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A344893 Fixed point of the morphism 1->1321, 2->0021, 3->1300, 0->0000 starting from 1.

%I A344893 #15 Nov 09 2023 08:52:04
%S A344893 1,3,2,1,1,3,0,0,0,0,2,1,1,3,2,1,1,3,2,1,1,3,0,0,0,0,0,0,0,0,0,0,0,0,
%T A344893 0,0,0,0,0,0,0,0,2,1,1,3,2,1,1,3,2,1,1,3,0,0,0,0,2,1,1,3,2,1,1,3,2,1,
%U A344893 1,3,0,0,0,0,2,1,1,3,2,1,1,3,2,1,1,3,0
%N A344893 Fixed point of the morphism 1->1321, 2->0021, 3->1300, 0->0000 starting from 1.
%C A344893 Loxton and van der Poorten give this morphism as a way to identify those n which can be represented in base 4 using only digits -1,0,+1 (A006288): n is a term of A006288 iff a(n) = 1 or 3.
%H A344893 Kevin Ryde, <a href="/A344893/b344893.txt">Table of n, a(n) for n = 0..8192</a>
%H A344893 John Loxton and Alf van der Poorten, <a href="https://doi.org/10.1515/crll.1988.392.57">Arithmetic Properties of Automata: Regular Sequences</a>, Journal für die Reine und Angewandte Mathematik, volume 392, 1988, pages 57-69.  Also <a href="http://web.archive.org/web/20040405090654id_/http://www-centre.mpce.mq.edu.au/alfpapers/a083.pdf">second author's copy</a>.  Section 1 example beta_n = a(n).
%H A344893 <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%H A344893 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F A344893 a(n) = 0 if n in base 4 has a digit pair 12, 13, 20, or 21; otherwise a(n) = 1,3,2,1 according as n == 0,1,2,3 (mod 4).
%t A344893 Nest[Flatten[ReplaceAll[#,{0->{0,0,0,0},1->{1,3,2,1},2->{0,0,2,1},3->{1,3,0,0}}]]&,{1},4] (* _Paolo Xausa_, Nov 09 2023 *)
%o A344893 (PARI) my(table=[9,8,9,0,0,8,6,2,4]); a(n) = my(s=2); if(n, forstep(i=bitor(logint(n,2),1),0,-1, (s=table[s-bittest(n,i)])||break)); s>>1;
%Y A344893 Cf. A006288, A344892, A007090 (base 4).
%K A344893 nonn,easy
%O A344893 0,2
%A A344893 _Kevin Ryde_, Jun 01 2021