Extensions and Refinements of Some Properties of Sums Involving Pell Numbers

Abstract

Falcón Santana and Díaz-Barrero [Missouri Journal of Mathematical Sciences, 18.1, pp. 33-40, 2006] proved that the sum of the first $4n+1$ Pell numbers is a perfect square for all $n \ge 0$. They also established two divisibility properties for sums of Pell numbers with odd index. In this paper, the sum of the first $n$ Pell numbers is characterized in terms of squares of Pell numbers for any $n \ge 0$. 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.