Question 56
Let $a_{n}$ be the number of ways to fill a $n \times n$ board with the digits 0 and 1, so that the sum in each row and each column is the same. For example, the $2 \times 2$ trays that satisfy this rule are:
| 0 | 0 |
| :--- | :--- |
| 0 | 0 |
| 1 | 0 |
| :--- | :--- |
| 0 | 1 |
| 0 | 1 |
| :--- | :--- |
| 1 | 0 |
| 1 | 1 |
| :--- | :--- |
| 1 | 1 |
Logo, $a_{2}=4$. Calculate the values of $a_{3}$ and $a_{4}$.
| 0 | 0 |
| :--- | :--- |
| 0 | 0 |
| 1 | 0 |
| :--- | :--- |
| 0 | 1 |
| 0 | 1 |
| :--- | :--- |
| 1 | 0 |
| 1 | 1 |
| :--- | :--- |
| 1 | 1 |
Logo, $a_{2}=4$. Calculate the values of $a_{3}$ and $a_{4}$.