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Use the enumitem package and its label* key-value option which appends it value to the parent label.

\documentclass{article}

\usepackage{enumitem}
\setlist[enumerate]{label*=\arabic*.}

\begin{document}

\begin{enumerate}    
  \item List 1

  \begin{enumerate}
    \item sub list 1
    \item sub list 2

    \begin{enumerate}
      \item subsub list 1
    \end{enumerate}

  \end{enumerate}

\end{enumerate}

\end{document}

The number of compositions of $n$ into odd parts is $F_n$, the $n$th Fibonacci number. See, e.g., page 259 of this paper by Hoggatt and Lind.