A007783 Mixed Van der Waerden numbers w(n, 3; 2).

3, 6, 9, 18, 22, 32, 46, 58, 77, 97, 114, 135, 160, 186, 218, 238, 279, 312, 349

Offset

1,1

Comments

This is the smallest number M such that if each integer 1, 2, ..., M is colored using one of two colors (say red and blue), then there must be an arithmetic progression of length 3 in one color (red) or an arithmetic progression of length n in the other color (blue). So the first term, w(1, 3; 2), is 3. - Donald Vestal, May 31 2005

Extended computations via SAT-solving in Ahmed, Kullmann, Snevily, 2011.

References

V. Chvatal, Some unknown Van der Waerden numbers, pp. 31-33 of R. K. Guy et al., editors, Combinatorial Structures and Their Applications (Proceedings Calgary Conference Jun 1969}), Gordon and Breach, NY, 1970.

Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Amer. Math. Soc., 2004.

Links

Table of n, a(n) for n=1..19.

T. Ahmed and O. Kullmann and H. Snevily, On the van der Waerden numbers w(2;3,t), arXiv:1102.5433 [math.CO], 2011-2014.

M. D. Beeler and P. E. O'Neil, Some new Van der Waerden numbers, Discrete Math., 28 (1979), 135-146.

Crossrefs

Cf. A002886 has the same definition but an incorrect first term.

Sequence in context: A324283 A161338 A047847 * A363128 A050625 A025614

Adjacent sequences: A007780 A007781 A007782 * A007784 A007785 A007786

Keyword

nonn,hard

Author

Matthew Klimesh (matthew(AT)engin.umich.edu)

Extensions

Entry revised by N. J. A. Sloane, Jun 01 2005

a(16)-a(19) from Oliver Kullmann, Oct 28 2011

Status

approved