login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A285962 Positions of 1 in A285960; complement of A285961. 3
1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 88 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Conjecture: a(n)/n -> 4/3.
From Michel Dekking, Apr 19 2022: (Start)
Kimberling's conjecture is equivalent to the property that the frequency of 1's in A285960 is equal to 3/4. This follows directly from the corresponding result in A285961.
But more is true. The first difference sequence (d(n)) = 1,1,2,1,2,1,1,1,2,1,1,2,1,... of (a(n)) is a morphic sequence.
From the representation of A285960 by the decoration A-> 11, B-> 10, C-> 110, D-> 1, we see that the differences between occurrences of 1's are also given by a decoration:
A->11, B->2, C->12, D->1.
This time one finds that (d(n)) can be obtained as a letter to letter image of a morphic sequence fixed point of a morphism mu on {a,b,c,d,e} given by
mu: a->ab, b->c, c->ade, d->ce, e->ad,
with the letter-to-letter map
lambda: a->1, b->1, c->2, d->2, e->1.
We have (d(n)) = lambda(z), where z is the fixed point z = abcade... of mu.
(End)
LINKS
EXAMPLE
As a word, A285960 = 111011011110111011..., in which 1 is in positions 1,2,3,5,6,8,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 8] (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"01" -> "1"}]
st = ToCharacterCode[w1] - 48 (* A285960 *)
Flatten[Position[st, 0]] (* A285961 *)
Flatten[Position[st, 1]] (* A285962 *)
CROSSREFS
Sequence in context: A039104 A184861 A183572 * A047450 A039073 A026359
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 05 2017
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)