Cake Number - General Formula

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 number is:

$C_n = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \frac{1}{6}(n^3 + 5n + 6).$

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