How to find equation for this sequence of numbers?

Question

I have a sequence of numbers 0, 1, 5, 19, .... This is the pseudocode to generate the sequence

$c = 0$

for $i=0, 1, 2, ...:$

$ c = 3c + 2^i$

Does anyone know how I would write an equation $f(i)$ that would generate this same sequence?

Thanks

Answer 1

The generating function is $$f(n)=3^n-2^n$$

Proof by induction : The base case $n=0$ is clear.

Now, we have $$3^{n+1}-2^{n+1}=3\cdot(3^n-2^n)+3\cdot 2^n-2^{n+1}=3\cdot(3^n-2^n)+2^n$$

So, we have $f(n+1)=3f(n)+2^n$, completing the proof.