Function not making sense in a Subsequences question?

Question

I'm studying the Subsequences and I've stumbled across a question which I can't seem to make any sense of?

I don't understand how they got $(0,1,2,0,1,2,\ldots)$? I don't even understand how to use the function or what it means? What does {$0 \text{ if } n= 3k − 2$} even mean?

Let $\left(a_n\right)_{n\in\mathbb{N}}$ be defined as $$a_n=\cases{0,\quad \text{if } n=3k-2 &for some $k\in\mathbb{N}$,\\1,\quad \text{if } n=3k-1 &for some $k\in\mathbb{N}$,\\2,\quad \text{if } n=3k &for some $k\in\mathbb{N}$,\\}$$ That is, $$\left(a_n\right)_{n\in\mathbb{N}}=(0,1,2,0,1,2,0,1,2,0,1,2,\ldots)$$ The accumulation points of $\left(a_n\right)_{n\in\mathbb{N}}$ are $0$, $1$, and $2$.

Answer 1

$$a_1 = 0$$ since there exists an natural number $k$, namely $k = 1$, such that $1 = 3 \cdot 1 - 2$.

$$a_2 = 1$$ since there exists an natural number $k$, namely $k = 1$, such that $2 = 3 \cdot 1 - 1$.

$$a_3 = 2$$ since there exists an natural number $k$, namely $k = 1$, such that $3 = 3 \cdot 1$.

$$a_4 = 0$$ since there exists an natural number $k$, namely $k = 2$, such that $4 = 3 \cdot 2 -2$.

I'm sure you can continue the pattern.

Answer 2

Your sequence depends on $n$. So basically the definition of the sequence means: if the remainder when $n$ is divided by 3 is 1, then the element of the sequence is 0, if the remainder is 2, then the element is 1, and if the remainder is 0, then the element is 2. Since this happens in a periodic manner, we clearly have $(0,1,2,0,1,2,\ldots)$.

Answer 3

For $k=1$, $3k-2 =1$, so $f(1)=0$, similarly, $3k-1=2$, so $f(2)=1$ and $3k=3$ so $f(3)=2$

Now we get the same pattern of $$\{0,1,2,0,1,2,...\}$$ for other values of $k$