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Using a method we have utilized previously, namely through a finite powerseries expansion which also sometimes is known as the "radix polynomial"representation of an integer, we find an upper bound for a van der Waerdennumber that has a recurrence property.

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