I've been thinking about this work
Great find. I found another way to visualize this. If you notice with each successive c for this you are adding up another odd number, much like constructing squares. Think of the value t which is always a+b-1 [think of ab like a rectangle and count the squares on 2 adjacent sides. side a = a, side b = b and they share one so -1]. This is always odd if a and b are the same parity. Since it's odd, we can represent it as 2n-1. If we solve for t=2i-1 we get i = (d+n). Then if we take this value t and remove it, then do the same until we have a one dimensional line, the length of this line is equal to 2(x+n)+1.
Example: 5*29 = 145
t = 5 + 29 - 1 = 33
33 = 2*i - 1 =34 = 2*i => i=17 = (d+n)
5*29 = 145
145 - t = 145 - 33 = 112
4*28 = 112
t = 4 + 28 - 1 = 33 - 2 = 31
112 - t = 112 - 31 = 81
3*27 = 81
t = 2 + 27 - 1 = 31 - 2 = 29
81 - t = 81 - 29 = 52
2*26 = 52
t = 2 + 26 - 1 = 29 - 2 = 27
52 - 27 = 25
1*25 = 25
2*j+1 = 25
2j = 24
j = 12
17^2 - 12^2 = 145