Square pyramidal number

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

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an $n \times n$ grid.

Formula

Play media

Six copies of a square pyramid with

n

steps can fit in a cuboid of size

n (n + 1)(2 n + 1)

The first few square pyramidal numbers are:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... (sequence A000330 in the OEIS).

These numbers can be expressed in a formula as

P_{n}=\sum _{k=1}^{n}k^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {2n^{3}+3n^{2}+n}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}.

This is a special case of Faulhaber's formula, and may be proved by a mathematical induction.^[1] Equivalent formulas are given by Archimedes^[2] and Fibonacci.^[3]

In modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial $L (P, t)$ of a polyhedron $P$ is a polynomial that counts the number of integer points in a copy of $P$ that is expanded by multiplying all its coordinates by the number $t$ . The Ehrhart polynomial of a pyramid whose base is a unit square with integer coordinates, and whose apex is an integer point at height one above the base plane, is $(t + 1)(t + 2)(2 t + 3) / 6 = P t + 1$ .^[4]

Squares in a square

A 5 by 5 square grid, with three of its 55 squares highlighted.

A common mathematical puzzle involves finding the number of squares in a large $n$ by $n$ square grid.^[5] This number can be derived as follows:

The number of 1 × 1 boxes found in the grid is $n 2$ .
The number of 2 × 2 boxes found in the grid is $(n - 1) 2$ . These can be counted by counting all of the possible upper-left corners of 2 × 2 boxes.
The number of $k \times k$ boxes $(1 \leq k \leq n)$ found in the grid is $(n - k + 1) 2$ . These can be counted by counting all of the possible upper-left corners of $k \times k$ boxes.

It follows that the number of squares in an $n \times n$ square grid is:

n^{2}+(n-1)^{2}+(n-2)^{2}+(n-3)^{2}+\ldots +1^{2}={\frac {n(n+1)(2n+1)}{6}}.

That is, the solution to the puzzle is given by the square pyramidal numbers.^[6]

The number of rectangles in a square grid is given by the squared triangular numbers.^[7]

Relations to other figurate numbers

A pyramid of cannonballs in the Musée historique de Strasbourg. The cannonball problem asks for the sizes of pyramids that can also be spread out to form a square array of cannonballs.

The cannonball problem asks which numbers are both square and square pyramidal. Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918.^[8]

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

P_{n}={\binom {n+2}{3}}+{\binom {n+1}{3}}.

The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive triangular numbers.^[9]

Square pyramidal numbers are also related to tetrahedral numbers in a different way:

P_{n}={\frac {1}{4}}{\binom {2n+2}{3}}.

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

Augmenting a pyramid whose base edge has $n$ balls by adding to one of its triangular faces a tetrahedron whose base edge has $n - 1$ balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism. This geometric dissection leads to another relation:

P_{n}=n{\binom {n+1}{2}}-{\binom {n+1}{3}}.

Another relationship involves the Pascal Triangle: Whereas the classical Pascal Triangle with sides (1,1) has diagonals with the natural numbers, triangular numbers, and tetrahedral numbers, generating the Fibonacci numbers as sums of samplings across diagonals, the sister Pascal with sides (2,1) has equivalent diagonals with odd numbers, square numbers, and square pyramidal numbers, respectively, and generates (by the same procedure) the Lucas numbers rather than Fibonacci.^{[citation needed]}

In the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes.

Also,

P_{n}={\binom {n+3}{4}}-{\binom {n+1}{4}}

which is the difference of two pentatope numbers.

This can be seen by expanding:

n(n+1)(n+2)(n+3)-(n-2)(n-1)n(n+1)=n(n+1)\left(n^{2}+5n+6-n^{2}+3n-2\right)=n(n+1)(8n+4)

and dividing through by 24.

