Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.4

Some More Van der Waerden Numbers

Tanbir Ahmed
Department of Computer Science and Software Engineering
Concordia University
Montréal, QC H3G1M8


The van der Waerden number w(k; t0, t1, ... , tk-1) is the smallest positive integer n such that every k-coloring of the sequence 1, 2, ... , n yields a monochromatic arithmetic progression of length ti for some color i ∈ {0, 1, ... , k-1}. In this paper, we propose a problem-specific backtracking algorithm for computing van der Waerden numbers w(k; t0, t1, ..., tk-1) with t0 = t1 = ... = tj-1 = 2, where kj+2, and ti ≥ 3 for ij. We report some previously unknown van der Waerden numbers using this method. We also report the exact value of the previously unknown van der Waerden number w(2; 5, 7).

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(Concerned with sequences A217005 A217007 A217008 A217037 A217058 A217059 A217060.)

Received September 28 2012; revised version received March 12 2013. Published in Journal of Integer Sequences, March 16 2013.

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