What is number sequence 3 4 6 9 13?

Answer 1

There are infinitely many rules based on polynomials of order 5 such that the first five numbers are as listed in the question. 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.

The simplest answer, based on a polynomial of order 2 (a quadratic) is

U(n) = (n^2 - n + 6)/2 for n = 1, 2, 3, ...

Answer 2

+1, +2, +3, +4...