어디 가져다 쓰기에는 많이 부족합니다. 그냥 참고만 하세요.

\documentclass{article}

\usepackage{mathtools}

\makeatletter

\newcount\m \newcount\n

\newcount\r \newcount\q \newcount\t

\def\hjmod#1#2{\m=#1 \n=#2

\ifnum\n=0 \else \r=\m

\divide\m by\n \q=\m \multiply\m by\n

\ifnum\m=\r \else\advance\q by 1

\t=\q \multiply\t by\n \m=\t \fi\advance\t-\r \fi}

\def\hjcfrac#1#2{\hjmod{#1}{#2} \number\q

\ifnum\t>0-{\strut1\hfill\over\displaystyle\hjcfrac\n\t}\fi}

\def\lhjcfrac[#1;#2]{\cfr@c#1,#2,\end}

\def\cfr@c#1,#2\end{\ifx#1\ldots\ddots\else#1\fi

  \ifx#2\end\else-{\strut1\hfill\over\displaystyle\cfr@c#2\end}\fi}

\makeatother

\begin{document}

\begin{align*}

\frac{19}{7}&=\hjcfrac{19}{7} \\

\frac{415}{93}&=\hjcfrac{415}{93} \\

[3;4,2]&=\lhjcfrac[3;4,2]

\end{align*}

\end{document}