INMO 2017 Problem 6

Let $n \geq 1$ be an integer and consider the sum

$$x=\sum_{k \geq 0}\binom{n}{2 k} 2^{n-2 k} 3^{k}=\binom{n}{0} 2^{n}+\binom{n}{2} 2^{n-2} \cdot 3+\binom{n}{4} 2^{n-4} \cdot 3^{2}+\cdots$$

Show that $2 x-1,2 x, 2 x+1$ form the sides of a triangle whose area and inradius are also integers.