Which math sequence gives the maximum pieces made by n straight cuts in a pancake?

Which math sequence gives the maximum pieces made by n straight cuts in a pancake? It is the lazy caterer’s sequence.

The lazy caterer’s sequence counts the greatest number of regions into which n straight cuts can divide a circular pancake or pizza. The maximum occurs when every new cut crosses all earlier cuts at distinct points inside the pancake, rather than passing through an existing intersection.

Its formula is 1 + n(n + 1)/2, or equivalently (n² + n + 2)/2. The sequence begins 1, 2, 4, 7, 11, 16, and 22 for zero through six cuts. Thus, three optimally placed cuts produce seven pieces—not merely six.

The sequence is also known as the central polygonal numbers. It is closely related to triangular numbers because each term is one more than a triangular number. A related three-dimensional problem asks for the maximum pieces made by planes cutting a solid cake; those values form the cake-number sequence.