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A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.

%I A287556 #5 May 31 2017 16:14:16
%S A287556 0,1,3,2,1,3,2,0,2,0,1,3,3,2,0,1,1,3,2,0,2,0,1,3,3,2,0,1,0,1,3,2,3,2,
%T A287556 0,1,0,1,3,2,1,3,2,0,2,0,1,3,2,0,1,3,3,2,0,1,0,1,3,2,1,3,2,0,1,3,2,0,
%U A287556 2,0,1,3,3,2,0,1,0,1,3,2,3,2,0,1,0,1
%N A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.
%C A287556 This is the fixed point of the morphism 0->0132, 1->1320, 2->3201, 3->2013 starting with 0.  Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4,  w(n)/n -> 4.   Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1.
%C A287556 In the following guide to related sequences, column 1 indexes fixed points on {0,1,2,3}, and column 2 indicates position sequences of 0, 1, 2, 3.  Those sequences therefore comprise a 4-way splitting of the positive integers.
%C A287556 Fixed points of morphisms:                        Position sequences:
%C A287556 A053839: 0->0123, 1->1230, 2->2301, 3->3012       A287552-A287555
%C A287556 A287556: 0->0132, 1->1320, 2->3201, 3->2013       A287557-A287560
%C A287556 A287561: 0->0213, 1->2130, 2->1302, 3->3021       A287562-A287565
%C A287556 A287566: 0->0231, 1->2310, 2->3102, 3->1023       A287567-A287570
%C A287556 A287571: 0->0312, 1->3120, 2->1203, 3->2031       A287572-A287575
%C A287556 A287576: 0->0321, 1->3210, 2->2103, 3->1032       A287577-A287580
%H A287556 Clark Kimberling, <a href="/A287556/b287556.txt">Table of n, a(n) for n = 1..20000</a>
%H A287556 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%e A287556 First three iterations of the morphism:
%e A287556 0132
%e A287556 0132132020133201
%e A287556 0132132020133201132020133201013232010132132020132013320101321320
%t A287556 s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9]; (* A287556 *)
%t A287556 Flatten[Position[s, 0]]; (* A287557 *)
%t A287556 Flatten[Position[s, 1]]; (* A287558 *)
%t A287556 Flatten[Position[s, 2]]; (* A287559 *)
%t A287556 Flatten[Position[s, 3]]; (* A287560 *)
%Y A287556 Cf. A287557, A287558, A287559, A287560.
%K A287556 nonn,easy
%O A287556 1,3
%A A287556 _Clark Kimberling_, May 31 2017