In how many ways can you cover a 2 × n chessboard by 2 × 1 dominoes (placed horizontally or vertically)? Find a recurrence relation and initial values.

Discrete Mathematics

Homework 7

Solve the following problems. Justify your answers. Solutions without justification will
not receive full credit.

1. How many binary strings of length n do not contain 10?

2. Find a recurrence relation for the number of length-n ternary strings (strings using values 0, 1, 2) without two consecutive 0s.What are the initial values?

3. In how many ways can you cover a 2 × n chessboard by 2 × 1 dominoes (placed horizontally or vertically)? Find a recurrence relation and initial values.

4. How many subsets are there of the set {1, 2, . . . , n} that DO NOT contain three consecutive integers? Find a recurrence relation and initial values.

5. Which is larger 2100 or F100?

6. Use induction to prove
F2 + F4 + F6 + ···+ F2n= F2n+1 −1.

7. Prove (without using induction)
F 2n+1 −F 2n= Fn−1Fn+2.