I want to define a kind of motion– a kind of motion in which an object starts from rest and always covers 3x as much distance during the 2nd half than the first half. Let’s say you cover 16 meters in some phase of a trip. By my rule, you would have had to have covered 12 m during the first leg and 4m during the second leg.
16 (total distance) = 12 (distance in 1st half) + 4 (distance in 2nd half).
I’m interested now in breaking down the 4 meters into two equal time parts. By my rule, you would have traveled 3 meters (in the 1st quarter) and 1 meter (in the second quarter)
16 = 12 (in the second half) + 3 (in the second quarter) + 1(in the first quarter)
The question now is, “How do I break apart either the 12 into quarter times slices?”
Let’s consider some candidates:
12 = 6 + 6 ? It turns out that this can’t work because this says that the ball would travel same distance in the last quarter than it did in the previous one, and we know that it’s traveling faster later than earlier.
12 = 7 + 5?
12 = 8 + 4?
12 = 9 +3? It turns out that this can’t work because 3 is the distance it traveled in the second quarter, and we know it had to have covered more distance.
So, ignoring non-integer possibilities, we have two options. It covers 5 meters and then 7 meters, or it covers 4 meters and then 8 meters. Two examine this further let’s look at the whole patterns we’d be proposing:
Pattern 1: 1, 3, 5, 7
Pattern 2: 1, 3, 4, 8
It looks like pattern 1 looks like it is increasing by same amount each time 2 more meters for every segment; Pattern 2 is harder to discern if there is a pattern, it changes by 2, then 1, and then 4.
If we were to guess what pattern 1 would go on as, we’d might say 1, 3, 5, 7, 9, 11,…
Remember, my rule is that if you go 3x as far during the last half of a trip, then first half. Well, let’s see if my rule still holds:
1+3+5 = 9 is the first half distance and 7+9+11 = 27 is the second half distance.
Thus, I’m feeling more confident that the pattern 1, 3, 5, 7, 9, 11, 13, 15, 17 describes the motion of an object that always covers 3x more distance in the second half of a trip than the first half.
All, I’ve just described is, of course, just constant acceleration motion.
Teach. Brian. Teach. 
If you start from rest, the velocity would be different in the first second than in the subsequent seconds where it is constant… I’m not sure if that has any effect on the scenario you are trying to create, but I just noticed that it’s not completely constant acceleration. Is there a reason why you wanted to develop a scenario like this?
One reason I suspect you might think my situation is constant velocity after the 2nd step is you interpreted my sequence of numbers as a list of the successive positions at successive times. But they are not intended to be; they are a list of successive distances covered within successive time intervals. The difference between position/distance is relevant here and the difference between time and time interval as well. I was probably not clear enough in my post.
In other words, i’m saying in between 0-1 seconds it went 1 meter, then between 1-2 seconds it went an additional 3 meters, in 2-3 seconds it went an additional 5 meters, between 3-4 seconds it went an additional 7 meters, between 4-5 seconds it went an additional 9 meters, in 5-6 seconds it went an additional 11 meters.
Thus, I simply wrote the list: 1, 3, 5, 7, 9, 11
If I were to list the sequence of positions (not distances), then they would be
0, 1, 4, 9, 16, 25, 36,… which you may recognize more clearly as the sequence of squared numbers, which you may recognize more clearly as accelerated motion from rest.
Why do this? Simply because the rule “3x distance in any 2nd leg” leads us to a sequence of odd numbers, which then leads us to series of squared numbers, which we see is just accelerated motion from rest with a =2 m/s/s. To me, understanding what I’ve done and why it works is essential to understanding why accelerated motion is quadratic.