We have a sequence of $\lfloor\alpha n\rfloor$, where $\alpha$ is a given irrational. I'm interested in closed-forms for $$ \sum_{n=1}^N\lfloor\alpha n\rfloor, \sum_{n=1}^Nn\lfloor\alpha n\rfloor, \mbox{and} \sum_{n=1}^N(\lfloor\alpha n\rfloor)^2. $$ Thanks.
EDIT: Recurrence relations would be good enough too, if they can be quickly evaluated for large $N$s, say on the order of $10^{15}$. I think I found such a recurrence for the first sum, but the latter two still stand unsolved.
EDIT2: To be precise, I need these formulas for one particular $\alpha = 1/\varphi$, which makes the sequence Hofstadter G-sequence, but I doubt it will help - there should be a universal approach for any $\alpha$.
