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This curious phenomenom requires an example first.Consider U(-2,3,-5)7 as representative of all U(a,b,c)F.
What is the relationship between this non-zero series, and the following pairs of zero series?
- U(0,1)7 - the primus of 7 - and U(25,20)7
- U(2,7)7 - the secundus of 7 - and U(-3,2)7
- U(1,1)7 - the tertius of 7 - and U(-1,19)7
- U(-1,1)7 - the quartus of 7 - and U(5,15)7
| U(-2,3,-5)7 | Primus 7 doubled | U(25,20)7 doubled | Secundus 7 doubled | U(-3,2)7 doubled | Tertius 7 doubled | U(-1,19)7 doubled | Quartus 7 doubled | U(5,15)7 doubled | |||||||||||||||||
| U-9 | -112287722 | = | -329 | * | 341300 | + | -22 | = | 2207 | * | -50878 | + | 24 | = | 1926 | * | -58301 | + | 4 | = | -2584 | * | 43455 | + | -2 |
| U-8 | -16382557 | = | -329 | * | 49795 | + | -2 | = | 2207 | * | -7423 | + | 4 | = | 1926 | * | -8506 | + | -1 | = | -2584 | * | 6340 | + | 3 |
| U-7 | -2390182 | = | -48 | * | 49795 | + | -22 | = | 322 | * | -7423 | + | 24 | = | 281 | * | -8506 | + | 4 | = | -377 | * | 6340 | + | -2 |
| U-6 | -348722 | = | -48 | * | 7265 | + | -2 | = | 322 | * | -1083 | + | 4 | = | 281 | * | -1241 | + | -1 | = | -377 | * | 925 | + | 3 |
| U-5 | -50877 | = | -7 | * | 7265 | + | -22 | = | 47 | * | -1083 | + | 24 | = | 41 | * | -1241 | + | 4 | = | -55 | * | 925 | + | -2 |
| U-4 | -7422 | = | -7 | * | 1060 | + | -2 | = | 47 | * | -158 | + | 4 | = | 41 | * | -181 | + | -1 | = | -55 | * | 135 | + | 3 |
| U-3 | -1082 | = | -1 | * | 1060 | + | -22 | = | 7 | * | -158 | + | 24 | = | 6 | * | -181 | + | 4 | = | -8 | * | 135 | + | -2 |
| U-2 | -157 | = | -1 | * | 155 | + | -2 | = | 7 | * | -23 | + | 4 | = | 6 | * | -26 | + | -1 | = | -8 | * | 20 | + | 3 |
| U-1 | -22 | = | 0 | * | 155 | + | -22 | = | 2 | * | -23 | + | 24 | = | 1 | * | -26 | + | 4 | = | -1 | * | 20 | + | -2 |
| U0 | -2 | = | 0 | * | 25 | + | -2 | = | 2 | * | -3 | + | 4 | = | 1 | * | -1 | + | -1 | = | -1 | * | 5 | + | 3 |
| U1 | 3 | = | 1 | * | 25 | + | -22 | = | 7 | * | -3 | + | 24 | = | 1 | * | -1 | + | 4 | = | 1 | * | 5 | + | -2 |
| U2 | 18 | = | 1 | * | 20 | + | -2 | = | 7 | * | 2 | + | 4 | = | 1 | * | 19 | + | -1 | = | 1 | * | 15 | + | 3 |
| U3 | 118 | = | 7 | * | 20 | + | -22 | = | 47 | * | 2 | + | 24 | = | 6 | * | 19 | + | 4 | = | 8 | * | 15 | + | -2 |
| U4 | 803 | = | 7 | * | 115 | + | -2 | = | 47 | * | 17 | + | 4 | = | 6 | * | 134 | + | -1 | = | 8 | * | 100 | + | 3 |
| U5 | 5498 | = | 48 | * | 115 | + | -22 | = | 322 | * | 17 | + | 24 | = | 41 | * | 134 | + | 4 | = | 55 | * | 100 | + | -2 |
| U6 | 37678 | = | 48 | * | 785 | + | -2 | = | 322 | * | 117 | + | 4 | = | 41 | * | 919 | + | -1 | = | 55 | * | 685 | + | 3 |
| U7 | 258243 | = | 329 | * | 785 | + | -22 | = | 2207 | * | 117 | + | 24 | = | 281 | * | 919 | + | 4 | = | 377 | * | 685 | + | -2 |
| U8 | 1770018 | = | 329 | * | 5380 | + | -2 | = | 2207 | * | 802 | + | 4 | = | 281 | * | 6299 | + | -1 | = | 377 | * | 4695 | + | 3 |
| U9 | 12131878 | = | 2255 | * | 5380 | + | -22 | = | 15127 | * | 802 | + | 24 | = | 1926 | * | 6299 | + | 4 | = | 2584 | * | 4695 | + | -2 |
The curious thing is that a non-zero series is split, in four different ways, in a peculiar 'product' of two zero series, called the operator, which is always one of the fabfour, and the output series.
