How to estimate the numbers of the integer points of $\frac{(m-1)(m-2)}{2}+k\leq N$, $a\leq\frac{k}{m}\leq b$

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How to estimate the numbers of the integer lattices of
$$\frac{(m-1)(m-2)}{2}+k\leq N$$ $$a\leq\frac{k}{m}\leq b$$ Here , m,k are the variables , b,a are selected, N can tend to the infinity, well, the result may be $$(b-a)N+O(\sqrt{N})$$