Test 2 for Math 405, Introduction to Combinatorics.

Name: October 27, 2000

Instructions: The test will be 50 minutes in length.

1. (10 pts) Assume that

h_{n} = 5h_{n−1} − 6h_{n−2}

for n ≥ 3 and that h_{1} = h_{2} = 1. Compute h_{n} for all integers n ≥ 3.

2. (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 ≥ 4.

3. (10 pts) Let

f(x) = \frac{1}{(x − 1)(x − 2)(x − 3)} = ∑_{n=0}^{∞} s_{n}x^{n}

be the generating function of a sequence. Find an explicit formula for s_{n}.

4. (10 pts) Solve explicitly the recurrence relation

h_{n+1} = 3h_{n} + 3^{n+1}, n ≥ 1.
h_{1} = 2.

5. (15 pts) What is the number of ways to place six non-attacking rooks on the following ‘pruned chess-board’?

6. (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.)

7. (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.

8. 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 ≤ d ≤ 2, 0 ≤ e ≤ 4.

a) (8 pts) Find the generating function of the sequence h_{n}.

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