Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of $1$ month and six new pairs of rabbits at the age of $2$ months and every month afterward. None of the rabbits ever die or leave the island.
- Find a recurrence relation for the number of pairs of rabbits on the island $n$ months after one newborn pair is left on the island.
- By solving the recurrence relation in $(A)$ determine the number of pairs of rabbits on the island $n$ months after one pair is left on the island.
