When the numbers are not too bad, we can just do a one-step calculation over a few times:
Here I am using decimals; for worse numbers like $\sqrt{61}$
Lubin posted an algorithm (using whole numbers only) that appears to be what Fermat used; I then programmed that. There is a method due (largely) to Gauss and Lagrange that produces a chain or cycle of indefinite quadratic form, I programmed that long ago. Again, no decimal, all whole number, allows for any $\frac{A+\sqrt B}{C}.$ Finally, Conway's Topograph method. For that, I can recommend Weissman for self study or parent-child team study of some sort
Full treatment of $\sqrt{19}$
Method described by Prof. Lubin at Continued fraction of $\sqrt{67} - 4$
$$ \sqrt { 19} = 4 + \frac{ \sqrt {19} - 4 }{ 1 } $$
$$ \frac{ 1 }{ \sqrt {19} - 4 } = \frac{ \sqrt {19} + 4 }{3 } = 2 + \frac{ \sqrt {19} - 2 }{3 } $$
$$ \frac{ 3 }{ \sqrt {19} - 2 } = \frac{ \sqrt {19} + 2 }{5 } = 1 + \frac{ \sqrt {19} - 3 }{5 } $$
$$ \frac{ 5 }{ \sqrt {19} - 3 } = \frac{ \sqrt {19} + 3 }{2 } = 3 + \frac{ \sqrt {19} - 3 }{2 } $$
$$ \frac{ 2 }{ \sqrt {19} - 3 } = \frac{ \sqrt {19} + 3 }{5 } = 1 + \frac{ \sqrt {19} - 2 }{5 } $$
$$ \frac{ 5 }{ \sqrt {19} - 2 } = \frac{ \sqrt {19} + 2 }{3 } = 2 + \frac{ \sqrt {19} - 4 }{3 } $$
$$ \frac{ 3 }{ \sqrt {19} - 4 } = \frac{ \sqrt {19} + 4 }{1 } = 8 + \frac{ \sqrt {19} - 4 }{1 } $$
Simple continued fraction tableau:
$$
\begin{array}{cccccccccccccccccc}
& & 4 & & 2 & & 1 & & 3 & & 1 & & 2 & & 8 & \\
\\
\frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 4 }{ 1 } & & \frac{ 9 }{ 2 } & & \frac{ 13 }{ 3 } & & \frac{ 48 }{ 11 } & & \frac{ 61 }{ 14 } & & \frac{ 170 }{ 39 } \\
\\
& 1 & & -3 & & 5 & & -2 & & 5 & & -3 & & 1
\end{array}
$$
$$
\begin{array}{cccc}
\frac{ 1 }{ 0 } & 1^2 - 19 \cdot 0^2 = 1 & \mbox{digit} & 4 \\
\frac{ 4 }{ 1 } & 4^2 - 19 \cdot 1^2 = -3 & \mbox{digit} & 2 \\
\frac{ 9 }{ 2 } & 9^2 - 19 \cdot 2^2 = 5 & \mbox{digit} & 1 \\
\frac{ 13 }{ 3 } & 13^2 - 19 \cdot 3^2 = -2 & \mbox{digit} & 3 \\
\frac{ 48 }{ 11 } & 48^2 - 19 \cdot 11^2 = 5 & \mbox{digit} & 1 \\
\frac{ 61 }{ 14 } & 61^2 - 19 \cdot 14^2 = -3 & \mbox{digit} & 2 \\
\frac{ 170 }{ 39 } & 170^2 - 19 \cdot 39^2 = 1 & \mbox{digit} & 8 \\
\end{array}
$$
$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$
$\sqrt{21}$ Simple continued fraction tableau:
4.58, 1.7, 1.39, 2.52, 1.89, 1.11,8.58
$$
\begin{array}{cccccccccccccccccc}
