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 n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... .

General formula

If n! denotes the factorial, and we denote the binomial coefficients by and 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 (k = 3) 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: ! !! 0 !! 1 !! 2 !! 3 ! rowspan="11" style="padding:0;"| !! Sum ! 0 ! 1 ! 2 ! 3 ! 4 ! 5 ! 6 ! 7 ! 8 ! 9

Other applications

In n spatial (not spacetime) dimensions, Maxwell's equations represent C_n different independent real-valued equations.