The problem presented involves arranging tiles of four different colors (red, blue, white, and yellow) in a row, with conditions on color selection and placement to achieve specific patterns. Here’s an analysis and breakdown of the key points:

1. Setup:

   • We have a sequence of $n$ tiles.
   • Only two colors are used for any given arrangement.
   • An example layout shows blue and white tiles alternating, with the last tile color depending on the total number $n$.

2. Objectives:

   • We need to find the total number of sequences of tiles that use exactly two colors from the four available colors.
   • There are functions defined, $a_n$ and $b_n$, which represent counts of arrangements:
     • $a_n$: Total number of ways to arrange $n$ tiles using exactly 2 colors.
     • $b_n$: Total number of ways to arrange $n$ tiles using exactly 3 colors.

3. Recurrence Relation:

   • The task includes expressing $a_{n+1}$ in terms of $a_n$, using recurrence relations.
   • Based on the diagram and accompanying text, there seems to be an emphasis on the tiles at the endpoints (first and last) to derive this recurrence.

Solution Steps:

1. Identifying the Base Cases:

   • $a_1$ and $a_2$ values are provided or can be calculated directly based on simple cases.
   • $a_1 = \text{one arrangement with two colors}$
   • $a_2 = \text{number of ways for two-tile arrangements using two colors}$

2. Recurrence Formula:

   • Use the constraints on color selection and placement to build a recurrence formula for $a_n$.
   • The analysis likely involves whether the first and last tiles are the same color or different, leading to a recursive count that relies on previous terms.

3. Counting with Combination:

   • For each color pair choice (from 4 colors, choose 2), apply the recurrence to count valid sequences for a total arrangement count.

Would you like detailed steps for setting up the recurrence relation or solving specific examples for $a_n$ or $b_n$?

Here are related questions for deeper exploration:

1. How can we derive the exact recurrence formula for $a_{n+1}$ in terms of $a_n$?
2. What are the specific values of $a_1$, $a_2$, and $a_3$, and how can they be calculated?
3. How does the constraint on endpoints influence the recurrence relation?
4. How many ways can we choose two colors from four, and how does it impact the total count?
5. How would we extend this pattern if more colors were added?

Tip: For complex combinatorial problems, defining clear base cases and small examples often helps in verifying recurrence relations.