Computing Off-Diagonal Van der Waerden Numbers: Improved Algorithm and New Exact Values

We present a backtracking algorithm with multiple pruning strategies for computing off-diagonal Van der Waerden numbers together with 55 previously unknown exact values. We distinguish between single-use colors and multiple-use colors. This structure can be exploited to significantly reduce the search space. Key optimizations include: a disallowed-colors forward propagation with stack-based backtracking, a pruning rule based on previously calculated Van der Waerden numbers, and a prefix-cache that memoizes the maximum certificate length attainable given any fixed prefix.