3, -2,  1, 18, 55, 118, 213

List them in a column, then form the differences to the right by
subtracting each number from the one just below it

  3  -5   8  6
 -2   3  14  6
  1  17  20  6
 18  37  26  6
 55  63  32
118  95
213

So we see that the 3rd differences are all 6's.

So we assume a 3rd degree polynomial general term:



There are 4 unknowns A,B,C,D so we will only need
the first 4 terms.

We substitute n=1 and a1=3
We substitute n=2 and a2=-2
We substitute n=3 and a3=1
We substitute n=4 and a4=18

So we have the system:



Solve by any method and get A=1, B=-2, C=-6, D=10

So the general equation is



Edwin