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A198684 Mixed palindromic van der Waerden numbers pdw(n,3;2)_1 ("satisfiable" part). 1
2, 3, 6, 15, 16, 30, 41, 52, 62, 93, 110, 126, 142, 174, 200, 232, 256, 299, 338, 380, 400, 444, 506, 568, 586, 634, 664 (list; graph; refs; listen; history; text; internal format)
The largest number M such that for all natural numbers 1 <= m <= M there exists a partitioning of {1,...,m} into 2 parts such that the first part contains no arithmetic progression of length 3, and the second part contains no arithmetic progression of size n, and such that the partitioning is palindromic, that is, written as 0-1 string is a palindrome (reading the same from both sides). So trivially pdw(1,3;2)=2 and pdw(2,3;2)=3.
pdw(n,3;2) is a pair of natural numbers, where the second part (the "unsatisfiable part") is given by A198685.
Tanbir Ahmed, Oliver Kullmann, and Hunter Snevily, On the van der Waerden numbers w(2;3,t), arXiv:1102.5433 [math.CO], 2011-2014; Discrete Applied Math., 174 (2014), 27-51.
For n=3 we have pdw(3,3;2)_1=6, because ({1,2,5,6},{3,4}) is a palindromic partitioning of {1,...,6} into two parts such that no part contains an arithmetic progression of size 3, while {1,...,7} has no such *palindromic* partitioning (though it has a nonpalindromic partitioning by w(3,3;2)=9), since one of the parts needs to contain the midpoint 4 and thus cannot contain any other number (by palindromicity we would get an arithmetic progression of size 3).
Cf. A007783.
Sequence in context: A091138 A335260 A090983 * A293534 A066653 A081945
Oliver Kullmann, Oct 28 2011

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