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\begin{center}
  {\large\bf Test 2 for Math 405, Introduction to Combinatorics.
    \vspace{12mm}\\ } {\LARGE\bf Name:}\vrule height .4pt depth 0
  pt width 8cm
  {\footnotesize \hfill October 27, 2000}\\
\end{center}
Instructions: The test will be 50 minutes in length.


\begin{enumerate}
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  \problem \hide Final Fall 95, \endhide %
  (10 pts) Assume that
  $$
  h_{n}=5h_{n-1} -6 h_{n-2}
  $$
  for $n\geq 3$ and that $h_1=h_2=1$. Compute $h_n$ for all
  integers $n\geq 3$.


\vfill

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  \problem \hide Final fall95 \endhide %
  (10 pts) Assume that
  $$
  h_{n+3}+ h_{n+1} +  h_n=0
  $$
  and that $h_1=0$, $h_2=0$ and $h_3=0$. Compute $h_n$ for all
  integers $n\geq 4$.



\vfill\newpage 



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  \problem \hide Final fall95 \endhide %
  (10 pts) Let
  $$
  f(x)=\frac{1}{(x-1)(x-2)(x-3)}=\sum_{n=0}^\infty s_nx^n
  $$
  be the generating function of a sequence. Find an explicit
  formula for $s_n$.  


\vfill

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  \problem \hide Final fall95 \endhide %
  (10 pts) Solve explicitly the recurrence relation
  \begin{eqnarray*}
    h_{n+1}&=&  3h_n+3^{n+1},\ \ n\geq 1.\\
     h_1&=&2.
  \end{eqnarray*}



\vfill\newpage 

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  \problem (15 pts) What is the number of ways to place six non-attacking
  rooks on the following `pruned chess-board'?

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\vfill\newpage 

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\problem (15 pts) Let $h_n$ denote the number of $n$-digit
sequences in which each digit is 0 or 1, no two consecutive 0's
being allowed.  Note that $h_1=2$ and $h_2=3$.  Find an explicit
formula for $h_n$. (Hint: establish a second order recurrence
relation.)



\newpage

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\problem (15 pts) Determine the number of $n$ digit numbers with
all digits odd, such that 1 and 3 each occur an even number of
times.


\newpage
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\problem For each $n$ let $h_n$ denote the number of
non-negative integral solutions of the equation
$$
a+3b+5c+d+e=n
$$
subject to the conditions:
$$
0\leq d \leq 2,\hspace{12mm} 0\leq e \leq 4.
$$

a) (8 pts) Find the generating function of the sequence $h_n$.

\vfill

b) (7 pts) When $n=100$ compute explicitly the number of solutions.
\vfill

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\end{enumerate}
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