Hi all, I'm new to this, so please accept my apologies if that post is obvious!!

At 3am this morning, my mind was buzzing so in order to stop my brain wandering all over the place, I thought about a board game I like playing called Settlers of Catan.

The game (which is very good) has 19 small hexagons that are positioned to form a larger hexagon. It got me thinking about how many smaller hexagons I would need to create an even larger hexagon. I realised that the number of smaller hexagons would always be a prime number and that:

For any odd natural number n,
n squared - (n squared - 1) / 4
is always prime.

Does anyone know if this is true? And if so, is there a proof?

As said above, if this is totally wrong or just plain obvious, please feel to let me know!