User blog comment:Plain'N'Simple/Using triangular numbers to create a Conway-Arrow Level Notation (probably not what you'd expect)/@comment-35434595-20191025155632/@comment-35470197-20191026235445

> I wasn't aware this was a standard function in the mathematical repertoire.

I note that the name "pair function" is used to several functions. In a course of set theory (or arithmetic), a teacher tell us at most one of them (because it suffices to know one) and might not tell an explicit one because it is possible to directly construct bijections N→Z and Z→Q associated to one.

(Although I guess that you already understand it,) the role of the pair function in arithmetic is the same as the role of the Hausdorff's pair formulation (x,y) = {{x},{x,y}} in set theory. It enables us to quantify "2-dimensional variables" in arithemetic and set theory, in which we are allowed only to use (1-dimensional) variables.

> I was really surprised to find how simple the final formula turned out.

I did not know that "the number immediately before n in the grid" can be expressed in that simple way. It is actually a novel discovery for me.