A

Step 1: Given the displacement equation in one dimension:
$$x^2 = a t^2 + 2bt + c$$
Step 2: Taking the second derivative of the displacement function will give us the acceleration as a function of time, which matches the given expression:
$$a = \frac{d^2x}{dt^2}$$
Since the displacement is quadratic in time, the acceleration is constant, which implies that the parameter related to time should yield an integer value as stated. Thus, if we set the equation appropriately considering units and dimensional analysis, we conclude with the integer relation based on the polynomial nature forming an integer exponent.
Therefore, based on the analysis, the integer value of \(n\) is determined as follows: \(n = 2\), leading us to understand the dependency accordingly.