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# Generating Functions

This combinatorial tool is used for solving recurrence relations and other difficult problems.

\(\bullet\) The number of \(\color{Purple}{Purple}\) onions can be any non-negative integer.

\(\bullet\) The number of \(\color{Green}{Green}\) onions is a multiple of 2.

\(\bullet\) The number of \(\color{Red}{Red}\) onions is a multiple of 3.

\(\bullet\) The number of \(\color{Blue}{Blue}\) onions is a multiple of 5.

If Cody has 23 onions, how many different distributions of colors can there be?

**Details and Assumptions**: Cody can't have a negative number of onions, but he may have 0 onions of a certain type(s).

A one year old bunny is sitting on the number 0 in the number line. His father, Bugs Bunny, is waiting for him on the number 10.

The bunny has to reach his father. At each minute, he can:

- Move one step forward;
- Move two steps forward;
- Stay still;
- Move one step backward;
- Move two steps backward.

There are \(N\) possible ways the bunny can be on the number 10 after 10 minutes. Find the last three digits of \(N\).

**Details and assumptions**

The bunny is allowed to hop beyond 10, and then come back.

There are no restrictions on the bunny entering the negative numbers. The bunny can go as much backwards as he wants.

Order does matter. For example, the steps \(\{0000022222\}\) and \(\{2020202020\}\) are considered distinct.

Consider the recurrence relation \( a_n = 2a_{n-1} + 3 a_{n-2} + 3^n \) for \(n \geq 2 \) with initial conditions \(a_0 = -1, a_1 = 1. \)

Given that \(a_{100}\) is in the form of

\[ \LARGE \frac {x\cdot y^z - w}{v} \]

where \(x,y,w,z\) are prime numbers and \(v\) as a perfect square, what is the value of \(w+v+x+y+z\)?

Find the general expression for \(c_n\), where

\[ c_n - 3 c_{n-1} = 2 \cdot 5^n . \]

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