Example 1 If a natural number $K$ can be expressed as
$$
\begin{aligned}
K & =V(a, b, c) \\
& =a^{3}+b^{3}+c^{3}-3 a b c \quad (a, b, c \in \mathbf{N})
\end{aligned}
$$

then $K$ is called a "Water Cube Number".
Now, arrange all different Water Cube Numbers in ascending order to form a sequence $\left\{K_{n}\right\}$, called the Water Cube Number sequence, and let $K_{0}=0$.
(1) Find $n$ such that $K_{n}=2016$;
(2) Determine $K_{2016}$ and $S_{2016}=\sum_{n=1}^{2016} K_{n}$.
This problem is the first part of a question from the 2016 China Southeast Mathematical Olympiad, where the Water Cube Number refers to the "watered-down" version of cube numbers.
【Analysis】Let $M=\{3 x \mid x \in \mathbf{N},(3, x)=1\}$.
Lemma A natural number $k$ is a water cube number if and only if
$k \in \mathbf{N} \backslash M$.
Proof Note that,
$$
\begin{array}{l}
V(a, b, c)=a^{3}+b^{3}+c^{3}-3 a b c \\
=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right) \\
=(a+b+c)^{3}-3(a+b+c)(a b+b c+c a) .
\end{array}
$$

Then when $3 \mid (a+b+c)$, $3^{2} \mid V(a, b, c)$;
and when $3 \nmid (a+b+c)$, $3 \nmid V(a, b, c)$.
Therefore, each number in the set $M$ is not a water cube number.
Next, we prove: Each natural number outside the set $M$ is a water cube number.
When $k \in \mathbf{N} \backslash M$:
If $k=0$, then $V(1,1,1)=0$;
If $k>0$, then
$$
k \in\{3 n+1,3 n+2,9(n+1) \mid k \in \mathbf{N}\} \text {, }
$$

Since for each natural number $n$, we have
$$
\begin{array}{l}
V(n, n, n+1)=3 n+1, \\
V(n, n+1, n+1)=3 n+2, \\
V(n, n+1, n+2)=9(n+1),
\end{array}
$$

Thus, $k$ is a water cube number.
The lemma is proved.
According to the lemma, we know that if a positive integer $k$ is not a water cube number, if and only if $k$ has the form $9 n+3$ or $9 n+6$, where $n$ is any natural number, i.e., in every nine consecutive positive integers, exactly seven are water cube numbers, and the other two are not water cube numbers.

Now list all positive integers in sequence, and divide them into segments of nine numbers each, from left to right:
$$
\begin{array}{l}
1,2, \cdots, 9 ; 10,11, \cdots, 18 ; \cdots ; \\
9 m-8,9 m-7, \cdots, 9 m ; \cdots
\end{array}
$$

In the first $m$ segments, each segment has exactly seven water cube numbers, and $9 m$ is the largest water cube number among them, then
$$
K_{7 m}=9 m \text {. }
$$
(1) According to equation (1),
$$
\begin{array}{l}
2016=9 \times 224=K_{7 \times 224}=K_{1568} \\
\Rightarrow n=1568 . \\
\text { Hence } S_{7 m}=\sum_{n=1}^{7 m} K_{n} \\
=\sum_{n=1}^{9 m} n-\sum_{n=0}^{m-1}(9 n+3)-\sum_{n=0}^{m-1}(9 n+6) \\
=\sum_{n=1}^{9 m} n-\sum_{n=0}^{m-1}(18 n+9) \\
=\frac{9 m(9 m+1)}{2}-9 \sum_{n=0}^{m-1}(2 n+1) \\
=\frac{9 m(9 m+1)}{2}-9 m^{2} \\
=\frac{9 m(7 m+1)}{2} .
\end{array}
$$
(2) Since 2016 is a multiple of 7 and 9, then $m=288$.
Thus, $K_{2016}=K_{7 \times 288}=9 \times 288=2592$,
$$
S_{2016}=\frac{9 m(7 m+1)}{2}=2614032 \text {. }
$$