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.
Distributions of Ulam Words up to Length 30
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abstract
We further explore the notion of Ulam words considered by Bade, Cui, Labelle, and Li, giving some lower bounds on how many there are of a given length. Gaps between words and words of special type also reveal remarkable structure. By substantially increasing the number of computed terms, we are also able to sharpen some of the conjectures made by Bade et al.
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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.