The generalized form of $n$ which satisfies $(n+2)^3+(n+1)^2+(n+0)^1 = m^2$ $(n \in \mathbb{N},m \in \mathbb{N})$

Question

I'm thinking about the generalized form of $n$ which satisfies the following equation.

$(n+2)^3+(n+1)^2+(n+0)^1 = m^2$ $(n \in \mathbb{N},m \in \mathbb{N})$

From some experiments, I expect that the form of $n$ is represented as $a(a+2)$ $(a \in \mathbb{N})$. But, I do not know how I can prove my guess.

How should I do this?

Answer 1

Note that $m^2=(n+2)^3+(n+1)^2+(n+0)^1=(n+1)(n+3)^2$, hence $n+1$ must be a perfect square. since $n+1\ge 1+1=2$, if we let $n+1=(a+1)^2$, where $a\in \mathbb{N}$, we get that $n=a(a+2)$ as OP found.