For the Ulam sequence, this paper proves a_n <= 1.454^n for large n and proves that min_{1<=k<=n} a_{k+1}/a_k <= 1 + 7 log n / n for large n.
Finch, On the regularity of certain 1-additive sequen ces
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Small gaps in the Ulam sequence
For the Ulam sequence, this paper proves a_n <= 1.454^n for large n and proves that min_{1<=k<=n} a_{k+1}/a_k <= 1 + 7 log n / n for large n.