Counting the number of k-term arithmetic progressions with fixed gap of 2m from [1,n]

Question

Following the proof of finding a lower bound using the probabilistic method for van der Waerden numbers, I'm trying to find the lower bound of a different family. The previously mentioned proof can be found here. Specifically, I'm looking at Lemma 3.2. So, my question is: if I'm looking at the family of arithmetic progressions with fixed gap of 2m, as opposed to the family of all arithmetic progressions, how can I count the number of k-term arithmetic progressions from [1,n], similar to this example?