How do you work out what number is next in this sequence 0 7 26 63 124?

Answer 1

Any number that you choose can be the next number. It is easy to find a rule based on a polynomial of order 5 such that the first five numbers are as listed in the question followed by the chosen next number. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.

You want 1 to be the next? Use the following rule:

t(n) = (-107*n^5 + 1605*n^4 - 9035*n^3 + 24075*n^2 - 29318*n + 12780)/60.

Want 10 to be next? Try

t(n) = (-41*n^5 + 615*n^4 - 3461*n^3 + 9225*n^2 - 11234*n + 4896)/24.

In fact, there is a simple answer here but you need to spot it and that is

t(n) = n^3 - 1