The cake numbers answer a tidy question: what is the greatest number of pieces you can cut a cake into with \(k\) straight planar slices? The answer is \(C_k = (k^3 + 5k + 6)/6\), giving 1, 2, 4, 8, 15, 26, 42, 64, 93, 130.
The first few look like the powers of two (1, 2, 4, 8) — a classic trap — but the sequence breaks away at the fourth cut (15, not 16). The cake numbers are the three-dimensional version of the [[lazy-caterer]] (pancake) numbers, and they are the sum of the first four entries in each row of Pascal's triangle: \(\binom{k}{0}+\binom{k}{1}+\binom{k}{2}+\binom{k}{3}\).