Also,

P_{n}={\frac {2n+1}{3}}{\binom {n+1}{2}}

References

^ Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2007), Introduction to Automata Theory, Languages, and Computation (3 ed.), Pearson/Addison Wesley, p. 20, ISBN 9780321455369
^ Archimedes, On Conoids and Spheroids, Lemma to Prop. 2, and On Spirals, Prop. 10. See "Lemma to Proposition 2", The Works of Archimedes, translated by T. L. Heath, Cambridge University Press, 1897, pp. 107–109
^ Fibonacci (1202), Liber Abaci, ch. II.12. See Fibonacci's Liber Abaci, translated by Laurence E. Sigler, Springer-Verlag, 2002, pp. 260–261, ISBN 0-387-95419-8
^ Beck, M.; De Loera, J. A.; Develin, M.; Pfeifle, J.; Stanley, R. P. (2005), "Coefficients and roots of Ehrhart polynomials", Integer points in polyhedra—geometry, number theory, algebra, optimization, Contemp. Math., 374, Providence, RI: Amer. Math. Soc., pp. 15–36, MR 2134759
^ Duffin, Janet; Patchett, Mary; Adamson, Ann; Simmons, Neil (November 1984), "Old squares new faces", Mathematics in School, 13 (5): 2–4, JSTOR 30216270
^ Robitaille, David F. (May 1974), "Mathematics and chess", The Arithmetic Teacher, 21 (5): 396–400, JSTOR 41190919
^ Stein, Robert G. (1971), "A combinatorial proof that $\textstyle \sum k^{3}=(\sum k)^{2}$ ", Mathematics Magazine, 44 (3): 161–162, doi:10.2307/2688231, JSTOR 2688231
^ Anglin, W. S. (1990), "The square pyramid puzzle", American Mathematical Monthly, 97 (2): 120–124, doi:10.2307/2323911, JSTOR 2323911
^ Beiler, A. H. (1964), Recreations in the Theory of Numbers, Dover, pp. 194, ISBN 0-486-21096-0

External links

Weisstein, Eric W., "Square Pyramidal Number", MathWorld

[hmu-1] Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2007), Introduction to Automata Theory, Languages, and Computation (3 ed.), Pearson/Addison Wesley, p. 20, ISBN 9780321455369

[archimedes-2] Archimedes, On Conoids and Spheroids, Lemma to Prop. 2, and On Spirals, Prop. 10. See "Lemma to Proposition 2", The Works of Archimedes, translated by T. L. Heath, Cambridge University Press, 1897, pp. 107–109

[fibonacci-3] Fibonacci (1202), Liber Abaci, ch. II.12. See Fibonacci's Liber Abaci, translated by Laurence E. Sigler, Springer-Verlag, 2002, pp. 260–261, ISBN 0-387-95419-8

[bddps-4] Beck, M.; De Loera, J. A.; Develin, M.; Pfeifle, J.; Stanley, R. P. (2005), "Coefficients and roots of Ehrhart polynomials", Integer points in polyhedra—geometry, number theory, algebra, optimization, Contemp. Math., 374, Providence, RI: Amer. Math. Soc., pp. 15–36, MR 2134759

[dpas-5] Duffin, Janet; Patchett, Mary; Adamson, Ann; Simmons, Neil (November 1984), "Old squares new faces", Mathematics in School, 13 (5): 2–4, JSTOR 30216270

[robitaille-6] Robitaille, David F. (May 1974), "Mathematics and chess", The Arithmetic Teacher, 21 (5): 396–400, JSTOR 41190919

[stein-7] Stein, Robert G. (1971), "A combinatorial proof that $\textstyle \sum k^{3}=(\sum k)^{2}$ ", Mathematics Magazine, 44 (3): 161–162, doi:10.2307/2688231, JSTOR 2688231

[anglin-8] Anglin, W. S. (1990), "The square pyramid puzzle", American Mathematical Monthly, 97 (2): 120–124, doi:10.2307/2323911, JSTOR 2323911

[beiler-9] Beiler, A. H. (1964), Recreations in the Theory of Numbers, Dover, pp. 194, ISBN 0-486-21096-0

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Square pyramidal number

Contents

Formula

Squares in a square

Relations to other figurate numbers

References

Further reading

External links