We check to see if it can be part of a polynomial sequence.
We write the sequence out:

1  7  17  31  49  71

We subtract every term from the term on its immediate right and write the
difference below and between the terms. This forms the sequence of FIRST
DIFFERENCES:

1  7  17  31  49  71
  6 10  14  18  22

Then we repeat. We subtract every term from the term on its immediate right
and write the difference below and between the terms. This forms the
sequence of SECOND DIFFERENCES:

1  7   17  31  49  71
  6  10  14  18  22
    4   4   4   4

This sequence of SECOND DIFFERENCES has all 4's. So we can find a SECOND
DEGREE polynomial expression for the nth term of the sequence:

So we assume this polynomial expression type for the nth term:



We need to find the unknowns A, B, and C.

We substitute n=1

 

We substitute 1 for t(1) and simplify the right side:



The first equation of our system is



-----------------------

We substitute n=2

 

We substitute 7 for t(2) and simplify the right side:



The second equation of our system is



-----------------------

We substitute n=3

 

We substitute 17 for t(3) and simplify the right side:



The third equation of our system is



-----------------------

We can stop here with 3 equations because we have only 3
unknowns, A,B, and C. So our system is:



Now we solve the system of three equations in three unknowns
by substitution, elimination, echelon method, or matrix method,
and we get:

We get A=2, B=0, C=-1

So we substitute these for the nth term in:



 

and simplify



Edwin

    1   7  17  31  49  71
      6  10  14  18  22
        4   4   4   4

     n  t(n) 2n^2  difference
                  t(n)-2n^2=bn+c
   -------------------------------
     1   1     2     -1
     2   7     8     -1
     3  17    18     -1