A198685 - OEIS

%I #15 Feb 03 2021 23:04:00

%S 3,6,9,16,21,31,44,57,77,94,113,135,155,183,205,237,279,312,347,389,

%T 405,463,507,593,607,643,699

%N Mixed palindromic van der Waerden numbers pdw(n,3;2)_2 ("unsatisfiable" part).

%C The smallest number M such that for all natural numbers m >= M there exists no 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 pdw(1,3;2)_2=3 and pdw(2,3;2)_2=6.

%C pdw(n,3;2) is a pair of natural numbers, where the first part (the "satisfiable part") is given by A198684.

%C pdw(n,3;2) is the palindromic version of w(n,3;2) which is A007783.

%H Tanbir Ahmed, Oliver Kullmann, and Hunter Snevily, <a href="http://arxiv.org/abs/1102.5433">On the van der Waerden numbers w(2;3,t)</a>, arXiv:1102.5433 [math.CO], 2011-2014; Discrete Applied Math., 174 (2014), 27-51.

%e pdw(3,3;2)_2=9, since ({2,4,5,7},{1,3,6,8}) is a palindromic partitioning of {1,...,8} such that no part contains an arithmetic progression of length 3, while there is no such a partitioning of {1,...,9} because of w(3,3;2)=9.

%Y Cf. A007783.

%K hard,nonn

%O 1,1

%A _Oliver Kullmann_, Oct 28 2011