How to find a closed form of $\displaystyle A_n=\sum_{k=1}^n\left\lfloor\varphi\sqrt k\right\rfloor$?

Question

When I was trying approximate this sum $\displaystyle A_n=\sum_{k=1}^n\left\lfloor\varphi\sqrt k\right\rfloor$, where $\varphi=\dfrac{1+\sqrt 5}{2}$ is the golden-ratio, I found that $B_n=\left\lfloor\dfrac{2\varphi\sqrt n(n+1)}{3}-\dfrac{\sqrt n(2\sqrt n+1)}{4}\right\rfloor$ is a good approximation. Can you help me proving that $A_n\simeq B_n$? Also, find a closed form of the sum $A_n$.