On the high-th mean of one special character sums modulo a prime

Using the elementary method of the classical Gauss sums and the properties of character sums, we study a linear recurrence formula about the form $G\left(n\right)=1+\sum _{a=1}^{p-1}\left(\frac{a^2+n{\overline{a}}^2}{p}\right)$ and about the mean value of $G(n)$. This is a further exploration of Yuan and Zhang's research in 2022, which help us to better understand the character sums wide range application.