Dilcue's Pizza (The Lazy Caterer Sequence)

This post is in memory of my high school algebra teacher, Larry Freeman, of Morgan Park High School, Chicago. Once he handed out a worksheet called Dilcue's Pizza in which we had to find the equation that produces the maximum number of pieces a pizza can be produced for a given number of cuts. The class had no idea this was his rewording of the classic "Lazy Caterer Sequence." Following his sense of humor, Dilcue is Euclid backwards.

(photo by https://www.instagram.com/enginakyurt/)

The sequence is as follows (note- pie-type cuts traditionally intersect in the center; however, if the cuts do not intersect, more pieces can be produced. Hence, it is called the Lazy Caterer Sequence):

Cuts        Max Pieces

0               1

1               2

2               4

3               7

4              11

5              16

6              22

The sequence is produced by the polynomial: (p = pieces, n = cuts)

p = (n² + n + 2)/2

The sequence can also be produced by the first entry of each row of a triangular arrangement of natural numbers, see Floyd's Triangle:

1

2  3

4  5  6

7  8  9  10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

Added note (November 27, 2023): The sequence is found in OEIS as  the Central polygonal numbers (the Lazy Caterer's sequence) https://oeis.org/A000124.

A 3-D version is also found in OEIS: Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake) https://oeis.org/A000125.