Exercise 12.

In their down time, ghost pirates enjoy stacking cannonballs in triangular based pyramids (aka, tetrahedrons), like those pictured here:

$A single shaded circle (meant to represent a cannonball)$

$Four overlapping circles, drawn to represent cannonballs stacked with a layer of three in a triangle with a single cannonball resting on top.$

$Overlapping circles drawn to represent a three-dimensional tetrahedron of balls consisting of a triangle of 6 balls supporting a triangle of 3, with a single ball balanced on top.$

Note, these are solid tetrahedrons, so there will be some cannonballs obscured from view (the picture on the right has one cannonball in the back not shown in the picture, for example)

The pirates wonder how many cannonballs would be required to build a pyramid 15 layers high (thus breaking the world cannonball stacking record). Can you help?

Let $P(n)$ denote the number of cannonballs needed to create a pyramid $n$ layers high. So $P(1) = 1\text{,}$ $P(2) = 4\text{,}$ and so on. Calculate $P(3)\text{,}$ $P(4)$ and $P(5)\text{.}$
Use polynomial fitting to find a closed formula for $P(n)\text{.}$ Show your work.
Answer the pirate's question: how many cannonballs do they need to make a pyramid 15 layers high?
Bonus: Locate this sequence in Pascal's triangle. Why does that make sense?