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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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Latest revision as of 16:07, 19 December 2014
Hi there. Let me begin by introducing the writer, her title is Myrtle Cleary. North Dakota is our birth location. One of the extremely best things in the globe for me is to do aerobics and I've been doing it for quite a whilst. For years I've been working as a payroll clerk.
my webpage :: http://blogzaa.com/blogs/post/9199