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In Chapter 4 (page 23, subsection "Somos sequence update") of his Tracking the Automatic Ant, David Gale discusses three families of recursively defined sequences of numbers, all due to Dana ...

The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...