A198685 Mixed palindromic van der Waerden numbers pdw(n,3;2)_2 ("unsatisfiable" part).

3, 6, 9, 16, 21, 31, 44, 57, 77, 94, 113, 135, 155, 183, 205, 237, 279, 312, 347, 389, 405, 463, 507, 593, 607, 643, 699

Offset

1,1

Comments

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.

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

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

Links

Table of n, a(n) for n=1..27.

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.

Example

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.

Crossrefs

Cf. A007783.

Sequence in context: A287554 A308777 A057855 * A070120 A070126 A351231

Adjacent sequences: A198682 A198683 A198684 * A198686 A198687 A198688

Keyword

hard,nonn

Author

Oliver Kullmann, Oct 28 2011

Status

approved