I'm not sure how much we've looked into it, but I was thinking about the jumps left and right. What we essentially do is increase, or decrease d for each jump. But we're also modifying a.
d = a + x and when we do our jumps, the x is locked. So I was wondering how it would look if we locked a instead.
It doesn't jump up or down in e, but instead it jumps up and down between n. So to jump one n up, you decrease d and x by 1. To jump one n down, you increase d and x by one.
Moving up would work like this:
d = d - 1
e = c - d^2
x = x - 1
Moving down would work like this:
d = d + 1
e = c - d^2
x = x + 1
From those values you can create a new record that holds a=a, b=b. I'm not sure if it's of any direct use yet. d - (n - 1) will, of course, put a, b at n=1, but if you have to iterate over these values, then it is of little use. Maybe someone else finds it interesting / useful. Maybe there are some shortcuts available too.
The part that intrigues me is that when we jump up or down like this, both (1, c) and (a, b) will end up on the same e's. Unlike our jumping left and right since then our n values will modify the length we jump.
So think of n as our marker/center. Jumping left and right moves left of that n, right moves right of that n. Jumping up moves the n up and jumping down moves the n down.
Jumping left decreases a, b but locks n. Jump right increases a, b but locks n.
Jumping up locks a, b but increases n. Jumping down locks a, b but decreases n.
Example time:
a = 7, b = 37, c = 259
The cell is: {3:6:16:9:7:37}
Say we want to move up. We then do
d = 16 - 1 = 15,
e = 259 - 15*15 = 34
x = 9 - 1 = 8
This will result in the cell:
{34:7:15:8:7:37}
Now with (1, c) the cell is: {3:114:16:15:1:259}
Then we do:
d = 16 - 1 = 15
e = 259 - 15*15 = 34
x = 15 - 1 = 14
This will result in the cell:
{34:115:15:14:1:259}
Now say we want to move down (decrease n):
The cell is: {3:6:16:9:7:37}
Say we want to move up. We then do
d = 16 + 1 = 17,
e = 259 - 17*17 = -30
x = 9 + 1 = 10
This will result in the cell:
{-30:5:17:10:7:37}
Now with (1, c) the cell is: {3:114:16:15:1:259}
Then we do:
d = 16 + 1 = 17
e = 259 - 17*17 = -30
x = 15 + 1 = 16
This will result in the cell:
{-30:113:17:16:1:259}
Note:
n will change (increase or decrease) as to balance out, since x is either increasing or decreasing. The easiest way to rationalize this is the b equation:
b = a + 2x + 2n
In order to keep b = 37 while decreasing x, n has to increase That is for moving up. For moving down, x increases and n decreases.