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Encyclopedia > Square pyramidal number

A pyramidal number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. The nth pyramidal number is

that is, it is the sum of the squares of the first n integers.


The first few pyramidal numbers are


1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819


Pyramidal numbers can be modelled in physical space with a given number of balls and a square frame that hold in place the number of balls forming the base, that is, n2. Besides 1, there is only one other number that is both a square and a pyramidal number, 4900. This fact was proven by G.N. Watson in 1918.


The sum of two consecutive square pyramidal numbers is an octahedral number.


See also: tetrahedral number


 
 

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