& & 4 & & 1 & & 1 & & 2 & & 1 & & 1 & & 8 & \\
\\
\frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 4 }{ 1 } & & \frac{ 5 }{ 1 } & & \frac{ 9 }{ 2 } & & \frac{ 23 }{ 5 } & & \frac{ 32 }{ 7 } & & \frac{ 55 }{ 12 } \\
\\
& 1 & & -5 & & 4 & & -3 & & 4 & & -5 & & 1
\end{array}
$$
$$
\begin{array}{cccc}
\frac{ 1 }{ 0 } & 1^2 - 21 \cdot 0^2 = 1 & \mbox{digit} & 4 \\
\frac{ 4 }{ 1 } & 4^2 - 21 \cdot 1^2 = -5 & \mbox{digit} & 1 \\
\frac{ 5 }{ 1 } & 5^2 - 21 \cdot 1^2 = 4 & \mbox{digit} & 1 \\
\frac{ 9 }{ 2 } & 9^2 - 21 \cdot 2^2 = -3 & \mbox{digit} & 2 \\
\frac{ 23 }{ 5 } & 23^2 - 21 \cdot 5^2 = 4 & \mbox{digit} & 1 \\
\frac{ 32 }{ 7 } & 32^2 - 21 \cdot 7^2 = -5 & \mbox{digit} & 1 \\
\frac{ 55 }{ 12 } & 55^2 - 21 \cdot 12^2 = 1 & \mbox{digit} & 8 \\
\end{array}
$$
=======================================
$ \sqrt{41}$
6.403, 2.48, 2.08, 12.40,2.48, 2.08, 12.39
$$
\begin{array}{cccccccccccccccc}
& & 6 & & 2 & & 2 & & 12 & & 2 & & 2 & & 12 & \\
\\
\frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 6 }{ 1 } & & \frac{ 13 }{ 2 } & & \frac{ 32 }{ 5 } & & \frac{ 397 }{ 62 } & & \frac{ 826 }{ 129 } & & \frac{ 2049 }{ 320 } \\
\\
& 1 & & -5 & & 5 & & -1 & & 5 & & -5 & & 1
\end{array}
$$
$$
\begin{array}{cccc}
\frac{ 1 }{ 0 } & 1^2 - 41 \cdot 0^2 = 1 & \mbox{digit} & 6 \\
\frac{ 6 }{ 1 } & 6^2 - 41 \cdot 1^2 = -5 & \mbox{digit} & 2 \\
\frac{ 13 }{ 2 } & 13^2 - 41 \cdot 2^2 = 5 & \mbox{digit} & 2 \\
\frac{ 32 }{ 5 } & 32^2 - 41 \cdot 5^2 = -1 & \mbox{digit} & 12 \\
\frac{ 397 }{ 62 } & 397^2 - 41 \cdot 62^2 = 5 & \mbox{digit} & 2 \\
\frac{ 826 }{ 129 } & 826^2 - 41 \cdot 129^2 = -5 & \mbox{digit} & 2 \\
\frac{ 2049 }{ 320 } & 2049^2 - 41 \cdot 320^2 = 1 & \mbox{digit} & 12 \\
\end{array}
$$
============================
$$
\begin{array}{cccc}
\frac{ 1 }{ 0 } & 1^2 - 21 \cdot 0^2 = 1 & \mbox{digit} & 4 \\
\frac{ 4 }{ 1 } & 4^2 - 21 \cdot 1^2 = -5 & \mbox{digit} & 1 \\
\frac{ 5 }{ 1 } & 5^2 - 21 \cdot 1^2 = 4 & \mbox{digit} & 1 \\
\frac{ 9 }{ 2 } & 9^2 - 21 \cdot 2^2 = -3 & \mbox{digit} & 2 \\
\frac{ 23 }{ 5 } & 23^2 - 21 \cdot 5^2 = 4 & \mbox{digit} & 1 \\
\frac{ 32 }{ 7 } & 32^2 - 21 \cdot 7^2 = -5 & \mbox{digit} & 1 \\
\frac{ 55 }{ 12 } & 55^2 - 21 \cdot 12^2 = 1 & \mbox{digit} & 8 \\
\end{array}
$$
http://www.maths.ed.ac.uk/~aar/papers/conwaysens.pdf
https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf
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