As we saw in Example 2.3.1, this sequence is not \(\Delta^k\)-constant for any \(k\text{.}\) Therefore the closed formula for the sequence is not a polynomial. In fact, we know the closed formula is \(a_n = 2^n\text{,}\) which grows faster than any polynomial (so is not a polynomial).
The sequence of first differences is \(7, 43, 133, 301, 571,\ldots\text{.}\) The second differences are: \(36, 90, 168, 270,\ldots\text{.}\) Third difference: \(54, 78, 102,\ldots\text{.}\) Fourth differences: \(24, 24, \ldots\text{.}\) As far as we can tell, this sequence of differences is constant so the sequence is \(\Delta^4\)-constant and as such the closed formula is a degree 4 polynomial.
This is the Fibonacci sequence. The sequence of first differences is \(0, 1, 1, 2, 3, 5, 8, \ldots\text{,}\) the second differences are \(1, 0, 1, 1, 2, 3, 5\ldots\text{.}\) We notice that after the first few terms, we get the original sequence back. So there will never be constant differences, so the closed formula for the Fibonacci sequence is not a polynomial.