Lazy caterer's sequence

The lazy caterer's sequence tells the maximum number of pieces of a pancake (or a circle) that can be made with the minimum number of cuts. For example, to cut a pancake into four pieces, four cuts could be made, each starting at the center and going to the edge. But it would be much simpler to make just two cuts to cut it into four pieces.

The maximum number of pieces that can be created with a given number of cuts n is given by the formula

$1 \over 2(n^{2}+n+2)$

giving the sequence (sequence A000124 in the OEIS)

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191

These are also called central polygonal numbers, and have applications in various other mathematical problems. Each of these numbers is 1 plus a triangular number.