A225183 Sequence defined by a mix-DFAO: Example 8 of Endrullis et al. (2013), written over the alphabet {0,1}.

0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0

Offset

0

Comments

From Dimitri Hendriks, Jun 29 2013: (Start)

The sequence is obtained as the solution of the variable S in the following set of recursive equations:

S = 0 : X

X = 1 : zip_2(X,Y)

Y = 0 : 1 : zip_3(Z,X,Y)

Z = 0 : zip_2(Y,X)

Here, for sequences a = a_0, a_1, a_2, .... and b = b_0, b_1, b_2, ..., the term zip_2(A,B) denotes their perfect shuffle, i.e., zip_2(a,b) = a_0, b_0, a_1, b_1, a_2, b_2, .... Likewise zip_3(a,b,c) = a_0, b_0, c_0, a_1, b_1, c_1, ... . The colon denotes concatenation of a letter to a sequence, e.g., 0 : a denotes the sequence 0, a_0, a_1, a_2, ... .

The sequence is produced by rewriting the equations from left to right, ad infinitum:

S = 0:X = 0:1:zip_2(X,Y) = 0:1:zip_2(1:zip_2(X,Y),Y) = 0:1:1:zip_2(Y,zip_2(X,Y)) = 0:1:1:zip_2(0:1:zip_3(Z,X,Y),zip_2(X,Y)) = 0:1:1:0:zip_2(zip_2(X,Y),1:zip_3(Z,X,Y)) = 0:1:1:0:zip_2(zip_2(1:zip_2(X,Y),Y),1:zip_3(Z,X,Y)) = 0:1:1:0:zip_2(1:zip_2(zip_2(Y,X),Y),1:zip_3(Z,X,Y)) = 0:1:1:0:1:zip_2(1:zip_3(Z,X,Y),zip_2(zip_2(Y,X),Y)) = 0:1:1:0:1:1:zip_2(zip_2(zip_2(Y,X),Y),zip_3(Z,X,Y)) = ...

Equivalently, the sequence is produced by a so-called mix-DFAO (deterministic finite state automaton with output); see further the LATA 2013 paper by Endrullis, Grabmayer, Hendriks, Example 8, page 268.

(End)

Links

Table of n, a(n) for n=0..249.

J. Endrullis, C. Grabmayer, D. Hendriks, Mix-Automatic Sequences, In: Proceedings Conference on Language and Automata Theory and Applications (LATA 2013), Springer, 2013, pages 262-274.

Prog

(Haskell)
zipl :: [[x]] -> [x]
zipl (s:ss) = head s : zipl (ss ++ [(tail s)])
a225183 = s where
  s = 0 : x
  x = 1 : zipl [x, y]
  y = 0 : 1 : zipl [z, x, y]
  z = 0 : zipl [y, x]
-- Dimitri Hendriks, Jun 29 2013

Crossrefs

Cf. A225182.

Sequence in context: A322829 A286400 A288478 * A082410 A189479 A260394

Adjacent sequences: A225180 A225181 A225182 * A225184 A225185 A225186

Keyword

nonn

Author

N. J. A. Sloane, May 03 2013

Extensions

Added more terms, Dimitri Hendriks, Jun 29 2013

Status

approved