Pancake Numbers

Yesterday, I turned 25652 days old. Among the many properties of this number is the one that I want to discuss in this post. Numbers Aplenty informed me that 25652 is a pancake number and the explanation of this term is contained in the screenshot from the website (Figure 1).

Figure 1

These numbers form OEIS A000124: central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts. In the OEIS link it's explained that:

The first line cuts the pancake into 2 pieces. For n > 1, the n-th line crosses every earlier line (avoids parallelism) and also avoids every previous line intersection, thus increasing the number of pieces by n. For 16 lines, for example, the number of pieces is 2 + 2 + 3 + 4 + 5 + ... + 16 = 137. These are the triangular numbers plus 1.

An example is shown in Figure 2 where examples of two cuts and three cuts are shown.

Figure 2

The day before I celebrated turning 25652 days old, I was inevitably 25651 days old and, as I duly noted, this was a triangular number and specifically the 226th such number. I didn't note the connection between the two numbers at the time but now I know that the maximum number of pancake slices that can be created from 226 slices is 25652.

OEIS A000217 is the sequence of triangular numbers that begin 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, ... and it can be seen that indeed the terms of OEIS A000124 are simply the terms of this sequence with 1 added. Figure 3 shows the first six triangular numbers.

Figure 3

Every triangular number (a particular type of figurate number) can also be represented as a polygonal number. For example, 6 can be represented as a hexagonal number (see Figure 4).

Figure 4

The pancake number is one more than the triangular number and is thus the centered hexagonal number 7 (see Figure 5).

Figure 5

Returning to our pancake slices once again, Figure 6 shows a nice animation of the situation (borrowed from Wikipedia).

Figure 6

The definition included at the beginning of this post refers to pancake numbers as being "bidimensional versions of cake numbers". So what are cake numbers? Well, Wikipedia describes them thus:

In mathematics, the cake number, denoted by \(C_n\) is the maximum 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. 

The values of \(C_n\) for increasing \(n ≥ 0\) are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, …(sequence A000125 in the OEIS) 

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.

Figure 7 shows an animation of the situation.

Figure 7: 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.

If \(n!\) denotes the factorial, and we denote the binomial coefficients by$$ {n \choose k} = \frac{n!}{k! \, (n-k)!}$$and we assume that \(n\) planes are available to partition the cube, then the number is:$$C_n = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \frac{1}{6}(n^3 + 5n + 6)$$The cake numbers up to 1000 are 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988.

There is another type of pancake number entirely and that is well described on this website that has very clear visuals and from which I'll include two screenshots (see Figures 8 and 9).

Figure 8

Figure 9

So these pancake numbers constitute OEIS A058986 and it is a short sequence: 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19.