Say you were to be shown a certain sequence, like $\{ ..., 1, 4, 7, 10, 13, ... \}$ and asked to find an algebraic function perfectly recreating the output. The obvious choice is $ y = 3x + k$ where $k$ is some starting point. After testing only, say, 5 members of the sequence, confidence in this function candidate would be nearing on 100%, but what's the probability that this function is wrong?
In a more notable example, the maximal number of regions obtained by joining $x$ points around a circle by straight lines matches the sequence $\{1, 2, 4, 8, 16,...\}$, and so it would be very reasonable to assume the correct function is $y = 2^{x}$. And yet keep testing further points, and the remainder of the sequence is revealed to be completely unrelated to consecutive powers of two:
$$\{ 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, ... \}$$
The actual function producing this output is $y = 2 + {x \choose 2} + {x \choose 4}$.
It occurred to me that it would be enormously helpful to have a way of analyzing the probability of a certain function producing a certain output given $x$ number of correct consecutive terms, and yet it is almost immediately apparent that finding such a probability function eludes all my years of mathematics knowledge, and I would hardly know where to begin.
How would one go about calculating, as generally as possible, the probability that a certain function produces an entire sequence of points given that it correctly produces $n$ consecutive points within the sequence?
Notes:
You'll notice I don't use the terms "model" or "modeling function" as I'm not referring to functions that closely approximate the output, but functions that produce it perfectly
Because of the nature of algebra, any sequence will have an infinite number of functions that perfectly produces it (the first sequence above can similarly be produced by $y = 3x - 3 + k$, among infinite other candidates). This complicates things, though if possible I'm looking for hints at arriving at this probability function in the most simplified case
( upgraded to an answer)Not high necessarily, see en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers. Within the OEIS, for example 1,4,7,10,13, in the search shows up 56 results. The problem, is there are a countable infinite of numbers, that could be on each side of this.
for example in the numbers 1,2,3,4,5,6,7,8,9,10,11,12,13 if you didn't care if repeats occur how many possibilities could you choose from for a sequence of 5 ...? 13^5 = 371293 that's right over 371293 sequences so assuming a sequence picks five from the range 1 to 13 ( it need not, and they need not be distinct) you have about a 1 in 371293 chance of matching all five in order.