Sequence detective
Type a few terms of a sequence. The page takes differences, then differences of the differences, until a row goes constant. That tells it the degree (the highest power of n in the formula), and from the first column it rebuilds the formula for the nth term.
What's going on
If the first differences are constant, the sequence is linear, like 4n + 3. If the second differences are constant, it is quadratic, and the n² coefficient is half that constant. Third differences constant means cubic, with the constant divided by 6, and so on: the divisor is k! for degree k.
The page uses Newton's forward-difference formula: the nth term equals the first term, plus the first difference times (n − 1), plus the second difference times (n − 1)(n − 2)/2, and so on down the left-hand edge of the table. Those pieces are then multiplied out into ordinary powers of n.
The trap: 1, 2, 4, 8, 16 looks like doubling, but a degree-4 polynomial passes through those five numbers too, and it continues with 31, not 32. Five terms can never pin down a sequence. Only a rule can.