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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)



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 SAT-solving in Ahmed, Kullmann, Snevily, 2011.


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

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.


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)


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




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


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

More terms for n=16,17,18,19 by Oliver Kullmann, Oct 28 2011



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Last modified December 19 10:16 EST 2014. Contains 252186 sequences.