Ruby, 5449 bytes
->n{*s=t=0;n.times{t+=1while*s=t=0;s|[t+=1]==s||s<<t<<t*2until s|[t]==s;s<<t<<t*2};s[n]s[n];s[n]}
1-indexed. Start with an array containing [0], buildadd pairs to the list with length 2narray until we have at least n+1 elements, then take the n+1th element.(1-based)