triangular number counting function

For a given nonnegative number $x$, the triangular number counting function counts how many triangular numbers are not greater than $x$. The formula is simple:

$$\lfloor \frac{-1+\sqrt{8x+1}}{2}\rfloor ,$$

in sharp contrast to the lack of a formula for the prime counting function $\pi (x)$. If one accepts 0 as a triangular number, the formula easily accommodates this by the mere addition of 1 after flooring the fraction.