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?

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 con- secutive 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)
F2 n+1 −F2 n = Fn−1Fn+2.