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

%I #15 Nov 09 2023 08:52:04

%S 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 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 1,3,0,0,0,0,2,1,1,3,2,1,1,3,2,1,1,3,0

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

%C 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 Kevin Ryde, <a href="/A344893/b344893.txt">Table of n, a(n) for n = 0..8192</a>

%H 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 <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%F 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 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 (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 Cf. A006288, A344892, A007090 (base 4).

%K nonn,easy

%O 0,2

%A _Kevin Ryde_, Jun 01 2021