

A007783


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


3



3, 6, 9, 18, 22, 32, 46, 58, 77, 97, 114, 135, 160, 186, 218, 238, 279, 312, 349
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Comments from Donald Vestal, May 31 2005: "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."
Extended computations via SATsolving in Ahmed, Kullmann, Snevily, 2011.


REFERENCES

M. D. Beeler and P. E. O'Neil, Some new Van der Waerden numbers, Discrete Math., 28 (1979), 135146.
V. Chvatal, Some unknown Van der Waerden numbers, pp. 3133 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)


CROSSREFS

Cf. A002886 has the same definition but an incorrect first term.
Sequence in context: A127644 A161338 A047847 * A050625 A025614 A182751
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
More terms for n=16,17,18,19 by Oliver Kullmann, Oct 28 2011


STATUS

approved



