= Continued fraction of the square root of nineteen
{title2=$\sqrt{19}=[4;\overline{2,1,3,1,2,8}]$}
For a <quadratic irrational> $\sqrt D$, write its <complete quotient> as $(\sqrt D+m_j)/d_j$. The recurrence $m_{j+1}=d_ja_j-m_j$, $d_{j+1}=(D-m_{j+1}^2)/d_j$, $a_{j+1}=\lfloor(\sqrt D+m_{j+1})/d_{j+1}\rfloor$ 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.
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