10. Observe the following equations: $C_{5}^{1}+C_{5}^{5}=2^{3}-2$;
$$
\begin{array}{l}
C_{9}^{1}+C_{9}^{5}+C_{9}^{9}=2^{7}+2^{3} ; \\
C_{13}^{1}+C_{13}^{5}+C_{13}^{9}+C_{13}^{13}=2^{11}-2^{5} ; \\
C_{17}^{1}+C_{17}^{5}+C_{17}^{9}+C_{17}^{13}+C_{17}^{17}=2^{15}+2^{7} ; \cdots \cdots,
\end{array}
$$

From the above equations, deduce the general conclusion:
For $n \in \mathbf{N}^{*}, C_{4 n+1}^{1}+C_{4 n+1}^{5}+C_{4 n+1}^{9}+\cdots+C_{4 n+1}^{4 n+1}=$ $\qquad$
$$
C_{4 n+1}^{1}+C_{4 n+1}^{5}+C_{4 n+1}^{9}+\cdots+C_{4 n+1}^{4 n+1}=2^{4 n-1}+(-1)^{n} \cdot 2^{2 n-1} .
$$

The solution is as follows:

$$
C_{4 n+1}^{1}+C_{4 n+1}^{5}+C_{4 n+1}^{9}+\cdots+C_{4 n+1}^{4 n+1}=2^{4 n-1}+(-1)^{n} \cdot 2^{2 n-1} .
$$