Extensions and refinements of some properties of sums involving Pell numbers (Q987946)

The Pell numbers \(P_n\) are defined by \(P_{n+1}= 2P_n+ P_{n-1}\) for \(n\geq 1\) and \(P_0= 0\), \(P_1= 1\). \textit{S. Falcón Santana} and \textit{J. L. Díaz-Barrero} [Missouri J. Math. Sci. 18, No. 1, 33--40 (2006; Zbl 1137.05009)] proved that the sum of the first \(4n+1\) Pell numbers is a perfect square for all \(n\geq 0\) and gave two divisibility properties for sums of Pell numbers with odd index. In the paper under review, the sum of the first \(n\) Pell numbers is characterized in terms of squares of Pell numbers for any \(n\geq 0\). Additional divisibility properties for sums of Pell numbers with odd index are also presented, and divisibility properties for sums of Pell numbers with even index are derived.