When showing that $\mathbb{N}\times\mathbb{N}$ is in bijection with $\mathbb{N}$, it seems standard to give a proof by picture that shows a way to systematically weave through all the points in $\mathbb{N}\times\mathbb{N}$ and label each one as you go.
I know there is a polynomial expression for this method, given by $ J(m,n)=[1+2+\cdots+(m+n)]+m=\frac{1}{2}[(m+n)^2+3m+n] $ where $m$ is the usual $x$-coordinate and $n$ the usual $y$-coordinate.
But how does one "see" how this formula is arrived at? I know how to manipulate the middle expression to arrive at the rightmost expression, but how does the middle expression relate to the weaving pattern through $\mathbb{N}\times\mathbb{N}$? Thank you.