Unimodality of Sequences Associated to Pell Numbers

Abstract

In this paper, we show that the sequences $p(n,k):=2^{n-2k}(\genfrac{}{}{0ex}{}{n-k}{k})$ and $q(n,k):=2^{n-2k}\frac{n}{n-k}(\genfrac{}{}{0ex}{}{n-k}{k})$, $k=0,\dots ,\lfloor \frac{n}{2}\rfloor$, are strictly log-concave and then unimodal with at most two consecutive modes. We localize the modes and the integers where there is a plateau. We also give a combinatorial interpretation of $p(n,k)$ and $q(n,k)$. These sequences are associated respectively to the Pell numbers and the Pell-Lucas numbers, for which we give some trigonometric relations.