Cake number

If n! denotes the factorial, and we denote the binomial coefficients by

:$\{n\; \backslash choose\; k\}\; =\; \backslash frac\{n!\}\{k!(n-k)!\},$

and we assume that n planes are available to partition the cube, then the n-th cake number is:{{cite book|last1=Yaglom |first1=A. M. |authorlink1=Akiva Yaglom |last2=Yaglom |first2=I. M. |authorlink2=Isaak Yaglom |title=Challenging Mathematical Problems with Elementary Solutions |volume=1 |location=New York |publisher=Dover Publications |year=1987}}

:$$

C_n = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \tfrac{1}{6}\!\left(n^3 + 5n + 6\right) = \tfrac{1}{6}(n+1)\left(n(n-1) + 6\right).

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.

{{Bernoulli_triangle_columns.svg|Cake numbers (blue)}}

The fourth column of Bernoulli's triangle (k = 3) gives the cake numbers for n cuts, where n ≥ 3.

File:cake_numbers_visual_proof.svg 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]]

The sequence can be alternatively derived from the sum of up to the first 4 terms of each row of Pascal's triangle:{{oeis|A000125}}

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

• Dividing a circle into areas (Moser's circle problem)
• Lazy caterer's sequence
• Pizza theorem

{{Reflist}}

• {{MathWorld |id=SpaceDivisionbyPlanes |title=Space Division by Planes |author=Eric Weisstein |access-date=January 14, 2021 |ref= }}
• {{MathWorld |id=CakeNumber |title=Cake Number |author=Eric Weisstein |access-date=January 14, 2021 |ref= }}

{{Classes of natural numbers}}

Category:Mathematical optimization

Category:Integer sequences

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