We can create a divided difference table as follows:
$$\begin{array}{c|cccccc}
x & f(x) & f[x_i, x_{i+1}] & f[x_i, x_{i+1}, x_{i+2}] & f[x_i, x_{i+1}, x_{i+2}, x_{i+3}] & f[x_i, x_{i+1}, x_{i+2}, x_{i+3}, x_{i+4}] \\
\hline
1 & 3 & 3 & 2 & -1 & 0.5 \\
2 & 6 & 13
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