I'll show you a procedure for this problem.
It's rather long, but it's simple and direct.
Can someone tell me what the algebra expression is for this pattern?
Take the difference of consecutive terms,
. . then take difference of the differences and so on.
We see that the second differences are constant.
Hence, the generating function is of the second degree . . . a quadratic.
The general quadratic function is: .
Use the first three terms of the sequence and set up a system of equations:
Therefore, the generating function is:
. . .