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| In [[mathematics]], the '''cake number''', denoted by ''C<sub>n</sub>'', is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly ''n'' [[plane (geometry)|plane]]s. 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.
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| The values of ''C<sub>n</sub>'' for increasing {{nowrap|1=''n'' ≥ 0}} are given by {{nowrap|1=1, 2, 4, 8, 15, 26, 42, 64, 93, …}}<ref>{{citeweb|url=http://oeis.org/A000125|author=The On-Line Encyclopedia of Integer Sequences|title=A000125: Cake Numbers|accessdate=August 19, 2010}}</ref>
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| 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.
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| == General formula ==
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| If ''n''! denotes the [[factorial]], and we denote the [[binomial coefficient]]s by
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| :<math> {n \choose k} = \frac{n!}{k! \, (n-k)!} , </math>
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| and we assume that ''n'' planes are available to partition the cube, then the number is:<ref>{{citeweb|url=http://mathworld.wolfram.com/SpaceDivisionbyPlanes.html|title=Space Division by Planes|author=Eric Weisstein|location=MathWorld − A Wolfram Web Resource|accessdate=August 19, 2010}}</ref>
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| :<math>
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| C_n = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \frac{1}{6}(n^3 + 5n + 6). </math>
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| == References ==
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| {{Reflist}}
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| [[Category:Mathematical optimization]]
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| {{combin-stub}}
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Revision as of 21:27, 10 February 2014
The title of the writer is Figures but it's not the most masucline name out there. For many years I've been operating as a payroll clerk. Puerto Rico is where he and his spouse live. To gather coins is one of the issues I love most.
My homepage ... www.ninfeta.tv