Pyramid number

Relations to other figurate numbers The pyramidal numbers can also be expressed as sums of binomial coefficients (or of two consecutive tetrahedral numbers) thus:

We can derive another relation between square pyramidal numbers and tetrahedral numbers: if we let Pn be the nth square pyramidal number then

where Tn is the nth tetrahedral number.

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

Besides 1, there is only one other number that is both a square and a pyramidal number, 4900, the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918.

[edit] Squares in a square A common mathematical puzzle involves finding the number of squares in a large n by n square grid. This number can be derived as follows:

The number of 1x1 boxes found in the grid is n2. The number of 2x2 boxes found in the grid is (n − 1)2. These can be counted by counting all of the possible upper-left corners of 2x2 boxes. The number of kxk boxes (1 =< k =< n) found in the grid is (n − k + 1)2). These can be counted by counting all of the possible upper-left corners of kxk boxes. It follows that the number of squares in a n by n square grid is:

or:

x = n(n + 1)(2n + 1) / 6 That is, the solution to the puzzle is given by the square pyramidal numbers.

[edit] See also Tetrahedral number Squared triangular number

[edit] References Abramowitz, M.; Stegun, I. A. (Eds.) (1964). Handbook of Mathematical Functions. National Bureau of Standards, Applied Math. Series 55, 813. ISBN 0486612724. Beiler, A. H. (1964). Recreations in the Theory of Numbers. Dover, 194. ISBN 0486210960. Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag, 260–261. ISBN 0-387-95419-8.

[edit] External links Eric W. Weisstein, Square Pyramidal Number at MathWorld. Retrieved from "https://secure.wikimedia.org/wikipedia/en/wiki/Square_pyramidal_number" Categories: Figurate numbers | Pyramids ViewsArticle Discussion Edit this page History Watch Personal toolsJoeldudesx My talk My preferences My watchlist My contributions Log out Navigation Main Page Contents Featured content Current events Random article Interaction About Wikipedia Community portal Recent changes Contact Wikipedia Donate to Wikipedia Help Search

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