Example 1 The table below gives an "arithmetic array":
where each row and each column are arithmetic sequences, $a_{i j}$ represents the number located at the $i$-th row and $j$-th column.
(I) Write down the value of $a_{45}$;
(II) Write down the formula for $a_{i i}$;
(III) Prove: A positive integer $N$ is in this arithmetic array if and only if $2 N+1$ can be factored into the product of two positive integers that are not 1.
Solution: (I) $a_{45}=49$.
(II) Grouping by rows, the first row, or the first group, is an arithmetic sequence with the first term 4 and a common difference of 3: $a_{ij} = 4 + 3(j-1)$,

the second group is an arithmetic sequence with the first term 7 and a common difference of 5:
$$a_{2j} = 7 + 5(j-1),$$
$\cdots$,
the $i$-th group is an arithmetic sequence with the first term $4 + 3(i-1)$ and a common difference of $2i + 1$, thus
$$\begin{array}{l}
a_{ij} = 4 + 3(i-1) + (2i + 1)(j-1) \\
= 2ij + i + j = i(2j + 1) + j.
\end{array}$$