Square pyramidal number

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Geometric representation of the square pyramidal number 1+4+9+16=30.

In mathematics, a pyramidal number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. Three ways of calculating the n-th pyramidal number are

$\sum_{k=1}^nk^2={(n^2 + n)(2n + 1) \over 6}={2n^3 + 3n^2 + n \over 6}$

that is, by adding up the squares of the first n integers, or by multiplying the nth pronic number by the nth odd number. By mathematical induction it is possible to derive one formula from the other. An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12).

The first few pyramidal numbers are:

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

(sequence A000330 in OEIS).

The pyramidal numbers can also be expressed as sums of binomial coefficients thus:

${{n + 2} \choose 3} + {{n + 1} \choose 3}$

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, the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918.

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

[edit] See also

[edit] References

Abramowitz, M.; Stegun, I. A. (Eds.) (1964). Handbook of Mathematical Functions. National Bureau of Standards, Applied Math. Series 55, 813. ISBN 0486612724.