Suppose you have n dollars and can buy coffee for $1, milk for $2, and orange juice for $2. Let C⁽ⁿ⁾ count the number of different ways of spending all your money.

(a) Explain why C⁽ⁿ⁾ = C⁽ⁿ⁻¹⁾ + 2C⁽ⁿ⁻²⁾, C⁽⁰⁾ = C⁽¹⁾ = 1.

(b) Find an explicit formula for C⁽ⁿ⁾.