What is the next number in the sequence 5 21 85 341?

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 4 such that the first four numbers are as listed in the question followed by your chosen number in the next place. 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.
One possible solution, using a polynomial of order 3, isU(n) = 39*n^3 - 210*n^2 + 373*n - 197 for n = 1, 2, 3, 4, 5 which gives U(5) = 1293.

Answer 2

It is: 1365 because 5+16 = 21, 21+64 = 85, 85+256 = 341 and 256+1024 = 1365