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

0,2

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

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).

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 *)

Prog

(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;

Crossrefs

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

Sequence in context: A250030 A316456 A370060 * A338946 A083716 A231820

Adjacent sequences: A344890 A344891 A344892 * A344894 A344895 A344896

Keyword

nonn,easy

Author

Kevin Ryde, Jun 01 2021

Status

approved