[peter_d_sherman]

>"My fascination with these sequences began in 1964 when I was a graduate student at Cornell University in Ithaca, NY, studying neural networks. I had encountered a sequence of numbers, 1, 8, 78, 944, 13800, . . ., and I badly needed a formula for the n-th term, in order to determine the rate of growth of the terms..."

Related Mathologer video:

Mathologer - "Why don't they teach Newton's calculus of 'What comes next?'"

https://www.youtube.com/watch?v=4AuV93LOPcE

[anderskaseorg]

The finite difference method of that video is only useful for finding polynomial sequences. Of course, any finite sequence can be extended to some polynomial, but in many cases (such as this one) that’s not the result you’re looking for.

[OscarCunningham]

https://johndonleyva.tripod.com/DifferenceTables.htm

> Robert Jackson suggests that if you've completed a difference table and still don't understand the sequence, you should turn the paper through an angle of 60 degrees, say, and start again and perhaps repeat this several times to make a fan of difference tables.

[peter_d_sherman]

Specifically, in this case, why isn't it?

[eesmith]

Because this sequences isn't polynomial. It's https://oeis.org/A000435 , with the explicit formula

   a(n) = (n-1)! * Sum_{k=0..n-2} n^k/k!

and the approximate form shows it's grows roughly as n^n:

   a(n) ~ sqrt(Pi/2)*n^(n-1/2)

Here's my Python implementation:

  from math import factorial
  from fractions import Fraction as F

  def A000435(n):
    return int(factorial(n-1) *
         sum(F(n**k, factorial(k)) for k in range(0, n-1)))

The video you linked to is on OEIS at https://oeis.org/A000127 and is a quartic:

  def A000127(n):
    return (n**4 - 6*n**3 + 23*n**2 - 18*n + 24)//24

[peter_d_sherman]

OK, I think I understand what you and anderskaseorg mean by polynomial/non-polynomial sequences...

If we think about a polynomial, say 3x^2 + 2x + 1 -- then that's basically an algorithm that says "take x, raise it to the second power, muliply it by 3, take the result of that, add it to x multiplied by 2, and then take the result of that, and add one to it".

In other words, in that algorithmic definition,

a) There is no recursion

(note that factorials imply recursion in an algorithm -- even though they could be computed by using a simple look-up table)

and,

b) There is no division

(which could result in non-integer values)

So, in the formulas you give, you are using both recursion and division to form your sequences.

OK, so number tables / triangles / fans (call them what you will) -- don't work for things like that.

I am willing to buy into that, prima facie, but "with the proverbial grain of salt"...

You see, there's something deeper about math -- that we're not understanding here...

To understand what it is or may be (I don't know what it is, all that follows is mathematically speculative reasoning, and might be wrong, might be quite wrong indeed!), then I would suggest the following:

First, consider the Fibonnaci Sequence: https://oeis.org/A000045

Why?

Because this is the simplest (AFAIK) recurrence relationship (AKA recursive, "defined using recursion") integer sequence -- that can be produced.

To recap, its definition is:

F(n) = F(n-1) + F(n-2)

(with F(0) = 0 and F(1) = 1)

Now let's create that integer sequence -- and a corresponding number table / difference table / triangle / fan (again, call it what you will) -- and let's see if that works...

Now, I don't have Python all set up to do this -- all I have is pen and paper.

But I tried it -- and lo and behold, it works!

What's very interesting about the Fibonnaci Sequence -- is that if you create a number table for it -- you'll see that it repeats (although each row is shifted to the right!) in descending rows!

In other words, that number table -- if we can spot that pattern -- is in fact showing us the recurrence/recursive relationship!

In other words, it's still working(!) -- for this simple recurrence/recursive formula!

But we know that it fails -- somewhere between this simple recurrence algorithm -- and the one you have presented!

My challenge to you then, as a fellow Mathematician (I haven't done this by the way, I'm lazy! <g>) -- is to figure out when/where/why the number table / difference table / triangle / fan -- fails -- between the simplest of all recurrence relationship formula, the Fibonnaci sequence -- and this one!

