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%I #17 Jul 21 2020 02:27:02
%S 2,5,6,11,14,15,18,29,32,33,38,41,42,45,54,83,86,87,92,95,96,99,110,
%T 113,114,119,122,123,126,135,162,245,248,249,254,257,258,261,272,275,
%U 276,281,284,285,288,297,326,329,330,335,338,339,342,353,356,357,362,365,366,369,378,405,486,731,734,735,740,743,744
%N Lexicographically first 3-free sequence on nonnegative integers not containing the Stanley sequence S(0,1), which is A005836.
%C Consider the lexicographically earliest sequence of nonnegative numbers that does not contain the arithmetic mean of any pair of terms (such sequences are called 3-free sequences as they do not contain 3-term arithmetic progressions): 0,1,3,4 and so on. This sequence is Stanley sequence S(0,1). Remove numbers in the Stanley sequence from nonnegative integers and repeat the process of finding the earliest 3-free sequence. The result is this sequence.
%C When written in base three these are numbers that contain exactly one 2 that might be followed by zeros.
%H Tanya Khovanova and Kevin Wu, <a href="https://arxiv.org/abs/2007.09705">Base 3/2 and Greedily Partitioned Sequences</a>, arXiv:2007.09705 [math.NT], 2020.
%e Removing the Stanley sequence from nonnegative integers we get sequence A074940: 2, 5, 6, 7, 8, 11, 14, 15, 16, 17, 18 (Numbers having at least one 2 in their ternary representation). Our new sequence starts with 2,5,6. It can't contain 7 as 5,6,7 form an arithmetic progression. It can't contain 8 as 2,5,8 form an arithmetic progression. The next term is 11.
%Y Cf. A005836, A074940, A323418, A323419.
%K nonn
%O 1,1
%A _Tanya Khovanova_ and PRIMES STEP Junior, Jan 13 2019
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