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A305369 Lexicographically earliest sequence of distinct positive integers such that for each 1 in the binary expansion of a(n), exactly one of a(n-1) and a(n+1) has a 1 in the same position.

%I A305369 #20 Apr 07 2022 12:39:20
%S A305369 1,3,2,4,5,9,8,6,7,17,16,10,11,21,20,32,33,13,12,18,19,37,36,24,25,35,
%T A305369 34,28,29,65,64,14,15,49,48,66,67,41,40,22,23,73,72,38,39,81,80,42,43,
%U A305369 69,68,26,27,97,96,30,31,129,128,44,45,83,82,132,133,51,50,76,77,131,130,52,53,75,74,144,145,47,46,192
%N A305369 Lexicographically earliest sequence of distinct positive integers such that for each 1 in the binary expansion of a(n), exactly one of a(n-1) and a(n+1) has a 1 in the same position.
%C A305369 This is to A280864 as A115510 is to A064413 (EKG) and A252867 is to A098550 (Yellowstone).
%D A305369 Empirical: a(4k) = 2*Q(2k), a(4k+1) = a(4k)+1, a(4k+2) = 2*Q(2k+1)+1, a(4k+3) = 2*Q(2k+1), where Q (for Quet) is A109812. Since Q has a simpler definition, there is hope for a proof of this connection.
%H A305369 N. J. A. Sloane, <a href="/A305369/b305369.txt">Table of n, a(n) for n = 1..10000</a>
%H A305369 N. J. A. Sloane, <a href="/A305369/a305369.txt">Maple program</a>
%e A305369 After a(1) = 1, a(2) is the smallest missing odd number, so a(2) = 3.
%e A305369 a(3) is then the smallest missing number of the form ...1*_2, so a(3) = 10_2 = 2.
%e A305369 After a(15) = 20 = 10100_2, a(16) is the smallest missing number of the form ...0*0**_2, which is 100000_2 = 32.
%Y A305369 Cf. A280864, A252867, A098550, A115510, A064413, A109812, A352578 (binary weight).
%Y A305369 The graphs of A109812, A252867, A305369, A305372 all have roughly the same, mysterious, fractal-like structure. - _N. J. A. Sloane_, Jun 03 2018
%K A305369 nonn,look
%O A305369 1,2
%A A305369 _N. J. A. Sloane_, Jun 02 2018