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2, 6, 8, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 38, 40, 44, 46, 50, 52, 54, 56, 60, 62, 66, 68, 70, 72, 76, 78, 82, 84, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 114, 116, 120, 122, 126, 128, 130, 132, 136, 138, 142, 144, 148, 150, 154, 156, 158, 160
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OFFSET
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1,1
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COMMENTS
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Conjecture: -2 < n*r - a(n) < 3 for n >= 1, where r = 1 + sqrt(3).
The formula a(n) = 2*A026363(n) follows from the two facts that (a(n)) gives the positions of 1 in A284751, fixed point of the morphism mu given by
mu: 0 -> 01, 1 -> 0001,
and A026363 gives the positions of 1 in the fixed point of the morphism tau given by
tau: 0->11, 1 -> 101.
The non-overlapping words of length 2 occurring in (a(n)) at odd positions are a:=00 and b:=01. Their occurrences can be read from the fixed point of the morphism nu induced by mu, and are given by
nu: a =00 -> 0101 = bb, b =01 -> 010001 = bab.
The fact that tau is obtained under the alphabet change {a,b} -> {0,1} proves that (a(n)) is twice the sequence of occurrences of 1 in the fixed point of tau.
Interestingly, the conjecture: -2 < n*(1+sqrt(3))- a(n) < 3 , implies a conjecture: -1 < n*(1 + sqrt(3))/2 - A026363(n) < 3/2, which is stronger than the conjecture -1 < n*(1 + sqrt(3))/2 - a(n) < 2 given in the comments of A026363.
The best bounds are not simple to obtain, but the fact that (n*(1+sqrt(3))- a(n)) is a bounded sequence follows by a general symbolic discrepancy result, Theorem 1 in Adamscewski's 2004 paper. To apply that theorem, one writes a(n) as a sum of differences 1 and 2, corresponding to the words 1 and 10 in the fixed point 10111101... of tau. The induced morphism is given by 1->21, 2->2111, with eigenvalues of its incidence matrix 1+sqrt(3) and 1-sqrt(3). The boundedness result is then implied by the fact that the second eigenvalue is smaller than 1 in absolute value.
(End)
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LINKS
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FORMULA
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EXAMPLE
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As a word, A284751 = 010001..., in which 1 is in positions 2,6,...
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MATHEMATICA
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s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 0, 1}}] &, {0}, 6] (* A284751 *)
Flatten[Position[s, 0]] (* A284752 *)
Flatten[Position[s, 1]] (* A284753 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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