I have a theory, (have no idea even if this has been discussed before).
In any set of numbers, you might find a pattern (given enough time), thus making it not random. Example: I was once given a set of numbers, asking to find the possible next number in the series, ‘2, 4, 9, 13, 18, 33, 37, _’
And the question was then forgotten, and I came up with a solution saying that the next number is “57” and showed the below explanation, it was later revealed that those were just random numbers given to me and there is no series. (The figure below was the one I used, so I posted it directly, it could be explained much better with a pyramid sort of structure, a triangle).
(Image 1)an attempt at solving a random sequence with a triangle/pyramid
Getting a solution here could just be chance; I know i haven’t provided enough set of examples to give statistically significant data proving there is very little chance for randomness. Given large enough set of numbers, you will find a pattern?
Like how I forced it to have a solution by placing the numbers in pyramid form, you can do that for almost any other kind of series like placing in figures, may be like a rhombus (given below a self made example, could be a rhombus or an octagon), and there are also 3d shapes, there are many possibilities even without considering any geometrical shapes, like reversing the number sequence and subtracting, adding the number of digits, etc.
Example 2:
“1, 4, 14, 29, 39, _”
(Image 2)solving a sequence using a rhombus/octagon
My so called “theory” may not be valid, given that I have not provided enough data, not enough examples proving this theory, but endless number of random sequences combining with endless methods of pattern forming: doesn’t that result in very low chances of “real random” numbers? (I’m not saying there could be no randomness at all)
What do you think? It’s not a question technically, I want to hear all the answers you have and possibly learn if there’s any study related to that where I can find more material
Every sequence of $n$ numbers fits exactly one polynomial of degree less than $n$. But a polynomial of degree $n-1$ has $n$ coefficients, so that's not surprising. If the polynomial turns out to have a very small degree, that is surprising, and probably significant.
Another question is whether the numbers are close to a simple formula. That is important because you always have noise that you want to remove. The simple formula is the signal, and noise is the difference between the simple formula and the data.