RO-NMO 2000 Problem 101

Let $\left( x_n\right)_{n\ge 1}$ be a sequence having $x_1=3$ and defined as $x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor ,$ for every natural number $n.$ Find all values $m$ for which the terms $x_m,x_{m+1},x_{m+2}$ are in arithmetic progression, where $\lfloor\rfloor$ denotes the integer part.