Does this mathematical problem have a name?

Question

Suppose we have a finite sequence of (seemingly) random numbers, such as $$ 1, 5, 3, 6, 11, 14, ... $$ Can we devise a function $f(x)$ such that for the $i^{th}$ value $x_i$ in the sequence $f(i) = x_i$? For example, for the above sequence, $$ f(1) = 1, \hspace{5mm} f(2) = 5, \hspace{5mm} ... $$

Does this problem have a name?

Answer 1

This question is pretty ambiguous.

Specifying the finite sequence of numbers, such as

$$ 1, 5, 3, 6, 11, 14 $$

(more precisely $f:=\{(1,1),(2,5),(3,6),(4,11),(5,14)\}$)

is a function definition, and this function satisfies your requirements, so there is no need to look further.

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But you probably want a formula (i.e. a closed-form expression), not a function, such that it evaluates to the given values at the given indexes.

Such a formula should allow you to compute values for other arguments, which is called interpolation when perfomed between the known indexes, or extrapolation when performed outside.

There are well-known methods to achieve this, for instance using a single polynomial (Lagrangian interpolation) or piecewise polynomials (cubic splines). Note that they can be good for interpolation, and are often very poor for extrapolation.

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Another understanding of your question, is if it is possible to find a simple expression that matches the numbers, for instance $2,3,5,9,17,33,65$ being described complactly as $f_i=2^i+1$. This is in relation with the concepts of pattern matching and minimum description length, and as far as I know, there is little theory available.

You might call that sequence identification.

Answer 2

Yes, we can always find such a function, and if we restrict our attention to polynomial functions $f$ there is in fact a canonical solution: Given a set of $n + 1$ data points $(x_i, y_i)$ (with pairwise distinct $x_i$), there is a unique polynomial of degree at most $n$ satisfying the interpolation condition $p(x_i) = y_i$---in our case we have $x_i = i$, $i = 1, \ldots, n + 1$.

This construction need not always give the most sensible function satisfying the condition, however: There are many sequences $(y_i)$ for which the (speaking deliberately imprecisely) simplest or most natural function $f$ satisfying $f(i) = y_i$ is not polynomial. For example, we can immediately recognize the terms of the finite sequence $(2, 4, 8, 16, 32)$ as successive powers of $2$, that is, that we can satisfy the condition with the simple function $$f(x) = 2^x .$$ This is, intuitively, much simpler that the polynomial interpretation of that data, $$p(x) = \tfrac{1}{12}(x^4 - 6 x^3 + 23 x^2 - 18 x + 24) .$$