Simplifying a complicated system of equations involving $\min$.

Question

$o_n = 2n + 1$

$e_n = (2n)^2$

$r_i = \mathop{\min} \{ n \mid e_n \ge i\}$

$a_n = n + e_n$

$q_i = \left\lfloor \frac{i - a_{r_i}}{o_{r_i}} \right\rfloor \mod 4$

Is there another way to write $q_i$ without having to compute an $\min$?

Answer 1

$r_i$ can be rewritten as $r_i = \left\lceil \frac {\sqrt i} 2 \right\rceil$

This removes the need to compute the minimum.