Fibonacci Function

Algebra Level 5

Let $f(x)$ be a quintic polynomial such that

$\begin{array} { r l } f(1) & = 1 \\ f(2) & = 1 \\ f(3) & = 2 \\ f(4) & = 3 \\ f(5) & = 5 \\ f(6) & = 8. \\ \end{array}$

Determine $f(7)$ .

Note: Many people are answering this incorrectly because they think it is the Fibonacci sequence, but this problem is asking about a quintic polynomial that passes through those points. That does not necessarily mean the next term behaves as the Fibonacci sequence would.