lazy caterer’s sequence

Given a pancake (or a circle), how can one cut n pieces (not necessarily of the same size) 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


which gives the lazy caterer’s sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, … (listed in A000124 of Sloane’s OEIS).

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

Shel Kaphan, in a remark to the OEIS writes that ”when constructing a zonohedron, one zone at a time, out of (up to) 3-D non-intersecting parallelepipedsMathworldPlanetmath, the nth element of this sequence is the number of edges in the nth zone added with the nth layer of parallelepipeds.”

Title lazy caterer’s sequence
Canonical name LazyCaterersSequence
Date of creation 2013-03-22 16:16:54
Last modified on 2013-03-22 16:16:54
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 51D20
Synonym lazy caterers sequence
Synonym circle cutting problem
Synonym pancake cutting problem
Defines central polygonal number