Because you see, I'll bet there's some interesting mathematical knowledge there!.

I'd do it myself -- but no time!

Besides, you have Python already set up and running and everything... I don't!

Anyway, I think it would be interesting to know this!

Also -- once the exact failure criteria are understood -- next question is, is it possible to construct an n-dimensional table (like maybe 2 or more interlinked/interrelated number/difference tables) -- where one maps to others, and you can get the correct answer for deeply recursive algorithms -- which include division?

[eesmith]

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

Yes, factorials can be defined through recursion. But so can multiplication of non-negative integers:

  def mult(a, b):
    if a == 0 or b == 0: return 0
    if a == 1: return b
    return mult(a-1, b) + b

  >>> mult(5, 9)
  45

Furthermore, factorials can alternatively be defined through the Gamma function, which supports more than just the integers and isn't defined recursively. https://en.wikipedia.org/wiki/Gamma_function .

> the simplest (AFAIK) recurrence relationship ...that can be produced.

The positive integers is even easier. For n >= 0:

  F(0) = 0
  F(n) = 1 + F(n-1)

> What's very interesting about the Fibonnaci Sequence

Another interesting thing about the Fibonnaci Sequence is that it has a closed-form solution, given in your OEIS link as:

  F(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5))

or see https://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_e... .

It is also non-polynomial. In the limit, it approaches a simple exponential.

Nor is it bounded by a polynomial, meaning that if you pick any positive integer p and any constant C, and define g(n) = C n^p then for a large enough n you will find that |g(n)| < |F(n)| .

Furthermore, A000435 , with its n^n - like form, grow even faster than exponential. It's similar to factorial growth.

> as a fellow Mathematician

I am not a mathematician. Neither are you.

> s to figure out when/where/why the number table / difference table / triangle / fan -- fails -- between the simplest of all recurrence relationship formula, the Fibonnaci sequence -- and this one

Consider f(n) = f(n-1) + 1 where f(0) = 0. This is the sequence 0, 1, 2, 3, .... This addition of 1 can be seen trivially in the difference table.

Consider f(n) = f(n-1) + f(n-1) with f(1) = 1. This is the sequence 1, 2, 4, 8, 16, 32, or 2^n, which is an exponential function.

Consider the Fibonacci function f(n) = f(n-1) + f(n-2) with f(1) = f(2) = 1. This uses an addition of the two previous numbers. It approaches an exponential function as n gets larger.

Consider f(n) = n * f(n-1) with f(1) = 1 This is the factorial. It grows faster than the exponential function.

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.

> deeply recursive algorithms

These are not deeply recursive. For an example of those, see the Ackermann function. https://en.wikipedia.org/wiki/Ackermann_function , which is not primitive recursive like functions we've discussed so far.

[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> :-)

[peter_d_sherman]

This is my second response (read the other one first...)

Upon deeper thought as to why a factorial might fail -- aren't these tables sort of like derivatives?

If so -- then we might want to consider thinking about the derivative of a factorial:

https://math.stackexchange.com/questions/300526/derivative-o...

Hey, the smart people StackExchange tell me that you can't take the derivative of a factorial (well, not in usual Calculus -- at least not without using the Gamma function, as you previously aluded to!)

>"I am not a mathematician. Neither are you."

Well then, one fellow, ahem, "Non-Mathematician" to another(!) -- let me lay out a mathematical speculation/conjecture for you...

I'll bet that somehow Mathologer's tables -- which are based on subtraction -- and fail to work with recursive functions that use multiplication -- could indeed work with even those functions(!) -- if they could somehow be constructed instead -- out of Gamma functions!

Why?

Well simply because you can take the derivative of a Gamma function -- whereas you can't for a factorial!

Now, if such a table as alluded to above could in fact be constructed -- that might be an interesting area of mathematical research!

I don't currently have the time or resources to prove or disprove that it could or couldn't...

But that it could is my bet, my speculation, my conjecture!

One "non-Mathematician" -- to another! <g> :-)

And we'll see if time proves it to be correct or incorrect!