Two Upper Bounds for both the van der Waerden Numbers $W(r, k + 1)$ and   $W(r + 1, k)$, that show the Existence of a Recurrence Relation
  Using a method we have utilized previously, namely through a finite power series expansion which also sometimes is known as the "radix polynomial" representation of an integer, we find an upper bound for a van der Waerden number that has a recurrence property. 
Robert J Betts
math.NT