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

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, 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, 1, 3, 0, 0, 0, 0, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 0

Offset: 0

Author

Kevin Ryde, Jun 01 2021

Keywords

• nonn
• easy

Comments

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.

Links

• Kevin Ryde, Table of n, a(n) for n = 0..8192
• John Loxton and Alf van der Poorten, Arithmetic Properties of Automata: Regular Sequences, Journal für die Reine und Angewandte Mathematik, volume 392, 1988, pages 57-69. Also second author's copy. Section 1 example beta_n = a(n).
• Index entries for 2-automatic sequences.
• Index entries for sequences that are fixed points of mappings

Crossrefs

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

Programs

Mathematica

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 *)

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;

Formula

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).