A214212 Number of right special factors of length n in the Thue-Morse sequence A010060.

1, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Author

N. J. A. Sloane, Jul 10 2012

Keywords

• nonn

References

• Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
• A. de Luca and S. Varricchio, Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups, Theoret. Comput. Sci. 63 (1989), 333-348.

Crossrefs

Cf. A010060, A005942, A212214.

Programs

Maple

ph:=proc(n) option remember;
if n=2 then 2 elif n<=3 then n+1 else if n mod 2 = 0 then ph(n/2) else ph((n+1)/2); fi;
fi; end;

Mathematica

ph[n_] := ph[n] = If[n == 2, 2, If[n <= 3, n+1, If[Mod[n, 2] == 0, ph[n/2], ph[(n+1)/2]]]];
ph /@ Range[0, 120] (* Jean-François Alcover, Jun 18 2020, after Maple *)

Formula

A005942(n+1) - A005942(n). - Michel Dekking, Sep 28 2020

Extensions

Name clarified by Michel Dekking, Sep 28 2020