Determining the autocorrelation sequence from an AR model
Looking at your definition of $\gamma_{xx}(m)$, it seems that you have 4 equations with 4 unknown (for $m=0,\ldots,3$):

\begin{align}
24\gamma_{xx}(0) &amp;= 14\gamma_{xx}(1) + 9\gamma_{xx}(2) - \gamma_{xx}(3) + 24\sigma_w^2  \\
24\gamma_{xx}(1) &amp;= 14\gamma_{xx}(0) + 9\gamma_{xx}(1) -\gamma_{xx}(2) \\ 
24\gamma_{xx}(2) &amp;= 14\gamma_{xx}(1) + 9\gamma_{xx}(0) - \gamma_{xx}(1)\\
24\gamma_{xx}(3) &amp;= 14\gamma_{xx}(2) + 9\gamma_{xx}(1) - \gamma_{xx}(0)
\end{align}

Solving the above system of equations is straight forward, and after that you can easily obtain $\gamma_{xx}(m)$ for $m=4,5$