Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$ We assume $k = 2$. We assume in this question that $(a_1, a_2) = (2, 3)$. Construct a function $\phi : \mathbb{Z} \rightarrow \mathbb{Z}$ of period 6 such that $P(n) = \frac{n + \phi(n)}{6}$ for all $n \in \mathbb{N}$.
Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$:
$$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$
We assume $k = 2$. We assume in this question that $(a_1, a_2) = (2, 3)$. Construct a function $\phi : \mathbb{Z} \rightarrow \mathbb{Z}$ of period 6 such that $P(n) = \frac{n + \phi(n)}{6}$ for all $n \in \mathbb{N}$.