In each split there are two constants involved, but they are outside the multiplication with the factor F, whereas the constant '-5' in the original series operates inside the multiplication.
Compare U(-2,3,-5)7 term by term with U(-2,3)7 to see its influence.
Note that the constant slows down the series in one direction, while accelerating it in the other.
| U(-2,3)7 | -37677 | -5497 | -802 | -117 | -17 | -2 | 3 | 23 | 158 | 1083 | 7423 | 50878 |
| U(-2,3,-5)7 | -50877 | -7422 | -1082 | -157 | -22 | -2 | 3 | 18 | 118 | 803 | 5498 | 37678 |
Yet this deviation created by the constant is somehow 'overcome' in the above splits.
Is this a freak example?
Unfortunately not. I can split any series U(a,b,c)F in the above four ways, using the fabfour as operators. So can you once you know the right way to put the primus-, secundus-, tertius- and quartus-splitvalues into the 'core' of the splittable, formed by the four rows U0 - U3.
Splitvalues?
Sixteen in total - four of them for each of the fabfour. They were not arrived at by any logic or reasoning. They were arrived at by painstakingly digging through multitudes of examples, in part found randomly in the course of other investigations, in part created for the very purpose of finding the patterns among the constantly shifting parameters.
It took me a couple of months in which I dug and kept digging, and calcguessing, and eventually these values emerged - they are what they are, what they were before God opened his eyes, what they will be after he shuts his mouth (which doesn't appear to be any time soon).
| U(a,b,c)F | Primus splitvalues | Secundus splitvalues | Tertius splitvalues | Quartus splitvalues |
| [1] | -Fa+2b-c | a+ c/(F-2) | (a+b)+ 2c/(F-2) | -(a-b) |
| [2] | Fb-2a+c | b+ c/(F-2) | Fb-(a-b)+ Fc/(F-2) | Fb-(a+b)+c |
| [3] | a | -a- 2c/(F-2) | -b- 2c/(F-2) | b |
| [4] | Fa-b+c | -Fa+b- Fc/(F-2) | -a- 2c/(F-2) | a |
I don't know how or why they work. They are not derived from anything, nor does the operation they make possible relate to anything known in elementary number theory. An enigma.
How does it work?
Consider the example. For U(-2,3,-5)7 the splitvalues are:
| U(-2,3,-5)7 | Primus splitvalues | Secundus splitvalues | Tertius splitvalues | Quartus splitvalues |
| [1] | 25 | -3 | -1 | 5 |
| [2] | 20 | 2 | 19 | 15 |
| [3] | -2 | 4 | -1 | 3 |
| [4] | -22 | 24 | 4 | -2 |
That wasn't too hard. Now place these values in exactly the above positions in the core of the splittable - the rows U0-U3 - and develop the series involved accordingly. That would be all, thank you.
The equalities will keep on track ad infinitum regardless of the choice of a, b, c and F.
What if (F-2) doesn't divide 'c'?
Then you will encounter fractions in the secundus' en tertius' splitvalues and consequently in the output series. Secundus- en tertius-based splits therefore aren't closed but jump into the U(p/r, q/r)F domain. It doesn't affect the equalities - they will still keep on track ad infinitum.
What if F=2?
Then, unless c=0, there are no secundus- and tertius-based splits possible. U(2,2)2, the secundus of 2, is a series of 'two's' while U(1,1)2, the tertius of 2, is a series of 'one's'.
What if F=2 and c=0?
Then the train remains on track: zero series can be split for F=2.