Cake number

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:[1] 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.

[1] 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:[2]

In n spatial (not spacetime) dimensions, Maxwell's equations represent

Three orthogonal planes slice a cake into at most eight ( C 3 ) pieces
Animation showing the cutting planes required to cut a cake into 15 pieces with 4 slices (representing the 5th cake number). Fourteen of the pieces would have an external surface, with one tetrahedron cut out of the middle.
Cake numbers (blue) and other OEIS sequences in Bernoulli's triangle
Proof without words that summing up to the first 4 terms on each row of Pascal's triangle is equivalent to summing up to the first 2 even terms of the next row