Suppose you have n dollars and can buy coffee for 1, milk...

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 $C^{(n)}=C^{(n-1)}+2 C^{(n-2)}, C^{(0)}=C^{(1)}=1$. (b) Find an explicit formula for $C^{(n)}$.

Key Concepts

Recurrence Relations

Recurrence relations are equations that define a sequence recursively: each term of the sequence is expressed in terms of previous terms. They are fundamental in enumerative combinatorics for modeling problems where the current state depends on a fixed number of preceding states. In this context, the recurrence relation is derived from analyzing the last choice in a sequence of decisions, thereby transforming a combinatorial counting problem into a recursive one.

Initial Conditions

Initial conditions provide the base values for a recurrence relation and allow the recursive definition to be fully determined. They are critical because they anchor the recurrence, ensuring the recurrence holds true from the very beginning of the sequence, and they often correspond to trivial or base case problems in combinatorial counting.

Characteristic Equation

The characteristic equation is an algebraic equation obtained by assuming a solution of a particularly simple form (usually exponential) for the recurrence relation. It transforms the recurrence into a polynomial whose roots determine the general form of the solution. This method is a standard tool for solving linear homogeneous recurrence relations with constant coefficients.

Solving Linear Recurrences

Solving linear recurrences involves finding the general solution to the recurrence relation defined by its characteristic equation. Once the roots of the characteristic equation are determined, they are used to express the general term of the sequence. Adjusting the solution with the initial conditions then provides an explicit formula for the sequence.

Combinatorial Counting

Combinatorial counting involves determining the number of ways an event can occur given a certain set of rules or constraints. In problems like this, where choices have associated costs or values, the goal is to count the number of sequences or compositions that sum to a target value, often by breaking the problem into smaller, more manageable parts that can be solved recursively.

Transcript

Please give Ace some feedback