; (*A287387*)
%t Flatten[Position[s, 2]]; (*A287388*)
%Y Cf. A287386, A287387, A287388.
%K nonn,easy
%O 1,3
%A _Clark Kimberling_, May 25 2017
%E Two entries in overview corrected by _Georg Fischer_, Sep 20 2021
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