I reached a result concerning displacement with quantized time intervals. Am I on to something? A few days ago, I realized a similarity between distance with constant acceleration, $d = v_i t + 1/2 a t^2$, and the sum of integers up to n, $(n^2 + n)/2$. This came up again today when I decided to work out some formulas for distance and velocity with constant acceleration updated at discrete intervals, as happens in physics simulations I've programmed. 

Discretely incremented $v$ and $x$ with constant acceleration, $v_i = 0$
  
  
*
  
*Add $a\mathrm{d}t$ to $v$ every tick
  
*Add $v\mathrm{d}t$ to $x$ every tick
  
  
  $$\begin{align}
v_f &= a\mathrm{d}t + a\mathrm{d}t + a\mathrm{d}t + \cdots \\
x_f &= v_1\mathrm{d}t + v_2\mathrm{d}t + v_3\mathrm{d}t + \cdots \\
v_2 &= (a\mathrm{d}t) + (a\mathrm{d}t) = 2(a\mathrm{d}t) \\
v_n &= n(a\mathrm{d}t) \\
x_f &= (a\mathrm{d}t)\mathrm{d}t + 2(a\mathrm{d}t)\mathrm{d}t + 3(a\mathrm{d}t)\mathrm{d}t + \cdots \\
x_n &= \sum_{i=0}^{n}i(a\mathrm{d}t)\mathrm{d}t = \frac{n^2 + n}{2}(a\mathrm{d}t)\mathrm{d}t \\
n &= \left\lfloor\frac{t_\text{total}}{\mathrm{d}t}\right\rfloor \\
x_f &= \frac{1}{2}\Biggl[\biggl(\frac{t_\text{total}}{\mathrm{d}t}\biggr)^2 + \frac{t_\text{total}}{\mathrm{d}t}\Biggr](a\mathrm{d}t^2) = \biggl(\frac{t_\text{total}^2}{2\mathrm{d}t^2} + \frac{t_\text{total}}{2\mathrm{d}t}\biggr)(a\mathrm{d}t^2) \\
x_f &= a\mathrm{d}t^2\times\frac{1}{2}\times\frac{t_\text{total}^2}{\mathrm{d}t^2} + \frac{1}{2}a\mathrm{d}t\frac{t_\text{total}}{\mathrm{d}t} \\
x_f &= \frac{1}{2}at_\text{total}^2 + a\mathrm{d}t\frac{t_\text{total}}{2} \\
\lim_{\mathrm{d}t\to 0} &= \frac{1}{2}at^2 = \text{normal $v_f$ equation where $v_i = 0$ and $\Delta a = 0$}
\end{align}$$

I'd like to know what relations to theory this or an idea like it has if any, and if it touches on anything else. 
 A: The result you've got would be better known as this:
$$\int_0^t\biggl(\int_0^{t'} a\mathrm{d}t''\biggr)\mathrm{d}t' = \frac{1}{2}at^2$$
In other words, it's a derivation of the formula for uniformly accelerated motion. This derivation, or something like it, is one of the first things students in a good calculus-based introductory physics class learn.
The only difference is that you've done it explicitly, using limits, rather than using the rules for integrating polynomials. That's a good thing! It will help you understand where the formula comes from and what it means, and if you continue to do more with numerical integration (as in your simulations), it's going to be very useful to know the details of how this stuff works.
Now, considering that this has been known for about 350 years, its applications have been pretty thoroughly explored. It's a part of classical kinematics, which is a branch of physics that analyzes simple motion without any quantum effects, so there is no special significance to the Planck time with respect to this equation.
