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?
Best Answer
Even your example of $1,2,4,8,16$ doesn't automatically mean that the sequence is uniquely defined by $a_i=2^{i-1}$
As humans, we would probably assume that was the sequence you meant, but we could also say that the sequence is defined by $$a_i=\frac{i^4}{24} - \frac{i^3}4+\frac{23i^2}{24}-\frac{3i}4+1$$ (which I found using WolframAlpha)
This then gives \begin{align}a_6&=\frac{6^4}{24} - \frac{6^3}4+\frac{23\times 6^2}{24}-\frac{3\times 6}4+1\\ &=31\end{align} as opposed to the $32$ you would expect.
Even if we then specify that the $6$th term is $32$, we then get a new generating function which then gives the $7$th term as $a_7=63$, again not $64$ as we expect.
So, the conclusion is that you can never uniquely define a sequence simply from its first $n$ terms, you can only uniquely define a sequence with its generating function