At my first attempt I was tempted to calculate the number of tennis balls using the formula for regular square based pyramid having $17$ rows in this case . However, on close inspection I realized that number of balls in some of the rows are in repetition and do not correspond to those in a regular square based pyramid . Therefore, $\biggl(\frac{(n)\cdot (n+1)\cdot (n+2)}{6} \biggl)$ for $n = 17$ rows will not give the result. Any hint how to move forward?
HINT
You have two different kinds of layers here, as it alternates between centered hexagonal numbers (e.g. the top layer is the first centered hexagonal number, i.e. 1, the third layer from the top is the second centered hexagonal number, i.e. 7, etc.), and layers where the sides alternate between lengths $n$ and $n+1$ (e.g. the second layer from the top of the pyramid is a 'hexagon' with sides $1$ and $2$, the fourth layer is a hexagon with sides $2$ and $3$, etc. ... I don't know if these have a name). That is, your tennis ball pyramid consists of the following layers:
So, use or derive formulas for those two kinds of layers, and add that all up.
Some further help: For the centered hexagonal numbers the formula for the $n$-th number is $3n(n-1)-1$ ... but even more useful is the fact that the sum of the first $n$ centered hexagonal numbers is $n^3$. Here is a nice 'proof by picture' of that result:
So, the numbers of tennis balls in all the odd layers of this pyramid is $9^3=729$
For the other layers: The formula for the 'hexagon' with sides $n$ and $n+1$ is $3n^2$. Here is a picture that should help make that clear:
Since you have $8$ of those layers, you have $3$ times the sum of the first $8$ square numbers, and the general sum of the first $n$ square numbers is $\frac{n(n+1)(2n+1)}{6}$, so that gives you $3\cdot \frac{8(8+1)(2\cdot 8+1)}{6} = 612$ more tennis balls, for a total of $729+612=1341$