the 1/4 does not need to be there since we are dealing with inequalities concerning integers. i believe what i originally had works correctly.
hi again,
my original answer is still correct.
we can write $\displaystyle m^2 - m + 1/4 < 2n <= m^2 + m + 1/4$ instead of $\displaystyle m^2 - m + 1/4 < 2n + 1/4 <= m^2 + m + 1/4$ since m and n are integers. that is, we only need to add 1/4 to the far left and right parts of the inequality.