1$; in this case we have $q=\alpha^{1/(\alpha-1)}$. If we now let $\alpha\to 1$ then $q\to e$, which we might expect given the construction of the sequence in Theorem~\ref{thm:alpha}. \section{Conclusion}\label{sec:conclusion} In this paper we have looked at the construction of cost-minimizing jump sequences. We have seen that when $w(i,j)=(i+j)/i^2$ that the corresponding jump sequence sieves to the Pell numbers. By choosing different weight functions we can generate different sequences; of course what is interesting are simple weight functions that can generate well-known sequences. Perhaps the most well known sequence is the Fibonacci numbers and these can be generated by a relatively simple weight function. \begin{theorem}\label{thm:fib} Let $w(i,j)=(ij+j^2)/i^3$. Then $\mathcal{A}_\infty=\{1,2,3,5,8,13,\ldots\}$ is the Fibonacci numbers. \end{theorem} The proof of Theorem~\ref{thm:fib} is done in the exact same manner as the proof of Theorem~\ref{thm:pell} and we will omit the proof. Applying the function $w(i,j)=(ij+j^2)/i^3$ to Theorem~\ref{thm:geometric} we get $q=\phi=(1+\sqrt5)/2$; while the Fibonacci numbers can be defined by letting $f_1=1$ and $f_n=\lfloor \phi f_{n-1}+0.5\rfloor$ for $n\geq 2$. It might appear that finding $\mathcal{A}_\infty$ reduces to finding the optimal real ratio and then ``round; multiply; repeat''. However this will not always work. For instance if we let $w(i,j)=(i^2+ij+j^2)/i^3$ then the optimal real ratio is the real root of $2q^3-2q^2-2q-1=0$, for which $q\approx 1.73990787\ldots$ . The ``round; multiply; repeat'' would predict the set $\{1,2,3,5,9,16,28,49,\ldots\}$ but instead we have $\mathcal{A}_\infty=\{1, 2, 4, 7, 12, 21, 37, 64,\ldots \}$. The problem is that instead of $3$ (as we would predict by rounding $3.47981574\ldots$) we got $4$, which then throws off the rest of the sequence. The problem of finding the set $\mathcal{A}_\infty$ in general remains elusive. However, it seems reasonable that the technique we have given here could work for cases when the ratio $q$ is a Pisot-Vijayaraghavan number, which has the property that $q^n$ is an almost integer while the power of any other root goes to $0$. We have limited ourselves to homogeneous functions (i.e., functions that can be expressed in the form $w(i,j)=i^{-\alpha}f(j/i)$). These functions have the benefit that when dealing with real sequences that the tails of the sequences also form cost-minimizing sequences, which was used in the proof of Theorem~\ref{thm:pell}. What can be said about functions thatare not homogeneous? (In \cite{paper} the authors considered the function $w(i,j)=(1-q^j)/i$ where $0