The van der Waerden number w(k; t₀, t₁, ... , t_k-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 t_i 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, ..., t_k-1) with t₀ = t₁ = ... = t_j-1 = 2, where k ≥ j+2, and t_i ≥ 3 for i ≥ j. 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).