Funny Fibonacci

Number Theory Level 4

The familiar Fibonacci sequence $(F_n) = (1, 1, 2, 3, 5, 8, \ldots)$ is defined recursively as $F_0 = 1,\ F_1 = 1,\ F_n = F_{n-1} + F_{n-2}\ \text{for}\ n \geq 2.$ Interestingly, $F_{25} = 121\:393$ and $F_{50} = 20\:365\:011\:074$ .

Now define a new sequence $(G_n)$ as follows: $G_0 = 1,\ G_1 = 2,\ G_n = 3G_{n-1} - G_{n-2}\ \text{for}\ n \geq 2.$ Determine $G_{25}$ without using a calculator.