# B Do numbers derived from successive differences have a name?

1. Dec 14, 2017

### ktoz

Hi

I was playing around with various sets of numbers while thinking about optimizing neural networks and was wondering if numbers derived as below have a specific name

given a set: [7, 5, 3, 8, 1, 5]
the difference set is [-2, -2, 5, -7, 4]
the difference set for that is [0, 7, -12, 11]
the difference set for that is [7, -19, 23]
the difference set for that is [-26, 42]
the difference set for that is [68]

What is the mathematical term/name for '68' in this construct? I'm calling it the 'root differential' , but was sure somebody has already come up with one.

As a sort of side note, I found that the count of these roots can be found with the following

For integer sets of width W > 0 and max value W, there are a finite number of unique 'roots' which can be found by: roots = (w - 1) * 2^(w - 1) + 1 (Cullen numbers)

For integer sets of width W > 0 and max value M there are a finite number of unique 'roots' which can be found by: roots = (w - 1) * 2^(M - 1) + 1 (Proth numbers)

I'm mostly interested in the correct name for my 'root differentials'.

Thanks for any help

2. Dec 14, 2017

### .Scott

I played around with those exact sets when I was a teenager. To my knowledge, they have no name.
The difference are like discrete derivatives. And I could use them to deduce polynomials to fit the original series.

For examples:

N^3 series:
0 1 8 27 64 125
1 7 19 37 61
6 12 18 24
6 6 6
0 0
0
Taking only the first column: 0 1 6 6 0 0

N^4 series:
0 1 16 81 256 625
1 15 65 175 369
14 50 110 194
36 60 84
24 24
0
Taking only the first column: 0 1 14 36 24 0

So if you have (N^4)/3 + (N^3) you will end up with this first column:
(0 1 14 36 24 0)/2 + 0 1 6 6 0 0
which will be: ...12 0
compare N^4: ... 24 0
So subtract out the N^4/2 part and you get:
0 1 6 6 0 0 (the N^3) term.

3. Dec 14, 2017

### Staff: Mentor

Not sure about the English term. In my language it's called (iterated) sequence of differences. Although there is a connection to derivatives and especially differential equations, I think the term root differential is problematic. Firstly, the depth and thus the root depend in general on the given number of elements in the original sequence and secondly, it is not really a differential. It's an iterated difference.

4. Dec 14, 2017

### ktoz

I've seen elsewhere the term "prime gaps' to describe the set of differences between successive primes, so perhaps 'root gap' would avoid any confusion with current usage of 'differential'.

5. Dec 14, 2017

### ktoz

Another sort of interesting thing about power series is that if you go the other way, you end up with Euler triangular numbers followed by difference of zero for every value V > P
for example:

Powers of 3
1 8 27 64 125 216
1 7 19 37 61 91
1 6 12 18 24 30
1 5 6 6 6 6
1 4 1 0 0 0

Powers of 4
1 16 81 256 625 1296
1 15 65 175 369 671
1 14 50 110 194 302
1 13 36 60 84 108
1 12 24 24 24 24
1 11 12 0 0 0

Powers of 5
1 32 243 1024 3125 7776
1 31 211 781 2101 4651
1 30 180 570 1320 2550
1 29 150 390 750 1230
1 28 121 240 360 480
1 27 94 119 120 120
1 26 67 25 1 0

Never bothered to take it any further, but it seemed interesting that no matter how large successive powers get, it always reduces to a difference of zero for n > p

6. Dec 14, 2017

### .Scott

I did the power of 3 and 4 in my previous post.
All you are doing here is forcing that first digit to 1. If you don't do that, it looks like this.
1 8 27 64 125 216
7 19 37 61 91
6 12 18 24
6 6 6 6
0 0 0

If you extend it, it looks like this:
-8 -1 0 1 8 27 64 125 216 343
7 1 1 7 19 37 61 91 127
-6 0 6 12 18 24 30 36
6 6 6 6 6 6 6
0 0 0 0 0 0

Notice that the 4th row should always be all sixes - and that the 5th an following rows are always all zeros.
Your 1 5 6 6 6 ... is just an artifact of forcing column 1 to all ones.

And if you do that, you get:
1 0 ...
1 -1 0 ...
1 -2 1 0 ...
1 -3 3 -1 0 ...
1 -4 6 -4 1 0 ...

7. Dec 14, 2017

### ktoz

So you did. Oops.
I “discovered” that thing about powers years ago and forget now why I was adding the “1” to each line, but I’m sure it was an excellent reason.
: )

8. Dec 17, 2017

### Stephen Tashi

The calculations you illustrate are part of the subject matter of The Calculus Of Finite Differences.

If we take successive differences of a series, we may or may not eventually reach a difference that is constant. In the event we reach a difference that is constant, then we can find a closed form expression for summing the series. This is analogous to the situation in (ordinary) Calculus where if we know a higher derivative of a function is a constant then we can express the function as a polynomial.

9. Jan 29, 2018

### chdia

1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 6227020800 87178291200 1307674368000 20922789888000 355687428096000 6402373705728000 121645100408832000 2432902008176640000 51090942171709440000...

1*2=2, 2*3=6, 6*4=24, 24*5=120, 120*6=720, ...

Does this sequence itself flatten out to a zero difference? I have always wondered this.