A036581 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

This sequence and A108694 are squarefree (they do not contain any substring XX). - Bill Gosper, Jul 22 2005

Trajectory of 1 under the morphism 0 -> 021, 1 -> 2 & 2 -> 01. - Robert G. Wilson v, Apr 06 2008

I believe that this is the sequence Cummings refers to as the Morse-Hedlund sequence. It can be constructed by starting with the Thue-Morse binary sequence A010060, 0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,..., reading successive pairs of digits: 01, 11, 10, 01, 10, 00, 01, 11, 10, 00, 01, ..., and mapping 01 to 0, 10 to 1, and both 00 and 11 to 2, getting 0,2,1,0,1,2,0,2,1,... - N. J. A. Sloane, Oct 17 2012

References

• L. J. Cummings, On the construction of Thue sequences, Proc. 9th S-E Conf. Combinatorics, Graph Theory and Computing, pp. 235-242. - From N. J. A. Sloane, Oct 17 2012
• M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.

Links

• R. J. Mathar, Table of n, a(n) for n = 0..10000
• A. Cobham, Uniform Tag Sequences, Mathematical Systems Theory, 6 (1972), 164-192.
• Marston Morse and Gustav A. Hedlund, Unending chess, symbolic dynamics and a problem in semigroups, Duke Math. J., Volume 11, Number 1 (1944), 1-7. - From _N. J. A. Sloane_, Oct 17 2012
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A036583, A108694.

Programs

Haskell

a036581 n = a036581_list !! n
a036581_list = zipWith (\u v -> if u /= v then 2 * u + v - 1 else 2)
                       a010060_list $ tail a010060_list
-- According to N. J. A. Sloane's comment from Oct 17 2012
-- Reinhard Zumkeller, Nov 30 2012

Maple

A036581 := proc(n)
    A010060(n+1)-A010060(n)-1 ;
    % mod 3 ;
end proc: # R. J. Mathar, Oct 17 2012

Mathematica

Nest[ # /. {0 -> {0, 2, 1}, 1 -> {2}, 2 -> {0, 1}} &, {0}, 7] // Flatten (* Robert G. Wilson v, Apr 06 2008 *)
a010060[n_]:=Mod[DigitCount[n, 2, 1], 2]; Table[Mod[a010060[n + 1] - a010060[n] - 1, 3], {n, 0, 100}] (* Indranil Ghosh, Apr 25 2017 *)
SubstitutionSystem[{0->{0,2,1},1->{2},2->{0,1}},{0},{7}][[1]] (* Harvey P. Dale, Dec 26 2021 *)

Python

def a010060(n): return bin(n)[2:].count("1")%2
def a(n): return (a010060(n + 1) - a010060(n) - 1)%3 # Indranil Ghosh, Apr 25 2017

Formula

a(n) = A010060(n+1) - A010060(n) - 1 mod 3. - Bill Gosper, Jul 22 2005

Extensions

More terms from Robert G. Wilson v, Apr 06 2008