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%I #9 Feb 27 2021 21:09:22
%S 2,6,8,10,12,16,18,22,24,28,30,34,36,38,40,44,46,50,52,54,56,60,62,66,
%T 68,70,72,76,78,82,84,86,88,92,94,98,100,104,106,110,112,114,116,120,
%U 122,126,128,130,132,136,138,142,144,148,150,154,156,158,160
%N Positions of 1 in A284751; complement of A284752.
%C Conjecture: -2 < n*r - a(n) < 3 for n >= 1, where r = 1 + sqrt(3).
%C From _Michel Dekking_, Feb 27 2021: (Start)
%C 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
%C mu: 0 -> 01, 1 -> 0001,
%C and A026363 gives the positions of 1 in the fixed point of the morphism tau given by
%C tau: 0->11, 1 -> 101.
%C 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
%C nu: a =00 -> 0101 = bb, b =01 -> 010001 = bab.
%C 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.
%C 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.
%C 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.
%C (End)
%H Clark Kimberling, <a href="/A284753/b284753.txt">Table of n, a(n) for n = 1..10000</a>
%H Boris Adamczewski, <a href="https://eudml.org/doc/116172">Symbolic discrepancy and self-similar dynamics</a>, Annales de l'Institut Fourier 54 (2004), 2201-2234.
%F a(n) = 2*A026363(n). - _Michel Dekking_, Feb 27 2021
%e As a word, A284751 = 010001..., in which 1 is in positions 2,6,...
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 0, 1}}] &, {0}, 6] (* A284751 *)
%t Flatten[Position[s, 0]] (* A284752 *)
%t Flatten[Position[s, 1]] (* A284753 *)
%Y Cf. A284751, A284752, A026363.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Apr 13 2017
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