Example 6 Find the value of $1^{2}+3^{2}+\cdots+(2 n-1)^2$
To construct a triangular number table, add 1 to the left: (1, 3, ..., 2n-1) as the first row. Then
$$
\begin{aligned}
S_{n}= & 2[(2 n-1)+((2 n-1)+(2 n \\
& -3))+\cdots+((2 n-1)+\cdots+3+
\end{aligned}
$$
$$
\begin{aligned}
& 1)]-[1+3+\cdots+(2 n-1)] \\
= & 2 \cdot \frac{1}{6} n(n+1)[2(2 n-1)+1] \\
& -n^{2} \\
= & \frac{1}{3} n\left(4 n^{2}-1\right) .
\end{aligned}
$$