

A341822


Length of the longest 2increasing sequence of positive integer triples with entries <= n.


0



1, 2, 4, 8, 10, 14, 17, 21, 27, 30, 35
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OFFSET

1,2


COMMENTS

A triple t=(a_1,a_2,a_3) is defined to be 2less 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 2increasing if for all i < j, t^(i) is 2less than t^(j).
Terms n <= 5 have been confirmed by bruteforce search (Table 1 of Gowers and Long (2021)).


REFERENCES

W. T. Gowers and J. Long, The length of an sincreasing sequence of rtuples, Combinatorics, Probability and Computing 30 (2021), 136.


LINKS



FORMULA

a(n) >= n^{3/2} when n is a perfect square.
It is conjectured that a(n) <= n^{3/2} for all n.


EXAMPLE

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 2increasing sequence of length 9 must contain a triple with some coordinate equal to 5.


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS

a(10)a(11) and confirmation of previous terms by Bert Dobbelaere, Mar 27 2021


STATUS

approved



