Define $f(n, k)$ as the number of ways to distribute $k$ chocolates to $n$ children such that each child receives 0, 1, or 2 chocolates. For example, $f(3,4)=6$, $f(3,6)=1$, and $f(3,7)=0$.

(a) List all 6 ways to distribute 4 chocolates to 3 children with each one receiving at most two chocolates.

(b) Considering 2015 children, verify that $f(2015, k)=0$ for all $k$ greater than or equal to an appropriate value.

(c) Show that the equation

$$
f(2016, k)=f(2015, k)+f(2015, k-1)+f(2015, k-2)
$$

is true for all positive integers $k$ greater than or equal to 2.

(d) Calculate the value of the expression

$$
f(2016,1)+f(2016,4)+f(2016,7)+\ldots+f(2016,4027)+f(2016,4030)
$$