



1, 3, 4, 5, 7, 8, 10, 12, 13, 14, 16, 18, 19, 21, 22, 23, 25, 26, 28, 30, 31, 33, 34, 35, 37, 39, 40, 41, 43, 44, 46, 48, 49, 50, 52, 54, 55, 57, 58, 59, 61, 63, 64, 65, 67, 68, 70, 72, 73, 75, 76, 77, 79, 80, 82, 84, 85, 86, 88, 90, 91, 93, 94, 95, 97, 98
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Conjecture: 3n/2  a(n) is in {0, 1/2, 1} for all n >= 1.
From Michel Dekking, Sep 03 2019: (Start)
Proof of the conjecture by Kimberling: more is true. Here follows a proof of the formula below. Let T be the transform T(01)=0, T(1)=0.
Consider the return word structure of A285949 for the word 0:
A285949 = 01 0 0 01 0 01 01 0 0 01 01 ....
[See Justin & Vuillon (2000) for definition of return word.  N. J. A. Sloane, Sep 23 2019]
The two return words are v:=0 and w:=01. Always v = T(1)and w = T(01) in this decomposition of the image T(A010060) of A010060 under the transform. It follows that the return words occur as the ThueMorse word 21121221211... on the alphabet {2,1}. But the lengths of the return words corresponds to the differences between the indices where the 0's occur in A285949, which generate (a(n)).
As the ThueMorse word is a concatenation of 12 and 21 which, considered as integers, both add to 3, it follows that a(2n+1) = 3n+1. Similarly, it follows that a(2n) = 3n  A010060(n).
(End)


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000
Jacques Justin and Laurent Vuillon, Return words in Sturmian and episturmian words, RAIROTheoretical Informatics and Applications 34.5 (2000): 343356.


FORMULA

a(2n) = 3n  A010060(n); a(2n+1) = 3n + 1.  Michel Dekking, Sep 03 2019


EXAMPLE

As a word, A285949 = 0100010010100010100100010..., in which 0 is in positions 1,3,4,5,7,...


MATHEMATICA

s = Nest[Flatten[# /. {0 > {0, 1}, 1 > {1, 0}}] &, {0}, 7] (* ThueMorse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" > "01", "1" > "0"}] (* A285949, word *)
st = ToCharacterCode[w1]  48 (* A285949, sequence *)
Flatten[Position[st, 0]] (* A285950 *)
Flatten[Position[st, 1]] (* A285951 *)


CROSSREFS

Cf. A010060, A003849, A285949, A285951, A285952.
Sequence in context: A270102 A026347 A187482 * A187689 A137292 A089358
Adjacent sequences: A285947 A285948 A285949 * A285951 A285952 A285953


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 02 2017


STATUS

approved



