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Math 491, Problem Set \#4 \\
(due 9/23/03)
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\begin{enumerate}
\item
\begin{itemize}
\item[(a)]
Does there exist a polynomial $p(t)$ of degree 3
such that the linear operator $p(T)$
annihilates the sequence whose $n$th term (for $n \geq 0$)
is $3^n+2^n+1^n$?
Exhibit such a polynomial or explain why none exists.
\item[(b)]
Same as (a), but with ``degree 3'' replaced by ``degree 4''.
\item[(c)]
Same as (a), but with ``degree 3'' replaced by ``degree 2''.
\end{itemize}
\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.
\end{enumerate}
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\noindent
Please be sure to write down how many hours you spent working on the problems,
and whom you worked with.
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