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%I #31 May 09 2023 15:36:19
%S 1,2,4,8,10,14,17,21,27,30,35
%N Length of the longest 2-increasing sequence of positive integer triples with entries <= n.
%C A triple t=(a_1,a_2,a_3) is defined to be 2-less than a triple u=(b_1,b_2,b_3) if a_i < b_i for at least two coordinates i. A sequence t^(j) of triples is 2-increasing if for all i < j, t^(i) is 2-less than t^(j).
%C Terms n <= 5 have been confirmed by brute-force search (Table 1 of Gowers and Long (2021)).
%D W. T. Gowers and J. Long, The length of an s-increasing sequence of r-tuples, Combinatorics, Probability and Computing 30 (2021), 1-36.
%H W. T. Gowers and J. Long, <a href="https://arxiv.org/abs/1609.08688">The length of an s-increasing sequence of r-tuples</a>, arXiv:1609.08688 [math.CO], 2016.
%H Po-Shen Loh, <a href="https://arxiv.org/abs/1505.07312">Directed paths: from Ramsey to Ruzsa and Szemerédi</a>, arXiv:1505.07312 [math.CO], 2015.
%F a(n) >= n^{3/2} when n is a perfect square.
%F It is conjectured that a(n) <= n^{3/2} for all n.
%e For n=4, the sequence (1,1,1), (1,2,2), (2,1,3), (2,2,4), (3,3,1), (3,4,2), (4,3,3), (4,4,4) has length a(4)=8 and every 2-increasing sequence of length 9 must contain a triple with some coordinate equal to 5.
%Y Cf. A000093.
%K nonn,hard,more
%O 1,2
%A _Marcel K. Goh_, Feb 20 2021
%E Edited by _N. J. A. Sloane_, Mar 21 2021
%E a(10)-a(11) and confirmation of previous terms by _Bert Dobbelaere_, Mar 27 2021
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