Let $a\in \mathbb{Z}^{n}$ and let infinity norm, defined as $ \|a\|_{\infty}= \max_{1\leq i\leq n}|a_{i}|$. I would like to know if there is an explicit formula to calculate the number of elements of the following set:
$$B_{k}=\{a\in \mathbb{Z}^{n}: \|a\|_{\infty}=k \}$$
where $k$ is a non-negative integer.
On
The norm infinity sphere is a cube. If the first number is equal to $k$ in absolute value, and all others are less than $k,$ then the number of options is $(2k-1)^{n-1}.$ since the same estimate works for the second number being equal to $k,$ that gives us $n (2k-1)^{n-1}$ points. If some two numbers are equal to $k$ we have $\binom{n}{2}(2k-1)^{n-2}$ points, and so on. To give us finally
$$\sum_{i=1}^n \binom{n}{i} (2k-1)^{n-i}.$$ Notice that the sum has a closed form thanks to the binomial theorem.
Unfortunately, my result is another. We have \begin{align} S_k&=\{x\in\mathbb Z^n :\|x\|=k\}\\ &=\{x\in\mathbb Z^n :\|x\|\leq k\}\setminus\{x\in\mathbb Z^n :\|x\|<k\}\\ &=\underbrace{\{x\in\mathbb Z^n :\|x\|\leq k\}}_{Q_k:=}\setminus\underbrace{\{x\in\mathbb Z^n :\|x\|\leq k-1\}}_{Q_{k-1}:=}\\ &=\{-k, -k+1, \ldots, k-1, k\}^n\setminus\{-k+1,-k+2 \ldots, k-2, k-1\}^n \end{align} Since $Q_k\supset Q_{k-1}$, we get $$|S_k|=|Q_k\setminus Q_{k-1}|=|Q_k|-|Q_{k-1}|=(2k+1)^n-(2k-1)^n.$$