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Math 192r, Problem Set \#3 \\
(due 9/26/01)
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\begin{enumerate}
\item
Let $F_n$ be the $n$th Fibonacci number,
as Wilf indexes them
(with $F_0=F_1=1$, $F_2=2$, etc.).
Give a simple homogeneous linear recurrence relation
satisfied by the sequence whose $n$th term is
\begin{itemize}
\item[(a)] $nF_n$;
\item[(b)] $1F_1+2F_2+...+nF_n$;
\item[(c)] $nF_1+(n-1)F_2+...+2F_{n-1}+F_n$;
\item[(d)] $F_n$ when $n$ is odd, and $2^n$ when $n$ is even.
\end{itemize}
In each case, an explanation should be included.
\item
The sequence of polynomials $f_n(x)$ in problem 2 of problem set 1
satisfies a second-order linear recurrence relation
with coefficients that are Laurent polynomials in $x$.
\begin{itemize}
\item[(a)]
Find it, and prove that it is correct.
(Note that this proves your conjectures from parts (a) through (c)
of that problem.)
\item[(b)]
Express $\sum_{n=0}^\infty f_n(x) y^n$ as a rational function of $x$ and $y$.
\end{itemize}
\end{enumerate}
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\noindent
Please be sure to write down how many hours you spent working on the assignment,
and whom you worked with.
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