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.