We make a difference table and see how many differences
it takes before the differences are all the same:

1     4     9    17    29   <--- original sequence
   3     5     8    12      <--- sequence of 1st differences 
      2     3     4         <--- sequence of 2nd differences
         1     1            <--- sequence of 3rd differences

Since the 3rd difference are all the same, a 3rd degree equation
will work. Let the 3rd degree equation be

 

We substitute the first 4 terms into that.











or






Do you know how to solve that system?  If not
post again asking how. The solutions are

A = , B = 0, C = , D = -1

So the equation is



or we could get a common denominator and write it



Edwin