An integer sequence from a rational recursion
The author proves that the terms of the sequence \(\{a_ i\}\), defined by the recurrence \(a_ n=(a_{n-3} a_{n-1}+a_{n-2}^ 2)/a_{n-4}\), for \(n>3\) with initial values \(a_ 0=a_ 1=a_ 2=a_ 3=1\), are integers. As a generalization of this result it is shown: If the initial values \(a_ 0\), \(a_ 1\), \(a_ 2\), \(a_ 3\) are rational numbers and \(a_ n=N_ n/D_ n\) where \(N_ n\) and \(D_ n\) are relatively prime integers, then \(p\mid D_ n\) implies the divisibility \[ p\mid\text{gcd}(N_ 2,N_ 3,N_ 4)\cdot D_ 0\cdot D_ 1\cdots D_ 7 \] for any prime \(p\). Some other generalizations are also investigated.
scientific article; zbMATH DE number 166367