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A285950 Positions of 0's in A285949; complement of A285951. 6
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 Thue-Morse 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 Thue-Morse 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, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356.

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]  (* Thue-Morse, 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

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Last modified July 5 23:36 EDT 2020. Contains 335475 sequences. (Running on oeis4.)