Let the first term of a series be T_1=6 and its r^th term T_r=3 T_r-1+6^r, r=2,3, …, n. If the sum of the first n terms of this series is 1/5(n^2-12…

Let the first term of a series be $\displaystyle T_1=6$ and its $\displaystyle r^{\text {th }}$ term $\displaystyle T_r=3 T_{r-1}+6^r, r=2,3, \ldots, n$. If the sum of the first $\displaystyle n$ terms of this series is $\displaystyle \frac{1}{5}\left(n^2-12 n+39\right)\left(4 \cdot 6^n-5 \cdot 3^n+1\right)$, then $\displaystyle n$ is equal to $\displaystyle \_\_\_\_$.