[peter_d_sherman]

>"Because the last one uses a multiplication instead of addition, a difference table (which is based on subtraction) won't show the pattern. The inverse of multiplication is division, so use a division table."

Well -- that looks like the problem then!

And the solution -- use a division table -- as you said.

Point is, is that the method shown in the Mathologer video -- does in fact work for every integer sequence defined by polynomials, including some of those that are non-polynomial and involve recursion -- as long as that recursion involves only addition and not multiplication...

In other words, sequence algorithms which are recursive and involve multiplication -- fail.

But then the next question must be -- is there a generalized algorithm for the construction of Mathologer's tables, which creates a subtraction table on the one hand and a divison table on the other?

Which points to the following problem/question:

Given an arbitrary integer sequence -- how can we know that the algorithm that generates it -- uses recursion and multiplication? (without knowing anything else about it -- other than the given sequence of numbers?)

?

>"I don't think you're thinking about these concepts in a useful way."

Define "useful". <g> :-)

[eesmith]

> does in fact work for every integer sequence defined by polynomials, including some of those that are non-polynomial and involve recursion

I went ahead and watched the video.

Your summary is incomplete. It works for anything which can be represented as a power series. A polynomial has a largest power. A power series can be seen as the extension of polynomials to have an infinite number of exponents, like how

   exp(x) = 1 + x + x^2/2! + x^3/3! + ... 

mentioned in the video.

This is why it can fit an exponential function.

And why it cannot fit a factorial/gamma function. (As only one of many classes of sequences it cannot fit.)

Multiplication is recursive addition. The phrase "recursive and involve multiplication" is equivalent to "recursive and involve addition".

> Define "useful". <g> :-)

Just because I can recognize that your misunderstanding of what "recursive" means is not a useful way to understand these topics doesn't mean I can describe what is useful.

I can know that the Eiffel Tower is not a fish even if I can't define what a fish is.

[peter_d_sherman]

>And why it cannot fit a factorial/gamma function. (As only one of many classes of sequences it cannot fit.)

We agree on half of this point -- that it cannot fit a factorial function.

I am (not sure/not convinced) currently (without a proof, one way or the other) on the other half of this point -- that (it would not/could not) fit a gamma function -- or some other function or functions which would allow us to differentiate the values at each point where an integer value would have existed .

If we think about the integer values in Mathologer's table -- they are created by simple subtraction, simple difference (of integers).

Now, the concept of "difference" -- not only exists for Integers -- but it also exists for functions...

In other words, f(a) = f(b) + f(c).

If you know that, and you know f(a) and f(b) -- you can solve for f(c).

Conversely, if you know f(a) and f(c) -- you can solve for f(b).

And if you know f(b) and f(c) -- you can solve for f(a).

In other words, it's just another 3-part trinity in Mathematics -- where something added to something else gives you a third thing -- well, if you know that that's what happens, and you have any two of the parts -- then you can solve for the third part.

A simple concept... yet mind-blowing in its ramifications...

Point is -- Mathologer's tables -- could be constructed with functions instead of integers -- and they could work and could work well if those functions are selected correctly to mirror the underlying geometric reality.

We can convert factorial functions into other types of functions (most notably gamma, but probably others as well) -- and if the Math is right -- then it should work...

Now, perhaps you don't get an integer sequence at the top row -- perhaps you get a series of other functions... and then you need to perform some other mathematical operations on them to get the integer sequence...

Also -- the starting point of all of this, should we desire to explore this in some depth -- would be the integer factorial sequence (as the initial integer sequence) -- without any additional polynomials or division.

Does that sequence work in Mathologer's table -- why or why not?

From there we could seek to define sequence values at specific points and in the table as functions -- and see if we could get that going correctly.

Point is -- while you might be very right, and I might be very wrong(!) -- I can't accept what you're telling me without an appropriate mathematical proof.

This is because we know without proof that Mathologer's tables are based on difference -- and that difference exists between functions as another function, and some functions are differentiable and some are not -- but those that are not differentiable can be converted into those that are, i.e. gamma function).

>"(As only one of many classes of sequences it cannot fit.)"

You moved the goalpost!

Originally, anderskaseorg said "The finite difference method of that video is only useful for finding polynomial sequences."

To recap, I tested that with the Fibonnaci Sequence (recursion with addition), and found that that viewpoint -- was not as large as it could be...

In other words, I showed that the set of possibilities was greater than what was expressed by anderskaseorg's viewpoint -- if only by a little bit.

What we're aiming for -- ultimately -- is to be able to rigorously define sequences mathematically -- which would give us clues as to how they are contructed, and answers to such questions as "what comes next".

I suggested that they might be defined in terms of functions instead of integers, and if these functions could not be mathematically manipulated one way in one form -- then it might be possible to do so if they were changed into another form.

You have shown me, over and over again, in your comments, that you wish to deal in limitations...

I have shown you, over and over again, in my comments, that I wish to deal with possibilities...

>Just because I can recognize that your misunderstanding of what "recursive" means is not a useful way to understand these topics doesn't mean I can describe what is useful.

If two wrongs don't make a right -- why do three left turns do?

[peter_d_sherman]

Maybe what we're looking to do is to convert factorial functions into logarithmic functions for our table -- and convert them back as needed.

As logarithmic functions -- difference could be computed between two of them more easily.

Maybe it wouldn't work -- but it would certainly be interesting to try...