How many numbers are required to define a sequence without stating a rule/function for generating the next term in the sequence? I'm wondering if there is some minimum number of numbers required to define a sequence, without explicitly stating the rule that generates the next term in the sequence. For instance if I write $(1,a_2,a_3,...)$, and hide the remaining numbers in the sequence behind $(a_2,a_3,...)$, we don't know what the sequence is or what rules define it. If I then write $(1,2,a_3,...)$, it still isn't clear. Is the rule for determining the next number in the sequence $a_{i+1}=2 a_i$? Is it $a_{i+1}=a_i+1$?
If I write $(1,2,4,8,16)$, it's clear the rule is $a_{i+1}=2a_i=2^{i-1}$. Could I even shorten this to $(1,2,4,...)$ and figure this out? Is this an example of the minimum number of numbers required to define the sequence of powers of $2$. As J.W. Tanner says in the comments, you can come up with a polynomial whose first terms are $1,2,4,8,16,23$, so apparently not.
How about the Fibonacci sequence? I think it's clear what the rule is if I write $(0,1,1,2,3,5,8,...)$, even if I hadn't learned of this sequence before. I can't learn anything from $(0,1)$. What about $(0,1,1)$? It's hard to decide if I can learn the rule from this or if I need more numbers from the sequence. Typically you would just say $a_0=0,a_1=1,$ and $a_{i} = a_{i-1} + a_{i-2}$ for $i>1$. But that defeats the point of the question. The point is to ask how many numbers we need in order to define/learn the sequence without explicitly stating the rule that generates the next term in the sequence, and writing $a_{i} = a_{i-1} + a_{i-2}$ is explicitly stating the rule.
How does this idea generalise?

There is an uncountably infinite number of sequences of natural numbers (even ones starting $a_0, a_1, a_2, a_3$, as anything whatsoever can follow). That this is so is simple to prove by Cantor's diagonalization argument.
