



2, 3, 5, 9, 11, 12, 14, 17, 19, 20, 23, 27, 29, 30, 33, 36, 38, 39, 41, 45, 47, 48, 50, 53, 55, 56, 59, 62, 64, 65, 67, 71, 73, 74, 77, 81, 83, 84, 86, 89, 91, 92, 95, 99, 101, 102, 105, 108, 110, 111, 113, 117, 119, 120, 123, 127, 129, 130, 132, 135, 137
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OFFSET

1,1


COMMENTS

a(n)  a(n1) is in {1,2,3,4} for n >= 2.
Conjecture: a(n)/n > 9/4.
From Michel Dekking, Aug 26 2017: (Start)
Proof of the conjecture. Let x = A010060 be the Thue Morse sequence, and let y = A284622 be the [0011>0]transform of x. Let a = A284626 be the positions of 1 in y. There are 3 steps in the proof.
Step 1. It is easily verified that a(n)/n > 9/4 if and only if f(1,a) = 4/9, where in general f(w,z) denotes the frequency of a word w in the infinite sequence z, if it exists.
Step 2. One has f(0011,x) = 1/12. It is wellknown that the frequencies of words in any fixed point of a primitive morphism exist. This is usually proved by PerronFrobenius theory. For a quick proof see the paper "On the ThueMorse measure".
Step 3. Let k(n) be the number of 1's in x(1)...x(n), and m(n) the number of 0011's in x(1)...x(n). Then the number of 1's in y(1)...y(n3m(n)) is equal to k(n)2m(n). But we know by Step 2 that m(n)/n > 1/12, and obviously k(n)/n > 1/2. So f(1,y) is equal to ((1/2  2/12)/(1  3/12) = 4/9. (End)


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000
Michel Dekking,On the ThueMorse measure, Acta Universitatis Carolinae. Mathematica et Physica 033.2 (1992), 3540.


EXAMPLE

As a word, A284622 = 011010001011010010..., in which 1 is in positions 2,3,5,9,11,...


MATHEMATICA

s = Nest[Flatten[# /. {0 > {0, 1}, 1 > {1, 0}}] &, {0}, 9] (* A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0011" > "0"}]
st = ToCharacterCode[w1]  48 (* A284622 *)
Flatten[Position[st, 0]] (* A284623 *)
Flatten[Position[st, 1]] (* A284626 *)


CROSSREFS

Cf. A010060, A284622, A284623.
Sequence in context: A058108 A174512 A056144 * A284847 A186776 A290475
Adjacent sequences: A284623 A284624 A284625 * A284627 A284628 A284629


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 10 2017


EXTENSIONS

Name corrected by Michel Dekking, Aug 26 2017


STATUS

approved



