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A286686
Positions of 0 in A286685; complement of A286687.
4
1, 3, 6, 7, 9, 12, 13, 15, 17, 20, 21, 23, 26, 27, 29, 31, 34, 35, 37, 40, 41, 43, 46, 47, 49, 51, 54, 55, 57, 60, 61, 63, 65, 68, 69, 71, 74, 75, 77, 80, 81, 83, 85, 88, 89, 91, 94, 95, 97, 99, 102, 103, 105, 108, 109, 111, 113, 116, 117, 119, 122, 123, 125
OFFSET
1,2
COMMENTS
a(n) - a(n-1) is in {1,2,3} for n>=2, and a(n)/n -> 2. These are also the positions of 1 in the {0->10, 1->01}-transform of the Pell word, A171588.
From Michel Dekking, Sep 19 2019: (Start)
Here is a precise description of the sequence of first differences.
Let tau be the map tau: 0->01, 1->10. By definition A286685 equals tau(b), where b is the Pell word. The words of length 2 occurring in b are 00, 01 and 10. These are mapped by tau to
tau(00) = 0101, tau(01) = 0110, tau(10) = 1001.
Each of these three four letter words contains exactly 2 0's, occurring among the first two letters and among the last two letters. It follows from this that the overlapping words of length 2 in the Pell word b induce distances between 0's in tau(b) of 2 for 00, of 3 for 01, and of 1 for 10. But then the difference sequence (a(n+1) - a(n)) = 2, 3, 1, 2, 3, 1, 2, ... is equal to the 1->3, 2->1, 3->2 permuted version of the 3-symbol Pell word A294180. (End)
LINKS
EXAMPLE
As a word, A286685 = 01011001011001010110010110..., in which 0 is in positions 1,3,6,7,9,12,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0}}] &, {0}, 6] (* A171588 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" -> "01", "1"->"10"}]
st = ToCharacterCode[w1] - 48 ; (* A286685 *)
p0 = Flatten[Position[st, 0]]; (* A286686 *)
p1 = Flatten[Position[st, 1]]; (* A286687 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 13 2017
STATUS
approved