Take
\[(2+3 \sqrt{5} )^n\]
and write \[a_n+b_n \sqrt{5} = (2+3 \sqrt{5} )^n\]
.Then
\[\begin{equation} \begin{aligned} a_{n+1}+b_{n+1} \sqrt{5} &= (2+3 \sqrt{5} )^{n+1} \\ &=(2+3 \sqrt{5} )(2+3 \sqrt{5} )^n \\ &=(2a_n+15b_n)+(3a_n+2b_n)\sqrt{5} \end{aligned} \end{equation}\]
.We can then define
\[(2+3 as
\[a_1=2, \: b_1=3\]
\[a_{n+1}=2a_n+15b_n, \: b_{n+1}=3a_n+2b_n\]
.