Cake number
In mathematics, the cake number, denoted by Cn, is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. The cake number is so called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence.
The values of Cn for are given by .
General formula
If n! denotes the factorial, and we denote the binomial coefficients byand we assume that n planes are available to partition the cube, then the n-th cake number is:
Properties
The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence. The difference between successive cake numbers also gives the lazy caterer's sequence.The fourth column of Bernoulli's triangle gives the cake numbers for n cuts, where n ≥ 3.
The sequence can be alternatively derived from the sum of up to the first 4 terms of each row of Pascal's triangle: