For all $n\in N,2\cdot 4^{2n+1}+3^{3n+1}$ is divisible by

For $n=1,$ we have

$2\cdot 4^{2n+1}+3^{3n+1}=2\times 4^3+3^4=209$ which is divisible by 11

For $n=2,$ we have

$2\cdot 4^{2n+1}+3^{3n+1}=2\times 4^5+3^7=4235$ which is divisible by 11

Similarly, it can be checked that ${2.4}^{2n+1}+3^{3n-1}$ is divisible by 11 for other values of $n$