Math Puzzle

By Sebastiaan

I came across a puzzle a few days ago. It is part of the (Dutch) National Science Quiz, my brother was reading the question out loud. I hadn't given it much thought, until now.

It is as follows: one person knows some sort of secret, he/she then tells three others. This takes an hour. Now these three others will each pass the secret on to again three others, which takes another hour. This process continues this way, until everyone in the world knows about the secret. Only the persons who were told about the secret last, can tell three more persons. How long will it take for the secret to be known by everyone in the world?

In the quiz, you can choose between a day, weekend or a week.

I won't give the answer to this problem, not just yet. I will wait until the 22^nd of December, for then the quiz doesn't take entries anymore. I will post the answer (at least, what I assume is the answer) before the 27^th, the day the official quiz answers are given.

So, let's take a look at the progress in this problem.

1 person knows about the secret. After 1 hour, 4 persons know about the secret. After 2 hours, 13 persons know about the secret.

So. in maths language:

n₀ = 1
n₁ = 4
n₂ = 13

Can we find a formula describing this process?

Please do not comment any answers.