Does the range of $L(r)=1+\sum^\infty_{b=1}\sum^{b}_{a=0}(2\lfloor\frac{r}{\sqrt{a^2+b^2}}\rfloor)$ span all natural numbers, where $a, b$ are coprime integers and r is any real number greater than 0?
I'm not sure if my current formulation of this question is correct (hence chose not to include the summation in the title in case it's complete nonsense), but I have a formula $2\lfloor\frac{r}{\sqrt{a^2+b^2}}\rfloor+1$ given the constants $a, b, r$, and wanted to sum it over all possible values of $a$ and $b$ for a single output. I was then wondering if the range of a function of this summation with a variable $r$ spanned all natural numbers. Based on my intuition I believe it should since the floor function can be reduced to $0$ if $\sqrt{a^2+b^2}>r$ and this can be caused after an arbitrary value with the right choice of $r$, but I don't know how to approach this kind of problem.