Let $t$ and $r$ be two integers with $r\geq 1, t\geq \frac{r}{2}$. Put $ f(r,t)=\lfloor 2(t^2+r)^{\frac{3}{2}}-(2t^3+3rt) \rfloor $
(here $\lfloor x \rfloor$ denotes the floor of $x$, i.e. the largest integer below $x$). Thus $f(1,.)$ is identically zero, the first two values of $f(2,.)$ are $2$ and $1$, followed by zeroes. It is easy to see that $f(r,t)=0$ whenever $t\geq \frac{3}{4}r^2$. Is it true that $f(r,.)$ is decreasing (as a function of $t$) for every $r \geq 1$ ?