Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods

We present a characterization of the algebraic integers with continued fraction expansions of the form $[a_0,\overline{a_1,\dots ,a_n,s}]$, where $(a_1,\dots ,a_n)$ is a palindrome and $s\in {\mathbb{N}}_{\ge 1}$. In particular, we focus on the special case where $(a_1,\dots ,a_n)=(m,\dots ,m)$, providing a detailed characterizations of the corresponding algebraic integers and $s$ in terms of Fibonacci polynomials. Then, we derive new expansions of square roots of integers with these periods, given $m$ and $n$. Moreover, we explicitly determine the fundamental solutions of both positive and negative Pell’s equations corresponding to this family of integers.