0

Prove that the number of $\mathbb{Z}^n$ vectors that their Taxicab norm is not greater than $m$ is equal to the number of $\mathbb{Z}^m$ vectors that their Taxicab norm is not greater than $n$

Let's define a recursive sequence $a(n_1,n_2)$ to count the number of $\mathbb{Z}^{n_2}$ vectors that their Taxicab norm is not greater than $n_1$ If we can find the recursive relation then the rest of it is straightforward. Is there anyone with an idea for $a(n_1,n_2)$?

user15550465
  • 19
  • The number of such vectors In $\Bbb{R}^{n_2}$ will be uncountably infinite. So no point in counting the number of vectors. The only way this might be shown is by establishing a bijection. – QED May 10 '21 at 16:41
  • Is this what you are talking about? https://math.stackexchange.com/questions/2743313/prove-that-the-number-of-answers-for-a-1a-2-a-k-le-n-is-equal-to-t/2744717#2744717 – Mike Earnest May 10 '21 at 22:25

0 Answers0