Continued fraction of the square root of nineteen ($\sqrt{19}=[4;\overline{2,1,3,1,2,8}]$)

For a quadratic irrational $\sqrt{D}$, write its complete quotient as $(\sqrt{D}+m_j)/d_j$. The recurrence $m_{j+1}=d_j{a}_j-m_j$, $d_{j+1}=(D-m_{j+1}^2)/d_j$, $a_{j+1}=\lfloor (\sqrt{D}+m_{j+1})/d_{j+1}\rfloor$ computes its continued fraction. For $D=19$, the successive pairs after the initial quotient are $(4,3),(2,5),(3,2),(3,5),(2,3),(4,1)$, and then the cycle repeats. They give the displayed period of length six.