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?

Consider the sequence $1,1,2,3.$  At first look, it seems to be the first terms of  Fibonacci numbers, but it's not true that the only sequence which starts with  $1,1,2,3,5$ are Fibonacci numbers.
Here we can say that these are the first terms of triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.
Or we can say these are the numbers of rooted trees with n vertices in which vertices at the same level have the same degree.
Or these are the terms generated by  $a_n=\lfloor(3^n / 2^n)\rfloor$.
In your example the terms of $1,2,4,8,16$ are not necessarily generated by $a_n=2^n$
For example we can say these are the coefficients of expansion of $$\frac{(1-x)}{(1-2^x)}$$ in powers of $x$.
Or these are the numbers of positive divisors of $n!$.
Or these are Pentanacci Numbers.
For more information look at  :oeis.org
