Pairing function ($⟨x,y⟩$)

A pairing function is a computable bijection from ${\mathbb{N}}_0^2$ to ${\mathbb{N}}_0$ whose inverse coordinate functions are also computable. Iterating it gives computable encodings of every fixed finite tuple of nonnegative integers.

The bracket enumerates the positive integers not divisible by $10$. Every positive integer has exactly one decomposition into a power of $10$ times such an integer, so this formula is a bijection from ${\mathbb{N}}^2$ to $\mathbb{N}$. Subtracting one in the codomain includes zero without requiring a trailing-zero convention for its decimal representation. For the inverse, write $N+1=10^{i}q$ with $10\nmid q$, then $j=q-1-\lfloor q/10\rfloor$.