SUMMER 2012 Assignment #1

Due May 23 by the end of the day.

1. [5 marks] 
   Install Sage on your computer.  Use it to compute Q(n) for n=1..2^10,
   where Q(n) = Q(n-Q(n-1))+Q(n-Q(n-2)), with Q(1) = Q(2) = 1.  Then
   plot n/2-Q(n) and turn in the result.

2. [5 marks]
   From the book: Problem 1.21 on page 20.  Of course, a solution
   is in the back of the book.  Your job is to find a solution of
   your own or to fill in the missing details of the book's 
   solution (i.e., why is it correct).

3. [5 marks]
   Compute J(99999999999999999999).  Of course, you can use Maple or Sage to 
   help you in your calculations.

4. [5 marks]
   Solve using two different methods.

           n
   a(n) = --- a(n-1) + 1,    a(0) = 1      
          n+1

5. [5 marks]
   Find a closed form expression for the following sum and prove that
   your answer is correct.   Your proof cannot use induction (or
   lookup or Maple).

            ---
             \ 
   a_n =      ) k^3 5^k
             /
            ---
          0<=k<=n   


6. [5 marks]
   Find the simplest expression that you can for the double sum

   ---      ---
    \        \
     )        )   max( j, k )
    /        /
   ---      ---
 1<=j<=n  1<=k<=n

   Also do the same for min instead of max.