Closed formulae for certain Fermat-Pell equations

Given positive integers j, k, with j >= 2, we show that there are positive integers d, e such that root d has continued fraction expansion root d = [e, k, ..., k, 2e], with period j, if and only if k is even or 3 inverted iota j, in which case we give closed formulae to find all such d, e as well as the smallest solution in positive integers to the Fermat-Pell equation X-2 - dY(2) = (-1)(1.)